An output stabilization of infinite dimensional semilinear systems

An output stabilization of infinite dimensional semilinear systems Abstract This article considers stabilization of infinite dimensional semilinear systems on a subregion $$\omega$$ of the evolution domain $${\it{\Omega}}$$. Under sufficient conditions, we give controls which ensure exponential, strong and weak stabilization of such systems on $$\omega$$. Also, we determinate a control which allows stabilization on $$\omega$$ minimizing a given functional cost. The developed results are illustrated by many examples and simulations. 1. Introduction Semilinear systems is an important subclass of nonlinear systems and numerous real-world problems have a semilinear structure. They include applications in nuclear, thermal, chemical, social processes, etc., (Mohler, 1973). The stabilization of infinite dimensional semilinear systems has been the subject of many studies using different approaches. More precisely, on $${\it{\Omega}}\subset\mathbb{R}^n$$ an open bounded domain with regular boundary $$\partial {\it{\Omega}}$$, we consider the system: \begin{equation}\label{1} \left\{\!\!\!\begin{array}{r@{\;}c@{\;}l} \displaystyle\frac{{\rm{d}}y(t)}{{\rm{d}}t} &=& Ay(t)+u(t)By(t)\\[12pt] y(0) &=& y_0, \end{array} \right. \end{equation} (1) where the operator $$A$$ generates a semigroup of contractions $$(S(t))_{t\geq 0}$$ on the Hilbert space $$L^2({\it{\Omega}})$$ with inner product $$\langle ., . \rangle$$ and norm $$\|.\|$$, $$u(t)\in L^2(0, +\infty)$$ is a scalar valued control and $$B$$ is a nonlinear operator mapping $$L^2({\it{\Omega}})$$ into itself such that $$B(0)=0$$. The problem of stabilizing system (1) on the whole domain $${\it{\Omega}}$$ has been considered in many works: In Ball & Slemrod (1979), it was shown that the control \begin{equation}\label{gkey2} u(t)=-\langle By(t), y(t)\rangle \end{equation} (2) weakly stabilizes system (1) provided that B is sequentially weakly continuous (i.e. $$B\varphi_k\rightharpoonup B\varphi$$ if $$\varphi_k \rightharpoonup \varphi$$) and the following condition holds \begin{equation}\label{key1} \langle BS(t)z, S(t)z\rangle = 0, \; \forall t\geq 0 \Longrightarrow z = 0. \end{equation} (3) Also, the control (2) allows the strong stabilization of system (1) under the condition: \begin{equation}\label{assum} \displaystyle \int^{T}_{0} |\langle BS(s)y,S(s)y\rangle|{\rm{d}}s \geq \mu \|y\|^2, \; (for \; some\; T, \mu >0).\; (see\; \text{Ouzahra (2008)}) \end{equation} (4) Moreover, in Bounit & Hammouri (1999), one showed that under specific conditions, the control \begin{equation} u(t) = \displaystyle-\frac{\langle By(t), y(t)\rangle }{1 + |\langle By(t), y(t)\rangle|} \end{equation} (5) realizes the strong stabilization of system (1). On the other hand, in Ouzahra (2011), author showed that the control \begin{equation}\label{key3} u(t) = \left\{ \begin{array}{lcl} \displaystyle-\frac{\langle By(t), y(t)\rangle }{\| y(t)\|^2},\; y(t)\neq 0 \\[6pt] 0, \hspace{2cm } ~~~~~~~y(t)=0 \end{array} \right. \end{equation} (6) stabilizes exponentially system (1) when (4) holds. For $$\omega\subset{\it{\Omega}}$$ an open subregion and of non-null measure, regional stabilization consists in choosing a control that stabilizes system (1) only on $$\omega$$ and we say that system (1) is regionally weakly (respectively strongly, exponentially) stabilizable on $$\omega$$, if for any initial condition $$ y_0\in L^2({\it{\Omega}})$$ the corresponding solution $$y(t)$$ of (1) is unique and $$\chi_{\omega}y(t)$$ converges to $$0$$ weakly (respectively strongly, exponentially) as $$t \longrightarrow +\infty$$ with $$\chi_{\omega}$$ is the restriction operator to $$\omega$$ (Zerrik & Ouzahra, 2003). This notion makes sense for the usual concept of stabilization taking into account the spatial variable and then it becomes closer to real-world problems, where one wishes to stabilize system (1) only on a critical subregion $$\omega$$ of $${\it{\Omega}}$$. The question of stabilizing regionally a linear or bilinear distributed system on a subregion $$\omega\subset{\it{\Omega}}$$ was introduced and developed by Zerrik and co-workers using different standard stabilization techniques. Then the relationship between regional exponential stabilization and the existence of an appropriate solution of the Riccati equation was examined (Zerrik & Ouzahra, 2003); also the problem was considered by means of a decomposition of the state-space using spectral properties of the dynamic of the considered system (Zerrik & Ouzahra, 2005). In Zerrik & Ezzaki (2017), we gave sufficient conditions for regional exponential, strong and weak stabilization for infinite bilinear systems. Furthermore, the properties and characterizations of a class of controls which ensure regional strong and weak stabilization of the gradient of system (1) with various illustrating examples were given in Zerrik & Ezzaki (2016). In Zerrik & Ouzahra (2007), it was shown that if the assumption \begin{equation}\label{assump} \langle KS(t)z, S(t)z\rangle\langle BS(t)z, S(t)z\rangle = 0, \; \forall t\geq 0 \Longrightarrow \chi_{\omega}z = 0 \end{equation} (7) holds, where $$K$$ is a continuous operator mapping $$L^2({\it{\Omega}})$$ into itself, then, the control \begin{equation}\label{cont1} u(t)=-\langle Ky(t), y(t)\rangle \end{equation} (8) regionally weakly stabilizes system (1) on $$\omega$$. Moreover, the control \begin{equation*} u(t)=-\langle i_{\omega}Pi_{\omega}By(t), y(t)\rangle \end{equation*} regionally weakly stabilizes system (1) on $$\omega$$ and is the unique solution of the minimization problem \begin{equation}\label{pr} \left\{\!\!\! \begin{array}{r@{\;}c@{\;}l} \min q(u)&=& \displaystyle\int^{+\infty}_0\langle i_{\omega} Ry(t), y(t)\rangle{\rm{d}}t + \displaystyle\int^{+\infty}_0|\langle By(t), i_{\omega}Pi_{\omega}y(t)\rangle|^2{\rm{d}}t + \displaystyle\int^{+\infty}_0|u(t)|^2{\rm{d}}t\\[6pt] v\in \mathcal{U}_{ad} &=& \{u|y(t)\; \text{is a global solution and}\; J(u)<+\infty\}, \end{array} \right. \end{equation} (9) where $$i_{\omega} = \chi_{\omega}^*\chi_{\omega}$$ with $$\chi_{\omega}^*$$ is the adjoint operator, $$R\in \mathcal{L}(L^2({\it{\Omega}}))$$ is a positive operator and $$P$$ is a positive operator verifying the following equation \begin{equation*} \langle Ay, i_{\omega}Pi_{\omega}y\rangle + \langle i_{\omega}Pi_{\omega}y, Ay\rangle + \langle y, i_{\omega} Ry\rangle=0, \; y\in \mathcal{D}(A). \end{equation*} In this article, we study regional exponential and strong stabilization of system (1). Furthermore, we study regional weak stabilization using a different control than (8). Moreover, we give a control that stabilizes regionally system (1) on $$\omega$$ minimizing a functional cost. The plan of the article is as follows: Second section is devoted to regional exponential, strong and weak stabilization of system (1), with some illustrating examples. In the third section, we establish a control that stabilizes regionally system (1) minimizing a functional cost. Finally, the developed results are illustrated by simulations. 2. Regional stabilization Here we deal with regional exponential, strong and weak stabilization of system (1). $$\chi_{\omega} : L^2({\it{\Omega}}) \longrightarrow L^2(\omega)$$ the restriction operator to $$\omega$$. Let us denote $$i_{\omega} = \chi_{\omega}^*\chi_{\omega}$$ with $$\chi_{\omega}^*$$ is the adjoint operator. Assume that \begin{equation}\label{pos} \langle i_{\omega} By, y\rangle \langle By, y\rangle \geq 0,\quad \forall y\in L^2({\it{\Omega}}) \end{equation} (10) and \begin{equation}\label{diss} \langle i_{\omega} Ay, y\rangle \leq 0,\quad \forall y\in \mathcal{D}(A). \end{equation} (11) The following result gives sufficient conditions for regional exponential stabilization of system (1). Theorem 2.1 Suppose that $$B$$ is locally Lipschitz such that \begin{equation}\label{sassum3} \displaystyle \int^{T}_{0} |\langle i_{\omega}BS(s)y, S(s)y\rangle|{\rm{d}}s \geq \alpha \|\chi_{\omega}y\|_{L^2(\omega)}, \; (for \; some\; T, \alpha >0) \end{equation} (12) then the control \begin{equation}\label{sskey21} u(t)=\left\{ \begin{array}{lll} -\displaystyle\frac{\langle i_{\omega}By(t), y(t)\rangle }{\|y(t)\|^2},\; y(t)\neq 0 \\[6pt] 0, \quad\!\!\hspace{2.8cm}y(t)=0 \end{array} \right. \end{equation} (13) regionally exponentially stabilizes system (1) on $$\omega$$. Proof. First, let us show that the map $$h : y\mapsto \frac{\langle i_{\omega}By, y\rangle}{\|y\|^2}By$$ is locally Lipschitz. Since $$B$$ is locally Lipshitz, for each $$R> 0$$ there exists a constant $$K>0$$ such that \begin{equation}\label{sblip} \| By-Bz \| \leq K\| y-z\|,\; \forall y, z \in L^2({\it{\Omega}}) : 0<\| y\| \leq \| z\| \leq R. \end{equation} (14) Thus $$\begin{array}{lll} h(z)Bz - h(y)By = \displaystyle\frac{\|y\|^2(\langle i_{\omega}Bz, z\rangle Bz - \langle i_{\omega}By, y\rangle By)}{\|z\|^2\|y\|^2} -\displaystyle\frac{(\|z\|^2-\|y\|^2)\langle i_{\omega}By, y\rangle By}{\|z\|^2\|y\|^2}. \end{array}$$ Since $$\chi_{\omega}$$ is continuous, there exists $$\delta>0$$ such that $$\begin{array}{l@{\;}ll} \|h(z)Bz - h(y)By\| & \leq \displaystyle\frac{\|\langle i_{\omega}Bz, z\rangle Bz - \langle i_{\omega}By, y\rangle By\|}{\|z\|^2} +K^2\delta|\|z\|^2-\|y\|^2|.\displaystyle\frac{\|y\|}{\|z\|^2}\\\\ & \leq \displaystyle\frac{\|\langle i_{\omega}Bz, z\rangle (Bz-By) + (\langle i_{\omega}Bz, z-y \rangle }{\|z\|^2}\\\\ &\quad{} -\displaystyle\frac{\langle i_{\omega}By-i_{\omega}Bz, y\rangle)By\|}{\|z\|^2} +2K^2\delta\| z-y\|\\\\ &\leq K^2\delta\|z\|^2\|z-y\|+ K^2\delta\|y\|\|z-y\|(\|y\|+\|z\|) + 2K^2\delta\| z-y\|\\\\ &\leq 3K^2\delta\|z\|^2\|z-y\|+2K^2\delta\| z-y\|. \end{array}$$ We conclude that for all $$y, z \in L^2({\it{\Omega}})$$, we have $$ \|h(z)-h(y)\|\leq L\|z-y\| $$ where $$L=3K^2\delta R^2 + 2K^2\delta$$. It follows that system (1) has a unique global mild solution $$y(t)$$ defined on a maximal interval $$[0,t_{\max}[$$ (see Pazy (1983)) and given by the variation of constants formula \begin{equation}\label{sconst} y(t) = S(t)y_0 + \int_{0}^{t}S(t-s)u(s)By(s){\rm{d}}s. \end{equation} (15) Let us consider the nonlinear semigroup $$N(t)y_0:=y(t)$$, we have \begin{equation*} \frac{{\rm{d}}}{{\rm{d}}t}\| N(t)y_0\|^2 = 2\langle AN(t)y_0, N(t)y_0\rangle -2 \frac{\langle i_{\omega}BN(t)y_0, N(t)y_0\rangle}{1 + |\langle i_{\omega}BN(t)y_0, N(t)y_0\rangle|}\langle BN(t)y_0, N(t)y_0\rangle. \end{equation*} Since $$S(t)$$ is a semigroup of contractions, then \begin{equation*} \frac{{\rm{d}}}{{\rm{d}}t}\| N(t)y_0\|^2 \leq -2 \frac{\langle i_{\omega}BN(t)y_0, N(t)y_0\rangle}{1 + |\langle i_{\omega}BN(t)y_0, N(t)y_0\rangle|}\langle BN(t)y_0, N(t)y_0\rangle. \end{equation*} Integrating this inequality, we get \begin{equation*} \| N(t)y_0\|^2 \leq \| y_0\|^2 -2 \int_0^t \langle h(N(s)y_0), N(s)y_0\rangle ds, \; \forall t\in[0, t_{max}[. \end{equation*} Using (10), we obtain \begin{equation}\label{sbound1} \| N(t)y_0\| \leq \| y_0\|. \end{equation} (16) For all $$y_0\in L^2({\it{\Omega}})$$ and $$t\geq 0$$, we have \begin{equation*} \begin{array}{l@{\;}l@{\;}l} \langle i_{\omega}BS(t)y_0, S(t)y_0\rangle &=& \langle i_{\omega}BS(t)y_0 - i_{\omega}BN(t)y_0, S(t)y_0 \rangle \\\\ &&- \langle i_{\omega}BN(t)y_0, N(t)y_0- S(t)y_0\rangle\\\\ &&+ \langle i_{\omega}BN(t)y_0, N(t)y_0\rangle. \end{array} \end{equation*} Using (14) and the continuity of $$\chi_{\omega}$$, we obtain \begin{equation}\label{sx13} \begin{array}{l@{\;}l@{\;}l} |\langle i_{\omega}BS(t)y_0, S(t)y_0\rangle|&\leq& 2\delta K\|N(t)y_0 - S(t)y_0\|\|y_0\|\\\\ && + |\langle i_{\omega}BN(t)y_0, N(t)y_0\rangle|. \end{array} \end{equation} (17) Moreover, from (15) we have \begin{equation*}\label{sx14} \|N(t)y_0 - S(t)y_0\|\leq K\int_{0}^{t}\displaystyle\frac{|\langle i_{\omega}BN(s)y_0, N(s)y_0\rangle|}{\|N(s)y_0\|}{\rm{d}}s. \end{equation*} For a fixed $$T\in]0, t_{max}[$$, Schwarz’s inequality yields \begin{equation}\label{sx15} \|N(t)y_0-S(t)y_0\|\leq K\left\{ T\int_{0}^{T}\displaystyle\frac{|\langle i_{\omega}BN(s)y_0, N(s)y_0\rangle|^2}{\|N(s)y_0\|^2}{\rm{d}}s\right\}^{\frac{1}{2}},\; \forall t\in [0,T]. \end{equation} (18) Using (16), we get \begin{equation}\label{sx16} |\langle i_{\omega}BN(t)y_0, N(t)y_0\rangle|\leq \displaystyle\frac{|\langle i_{\omega}BN(t)y_0, N(t)y_0\rangle|}{\|N(t)y_0\|}\|y_0\|,\; \forall t\in [0,T]. \end{equation} (19) Integrating (17) over the interval $$[0, T]$$ and using (18) and (19), we obtain \begin{equation}\label{sx17} \begin{array}{l@{\;}l@{\;}l} \displaystyle\int_{0}^{T}|\langle i_{\omega}BS(s)y_0, S(s)y_0\rangle|{\rm{d}}s &\leq 2\delta K^2T^{\frac{3}{2}}\|y_0\| \left\{ \displaystyle\int_{0}^{T}\displaystyle\frac{|\langle i_{\omega}BN(s)y_0, N(s)y_0\rangle|^2}{\|N(s)y_0\|^2}{\rm{d}}s\right\}^{\frac{1}{2}}\\\\ &\quad{}+ \sqrt{T}\|y_0\|\left\{ \displaystyle\int_{0}^{T}\displaystyle\frac{|\langle i_{\omega}BN(s)y_0, N(s)y_0\rangle|^2}{\|N(s)y_0\|^2}{\rm{d}}s\right\}^{\frac{1}{2}}. \end{array} \end{equation} (20) Replacing $$y_0$$ by $$N(t)y_0$$ in (20), we get \begin{equation*}\label{sx18} \begin{array}{l@{\;}ll} \displaystyle\int_{0}^{T}|\langle i_{\omega}BS(s)N(t)y_0, S(s)N(t)y_0\rangle|{\rm{d}}s &\leq 2\delta K^2T^{\frac{3}{2}}\|N(t)y_0\|\\\\ &\quad\times\left\{ \displaystyle\int_{t}^{t+T}\displaystyle\frac{|\langle i_{\omega}BN(s)y_0, N(s)y_0\rangle|^2}{\|N(s)y_0\|^2}{\rm{d}}s\right\}^{\frac{1}{2}}\\\\ &\quad+ \sqrt{T}\|N(t)y_0\|\left\{ \displaystyle\int_{t}^{t+T}\displaystyle\frac{|\langle i_{\omega}BN(s)y_0, N(s)y_0\rangle|^2}{\|N(s)y_0\|^2}{\rm{d}}s\right\}^{\frac{1}{2}}. \end{array} \end{equation*} By (12) and (16), we obtain \begin{equation*} \alpha\|\chi_{\omega}N(t)y_0\|_{L^2(\omega)}\leq \beta\left\{ \int_{t}^{t+T}\displaystyle\frac{|\langle i_{\omega}BN(s)y_0, N(s)y_0\rangle|^2}{\|N(s)y_0\|^2}{\rm{d}}s\right\}^{\frac{1}{2}}, \end{equation*} where $$\beta=(2\delta K^2T + 1)\sqrt{T}\|y_0\|$$. Then \begin{equation}\label{sx19} \alpha^2\|\chi_{\omega}N(t)y_0\|^2_{L^2(\omega)}\leq \beta^2 \int_{t}^{t+T}\displaystyle\frac{|\langle i_{\omega}BN(s)y_0, N(s)y_0\rangle|^2}{\|N(s)y_0\|^2}{\rm{d}}s. \end{equation} (21) Using (11), we get \begin{equation}\label{sx20} \frac{{\rm{d}}}{{\rm{d}}t}\|\chi_{\omega}N(t)y_0\|^2_{L^2(\omega)}\leq -2\displaystyle\frac{|\langle i_{\omega}BN(t)y_0, N(t)y_0\rangle|^2}{\|N(t)y_0\|^2}. \end{equation} (22) Integrating (22) from $$mT$$ to $$(m+1)T$$, $$(m\in\mathbb{N})$$, we obtain \begin{equation*} \|\chi_{\omega}N(mT)y_0\|^2_{L^2(\omega)} - \|\chi_{\omega}N((m+1)T)y_0\|^2_{L^2(\omega)} \geq 2 \int_{mT}^{(m+1)T}\displaystyle\frac{|\langle i_{\omega}BN(s)y_0, N(s)y_0\rangle|^2}{\|N(s)y_0\|^2}{\rm{d}}s. \end{equation*} By (21) and (22), we deduce \begin{equation*} \left(1 + 2\left(\frac{\alpha}{\beta}\right)^2\right)\|\chi_{\omega}N((m+1)T)y_0\|^2_{L^2(\omega)} \leq \|\chi_{\omega}N(mT)y_0\|^2_{L^2(\omega)}. \end{equation*} Then $$\|\chi_{\omega}N((m+1)T)y_0\|_{L^2(\omega)} \leq \rho\|\chi_{\omega}N(mT)y_0\|_{L^2(\omega)}$$ where $$\rho=\frac{1}{(1 + 2(\frac{\alpha}{\beta})^2)^{\frac{1}{2}}}$$. By recurrence, we show that $$ \|\chi_{\omega}N(mT)y_0\|_{L^2(\omega)} \leq \rho^m\|\chi_{\omega}y_0\|_{L^2(\omega)}$$. Taking $$m=E(\frac{t}{T})$$ the integer part of $$\frac{t}{T}$$ and remarking that $$m\geq \frac{t}{T}-1$$, we get \begin{equation*} \|\chi_{\omega}N(t)y_0\|_{L^2(\omega)} \leq\rho^{\frac{t}{T}-1} \|\chi_{\omega}y_0\|_{L^2(\omega)}, \; \forall t\geq 0. \end{equation*} It follows that \begin{equation*} \|\chi_{\omega}N(t)y_0\|_{L^2(\omega)} \leq Me^{-\sigma t}\|y_0\|, \; \forall t\geq 0 \end{equation*} where $$M = \delta(1 + 2(\frac{\alpha}{\beta})^2)^{\frac{1}{2}}$$ and $$\sigma = \frac{\ln(1 + 2(\frac{\alpha}{\beta})^2) }{2T}$$, which shows the regional exponential stability of system (1) on $$\omega$$. □ Example 2.2 Let $${\it{\Omega}} = ]0,1[$$, we consider the wave equation defined by \begin{equation}\label{bexp23} \left\{\!\!\!\!\!\! \begin{array}{lll} &\displaystyle\frac{\partial^2 y}{\partial t^2}(x, t) = {\it{\Delta}} y(x, t) + u(t)\displaystyle\frac{\partial y}{\partial t}(x, t) \quad on \; {\it{\Omega}} \times ]0, +\infty[\\ & y(x, 0) = y_0, \; \displaystyle\frac{\partial y}{\partial t}(x, 0)=y_1 \hspace{2.8cm} on\; {\it{\Omega}}\\ & y(0,t) = y(1,t) = 0 \hspace{4.7cm} on\; [0, +\infty[. \end{array}\right. \end{equation} (23) This system has the form of (1) if we set $$A = \begin{pmatrix} 0 & I \\ {\it{\Delta}} & 0 \end{pmatrix}\; and\; B = \begin{pmatrix} 0 & 0 \\ 0 & I \end{pmatrix}\!. $$ Let $$\tilde{A}= -{\it{\Delta}}$$, with domain $$\mathcal{D}(\tilde{A})=H^2({\it{\Omega}})\cap H^1_0({\it{\Omega}})$$. Consider $$H = H^1_0({\it{\Omega}})\times L^2({\it{\Omega}})$$ the state space endowed with the inner product $$\langle (y_1, z_1), (y_2, z_2)\rangle = \langle \tilde{A}^\frac{1}{2}y_1, \tilde{A}^\frac{1}{2}y_2\rangle_{L^2({\it{\Omega}})} + \langle z_1, z_2\rangle_{L^2({\it{\Omega}})}$$. The eigenvalues of $$\tilde{A}$$ are $$\lambda_k = (k\pi)^2$$, corresponding to eigenfunctions $$\phi_k(x) = \sqrt{2}\sin(k\pi x),\; \forall k\,{\in}\,\mathbb{N}^*$$. We set $$y=(y_1, y_2)\in H$$ with $$y_1 = \sum_{k=1}^{\infty}a_k\phi_k$$ and $$y_2 = \sum_{k=1}^{\infty}\sqrt{\lambda_k}b_k\phi_k$$, where $$(a_k, b_k)\in \mathbb{R}^2$$$$\forall k\geq 1$$, and the semigroup is given by $$S(s)y = \sum_{k=1}^{\infty}\begin{pmatrix} a_k\cos(k\pi s) + b_k\sin(k\pi s)\\ b_k k\pi\cos(k\pi s) - a_k k\pi\sin(k\pi s) \end{pmatrix}\phi_k, \quad \forall s\geq 0. $$ For $$\omega = ]0,\frac{1}{2}[$$, we have \begin{equation*} \begin{array}{lll} \langle i_{\omega}BS(s)y, S(s)y\rangle = \displaystyle\sum_{k=1}^{\infty} \frac{(k\pi)^2}{2}\big(a_k^2\sin^2(k\pi s) - b_k a_k\sin(2k\pi s) + b_k^2\cos^2(k\pi s) \big) . \end{array} \end{equation*} Taking $$T=2$$, we get \begin{equation*} \begin{array}{lll} \displaystyle\int_{0}^{2} |\langle i_{\omega}BS(s)y, S(s)y\rangle|{\rm{d}}s = \displaystyle\sum_{k=1}^{\infty}\frac{(k\pi)^2}{2}(a_k^2 + b_k^2) \geq \alpha \|\chi_{\omega}y\|_{L^2(\omega)}, \end{array} \end{equation*} where $$\alpha = \frac{\pi}{2}\sqrt{\eta}$$, $$(\eta=\min\{(a_k^2+b_k^2)|k\in\mathbb{N}^*\})$$, thus (12) holds and the control \begin{equation*} u(t) = -\frac{\|\chi_{\omega}\partial_ty\|^2_{L^2(\omega)}}{\|y(t)\|^2_{H^1_0({\it{\Omega}})}+\|\partial_t y(t)\|^2} \end{equation*} regionally exponentially stabilizes system (23) on $$\omega$$. The next result gives sufficient conditions for regional strong stabilization. Theorem 2.3 Assume that $$B$$ is locally Lipschitz and satisfies \begin{equation}\label{ss3} \displaystyle \int^{T}_{0} |\langle i_{\omega}BS(s)y, S(s)y\rangle|{\rm{d}}s \geq \alpha \|\chi_{\omega}y\|^2_{L^2(\omega)}, \; ( T, \alpha >0) \end{equation} (24) then the control \begin{equation}\label{skey21} u(t)= -\frac{\langle i_{\omega}By(t), y(t)\rangle}{1 + |\langle i_{\omega}By(t), y(t)\rangle|} \end{equation} (25) regionally strongly stabilizes system (1) on $$\omega$$. Proof. By the same arguments using in the proof of Theorem 2.1, we can show that the map $$g : y\mapsto \frac{\langle i_{\omega}By, y\rangle}{1+|\langle i_{\omega}By, y\rangle|}By$$ is locally Lipschitz and we deduce that system (1) has a unique global mild solution $$y(t)$$ (see Pazy (1983)). It remains to show that $$\chi_{\omega} y(t) \longrightarrow 0\quad as\quad t\longrightarrow +\infty$$. For all $$y_0\in L^2({\it{\Omega}})$$, we have \begin{equation*} \begin{array}{lll} \langle i_{\omega}BS(t)y_0, S(t)y_0\rangle = \langle i_{\omega}BS(t)y_0 - i_{\omega}By(t), S(t)y_0\rangle + \langle i_{\omega}By(t), S(t)y_0\rangle. \end{array} \end{equation*} Using formula (15), we obtain \begin{equation*} \begin{array}{l@{\;}l@{\;}l} \langle i_{\omega}BS(t)y_0, S(t)y_0\rangle &=& \langle i_{\omega}BS(t)y_0 - i_{\omega}By(t), S(t)y_0\rangle - \langle i_{\omega}By(t), {\it{\Phi}}(t)\rangle\\\\ &&\quad{} + \langle i_{\omega}By(t), y(t)\rangle, \end{array} \end{equation*} where $${\it{\Phi}}(t)=\int_{0}^{t} S(t-s)u(s) By(s){\rm{d}}s$$. By (14) and the continuity of $$\chi_{\omega}$$, we deduce that \begin{equation}\label{sk4} \begin{array}{l@{\;}l@{\;}l} |\langle i_{\omega}BS(t)y_0, S(t)y_0\rangle| & \leq \delta K\| {\it{\Phi}}(t)\|(\| S(t)y_0\| + \| y(t)\|) +| \langle i_{\omega}By(t), y(t)\rangle |. \end{array} \end{equation} (26) Then \begin{equation*} |\langle i_{\omega}BS(t)y_0, S(t)y_0\rangle| \leq 2\delta K\| {\it{\Phi}} (t)\|\| y_0 \| + |\langle i_{\omega}By(t), y(t)\rangle|, \forall t\in [0, T]. \end{equation*} Moreover, Schwarz’s inequality gives \begin{equation}\label{sk5} \| {\it{\Phi}}(t)\| \leq K\|y_0\|\sqrt T\left(\int_{0}^{T}|\langle i_{\omega}By(s), y(s)\rangle|^2 {\rm{d}}s\right)^{\frac{1}{2}}\!. \end{equation} (27) Integrating (26) over the interval $$[0, T]$$ and using (27), we obtain \begin{equation*} \begin{array}{l@{\;}l@{\;}l} \displaystyle\int^T_0 |\langle i_{\omega}BS(s)y_0, S(s)y_0\rangle |{\rm{d}}s &\leq \displaystyle 2\delta T^\frac{3}{2}K\|y_0\|^2 \left(\displaystyle\int_{0}^{T}|\langle i_{\omega}By(s), y(s)\rangle|^2{\rm{d}}s\right)^{\frac{1}{2}}\\\\ &\quad +\sqrt{T}\|y_0\|\left(\displaystyle\displaystyle\int^T_0 |\langle i_{\omega}By(s), y(s)\rangle|^2{\rm{d}}s\right)^{\frac{1}{2}}\!. \end{array} \end{equation*} Replacing $$y_0$$ by $$y(t)$$, we get \begin{equation}\label{sineq} \begin{array}{l@{\;}ll} \displaystyle\int^T_0 |\langle i_{\omega}BS(s)y(t), S(s)y(t)\rangle |{\rm{d}}s & \leq \displaystyle 2\delta T^\frac{3}{2}K\|y_0\| \left(\displaystyle\int_{t}^{t+T}|\langle i_{\omega}By(s), y(s)\rangle|^2{\rm{d}}s\right)^{\frac{1}{2}}\\\\ &\quad +\sqrt{T}\|y_0\|\left(\displaystyle\displaystyle\int^{t+T}_t |\langle i_{\omega}By(s), y(s)\rangle|^2{\rm{d}}s\right)^{\frac{1}{2}}\!. \end{array} \end{equation} (28) Furthermore, using (11), we obtain \begin{equation*} \frac{d}{dt}\|\chi_{\omega}y(t)\|^2_{L^2(\omega)} \leq -2 \displaystyle\frac{|\langle i_{\omega}By(t), y(t)\rangle|^2}{1 + |\langle i_{\omega}By(t), y(t)\rangle|},\; \forall t\geq 0. \end{equation*} Integrating this inequality, we get \begin{equation*} \int_0^t |\langle i_{\omega}By(s), y(s)\rangle|^2 ds \leq \frac{L_{\|y_0\|}}{2}\|\chi_{\omega}y_0\|^2_{L^2(\omega)}, \; \forall t\geq 0 \end{equation*} with $$L_{\|y_0\|} = \sup_{\| y\| \leq \|y_0\|}(1 + |\langle i_{\omega}By, y\rangle|)$$. Hence \begin{equation}\label{sfini} \int_0^{+\infty} |\langle i_{\omega}By(s), y(s)\rangle|^2 ds < +\infty. \end{equation} (29) From (29) and (28), we deduce that \begin{equation}\label{ss7} \int^T_0 |\langle i_{\omega}BS(s)y(t), S(s)y(t)\rangle |{\rm{d}}s \longrightarrow 0,\; as\; t \longrightarrow +\infty. \end{equation} (30) It follows from (24) and (30) that $$\|\chi_{\omega}y(t)\|_{L^2(\omega)} \longrightarrow 0, \: as\; t \longrightarrow +\infty$$, which completes the proof. □ Example 2.4 On $${\it{\Omega}} = ]0,1[$$, we consider the following beam equation \begin{equation}\label{exp23} \left\{\!\! \begin{array}{l@{\;}l@{\;}l} \displaystyle\frac{\partial^2 y}{\partial t^2}(x, t) &=& -\displaystyle\frac{\partial^4 y}{\partial x^4}(x, t) + u(t)\displaystyle\frac{\partial y}{\partial t}(x, t) \quad on \; {\it{\Omega}} \times ]0, +\infty[\\[12pt] y(x, 0) &=& y_0,\; \displaystyle\frac{\partial y}{\partial t}(x, 0)=y_1 \hspace{3cm} on\; {\it{\Omega}}\\[12pt] y(\xi, t) &=& \displaystyle\frac{\partial^2 y}{\partial x^2}(\xi, t) = 0, \;\xi = 0, 1 \hspace{1.8cm} on\; ]0, +\infty[. \end{array}\right. \end{equation} (31) Let $$\tilde{A}= \frac{\partial^4 y}{\partial x^4}$$, with $$\mathcal{D}(\tilde{A}) = \{y\in L^2({\it{\Omega}})/\frac{\partial^4 y}{\partial x^4}\in L^2({\it{\Omega}}), y(\xi, t) = \frac{\partial^2 y}{\partial x^2}(\xi, t) = 0, \;\xi = 0, 1 \}$$. Setting $$H = (H^2({\it{\Omega}})\cap H^1_0({\it{\Omega}}))\times L^2({\it{\Omega}})$$ the state space endowed with the inner product $$\langle (y_1, z_1), (y_2, z_2)\rangle = \langle \tilde{A}^{\frac{1}{2}}y_1, \tilde{A}^{\frac{1}{2}}y_2\rangle_{L^2({\it{\Omega}})} + \langle z_1, z_2\rangle_{L^2({\it{\Omega}})}$$. The eigenvalues of $$\tilde{A}$$ are $$\lambda_j = (j\pi)^4$$, corresponding to eigenfunctions $$\varphi_j(x) = \sqrt{2}\sin((j\pi)^2 x),\; \forall j\,{\in}\,\mathbb{N}^*$$. System (31) has the form of (1) if we take $$A = \begin{pmatrix} 0 & I \\ -\tilde{A} & 0 \end{pmatrix}\; and\; B = \begin{pmatrix} 0 & 0 \\ 0 & I \end{pmatrix}. $$ $$B$$ satisfies condition (24), indeed, for $$y\in H$$, we have $y = \sum_{j=1}^{\infty}\begin{pmatrix} \alpha_j\\ \lambda_j^{\frac{1}{2}}\beta_j \end{pmatrix} \varphi_j$, where $$(\alpha_j, \beta_j)\in \mathbb{R}^2\; \forall j\geq 1$$ and the semigroup is given by $$S(s)y = \sum_{j=1}^{\infty}\begin{pmatrix} \alpha_j\cos(\lambda_j^{\frac{1}{2}} s) + \beta_j\sin(\lambda_j^{\frac{1}{2}}s)\\\\ \beta_j\lambda_j^{\frac{1}{2}}\cos(\lambda_j^{\frac{1}{2}} s) - \alpha_j\lambda_j^{\frac{1}{2}}\sin(\lambda_j^{\frac{1}{2}}s) \end{pmatrix}\varphi_j, \quad \forall s\geq 0. $$ For $$\omega = ]0,\frac{1}{2}[$$, we have \begin{equation*} \begin{array}{l@{\;}l@{\;}l} \langle i_{\omega}BS(s)y, S(s)y\rangle &=& \displaystyle\sum_{j=1}^{\infty} \frac{\lambda_j}{2}\big\{\alpha_j\sin(\lambda_j^{\frac{1}{2}} s) - \beta_j\cos(\lambda_j^{\frac{1}{2}}s)\big\}^2\\\\ &=& \displaystyle\sum_{j=1}^{\infty} \frac{\lambda_j}{2}\big\{\alpha_j^2\sin^2(j\pi s) + \beta_j^2\cos^2(j\pi s) - \alpha_j\beta_j\sin(2j\pi s)\big\} . \end{array} \end{equation*} Integrating this relation over the time interval $$[0, 2]$$, we obtain \begin{equation*} \begin{array}{lll} \displaystyle\int_{0}^{2} |\langle i_{\omega}BS(s)y, S(s)y\rangle|{\rm{d}}s = \displaystyle\sum_{j=1}^{\infty}\frac{\lambda_j}{2}(\alpha_j^2 + \beta_j^2) = \|\chi_{\omega}y\|^2 \end{array} \end{equation*} then (24) holds. We conclude that the control \begin{equation*} u(t) = -\frac{\| \chi_{\omega} \partial_t y(., t)\|^2_{L^2(\omega)}}{1+\| \chi_{\omega} \partial_t y(., t)\|^2_{L^2(\omega)}} \end{equation*} regionally strongly stabilizes system (31) on $$\omega$$. The following result gives sufficient conditions for regional weak stabilization. Theorem 2.5 Let $$B$$ be locally Lipschitz and weakly sequentially continuous such that \begin{equation}\label{s12} \langle i_{\omega}BS(t)z, S(t)z\rangle =0, \; \forall t\geq 0 \Longrightarrow \chi_{\omega} z=0 \end{equation} (32) then control (25) regionally weakly stabilizes system (1) on $$\omega$$. Proof. Consider the nonlinear semigroup $$N(t)y_0=y(t)$$ and let $$(t_k)$$ be a sequence of real numbers such that $$t_k\longrightarrow +\infty$$ as $$k\longrightarrow +\infty$$. From (16), $$N(t_ {k})y_0$$ is bounded in $$L^2({\it{\Omega}})$$, then there exists a subsequence $$(t_ {\phi(k)})$$ of $$(t_k)$$ and $$z\in L^2({\it{\Omega}})$$ such that $$N(t_ {\phi(k)})y_0 \rightharpoonup z\quad as\quad k\longrightarrow +\infty$$. For all $$y_0\in L^2({\it{\Omega}})$$ and $$t\geq 0$$, we have \begin{equation*} \begin{array}{l@{\;}l@{\;}l} \langle i_{\omega}BS(t)y_0, S(t)y_0\rangle &=& \langle i_{\omega}BN(t)y_0, N(t)y_0\rangle - \langle i_{\omega}BN(t)y_0, N(t)y_0 - S(t)y_0 \rangle \\\\ &&{}+ \langle i_{\omega}BS(t)y_0 - i_{\omega}BN(t)y_0, S(t)y_0\rangle. \\\\ \end{array} \end{equation*} Using (14) and the continuity of $$\chi_{\omega}$$, we get \begin{equation}\label{s13} \begin{array}{l@{\;}l@{\;}l} |\langle i_{\omega}BS(t)y_0, S(t)y_0\rangle|&\leq& |\langle i_{\omega}BN(t)y_0, N(t)y_0\rangle|\\\\ &&{} + 2\delta K\|N(t)y_0 - S(t)y_0\|\|y_0\|,\; (\text{for some}\; \delta > 0). \end{array} \end{equation} (33) From (15) and Schwarz’s inequality, we have \begin{equation}\label{s15} \|N(t)y_0-S(t)y_0\|\leq K\|y_0\|\sqrt{Tf(0)},\quad \forall t\in[0,T] \end{equation} (34) where $$f(t) = \int_{t}^{ t + T}|\langle i_{\omega}BN(s)y_0, N(s)y_0\rangle|^2{\rm{d}}s. $$ Integrating (33) over the interval $$[0, T]$$ and using (34), we obtain \begin{equation}\label{s17} \int_{0}^{T}|\langle i_{\omega}BS(s)y_0, S(s)y_0\rangle|{\rm{d}}s\leq \beta \|y_0\|\sqrt{f(0)} \end{equation} (35) with $$\beta = (2\delta K^2T\|y_0\|+1)\sqrt{T}$$. Replacing $$y_0$$ by $$N(t_ {\phi(k)})y_0$$ in (35), we get \begin{equation*}\label{s18} \int_{0}^{T}|\langle i_{\omega}BS(s)N(t_ {\phi(k)})y_0, S(s)N(t_ {\phi(k)})y_0\rangle|{\rm{d}}s\leq \beta\|y_0\|\sqrt{f(t_ {\phi(k)})}. \end{equation*} It follows that \begin{equation*} \int_{0}^{T}|\langle i_{\omega}BS(s)N(t_ {\phi(k)})y_0, S(s)N(t_ {\phi(k)})y_0\rangle|{\rm{d}}s \longrightarrow 0 \quad as \quad k\longrightarrow 0. \end{equation*} Since $$B$$ is weakly sequentially continuous and using the dominated convergence theorem, we deduce that $$\langle i_{\omega}BS(s)z, S(s)z\rangle = 0,\quad \forall s\in[0,T]$$. From (32), it follows that $$\chi_{\omega}N(t_{\phi(k)})y_0 \rightharpoonup 0 \quad as\quad k\longrightarrow +\infty$$, and then $$\forall \phi\in L^2({\it{\Omega}})$$, $$\langle \chi_{\omega}N(t_k)y_0, \phi\rangle \longrightarrow 0$$ as $$k\longrightarrow +\infty$$, hence $$\chi_{\omega} N(t)y_0 \rightharpoonup 0\quad as\quad t\longrightarrow +\infty.$$ In other words $$\chi_{\omega} y(t)$$ converges weakly to $$0$$ as $$t\longrightarrow +\infty$$, and then system (1) is regionally weakly stabilizable on $$\omega$$. □ Example 2.6 On $${\it{\Omega}} = ]0,1[$$, we consider the following equation \begin{equation}\label{2exp23} \left\{\!\! \begin{array}{l@{\;}l@{\;}l} \displaystyle\frac{\partial^2 y}{\partial t^2}(x, t) &=& -\displaystyle\frac{\partial^4 y}{\partial x^4}(x, t) + u(t)\displaystyle\frac{\partial y}{\partial t}(x, t) \quad on \; {\it{\Omega}} \times ]0, +\infty[\\ y(x, 0) &=& y_0, \; \displaystyle\frac{\partial y}{\partial t}(x, 0)=y_1 \hspace{2.7cm} on\; {\it{\Omega}}\\ y(\xi, t) &=& \displaystyle\frac{\partial^2 y}{\partial x^2}(\xi, t) = 0, \;\xi = 0, 1 \hspace{1.7cm} on\; ]0, +\infty[ \end{array}\!.\right. \end{equation} (36) The state space is $$H = (H^2({\it{\Omega}})\cap H^1_0({\it{\Omega}}))\times L^2({\it{\Omega}})$$, this system has the form of (1) if we set $$A = \begin{pmatrix} 0 & I \\ -{\it{\Delta}}^2 & 0 \end{pmatrix}\; and\; B = \begin{pmatrix} 0 & 0 \\ 0 & I \end{pmatrix}\!. $$ For $$\omega = ]0,\frac{1}{2}[$$, by the same arguments using in Example 2.4, we show that \begin{equation*} \begin{array}{lll} \displaystyle\int_{0}^{2} |\langle i_{\omega}BS(s)y, S(s)y\rangle|{\rm{d}}s = \|\chi_{\omega}y\|^2 \end{array}. \end{equation*} It follows that \begin{equation*} \langle i_{\omega}BS(t)y, S(t)y\rangle =0 \; \Longrightarrow \chi_{\omega} y=0. \end{equation*} Then, condition (32) holds. We conclude that the control \begin{equation*} u(t) = -\frac{\| \chi_{\omega} \partial_t y(., t)\|^2_{L^2(\omega)}}{1+\| \chi_{\omega} \partial_t y(., t)\|^2_{L^2(\omega)}} \end{equation*} regionally weakly stabilizes system (36) on $$\omega$$. 3. Regional stabilization problem This section deals with regional stabilization of system (1) by considering the following minimization problem \begin{equation}\label{p23} \left\{\!\! \begin{array}{r@{\;}c@{\;}l} \min J(u)&=& \displaystyle\int^{+\infty}_0\frac{\langle P_{\omega}By(t), y(t)\rangle^2}{\|y(t)\|^2}{\rm{d}}t + \displaystyle\int^{+\infty}_0\langle i_{\omega} Ry(t), y(t)\rangle{\rm{d}}t + \displaystyle\int^{+\infty}_0\|y(t)\|^2|u(t)|^2{\rm{d}}t\\[6pt] u\in \mathcal{U}_{ad} &=& \{u\in L^2(0,+\infty)|y(t)\; \text{is}~ \text{a}~ \text{global}~ \text{solution}~ \text{and}~ J(u)<+\infty\}, \end{array} \right. \end{equation} (37) where $$B$$ is bounded, $$A$$ satisfies (11) and $$P_{\omega}=i_{\omega}Pi_{\omega}$$ with $$P\in \mathcal{L}(L^2({\it{\Omega}}))$$ is a positive and bounded operator satisfying the equation \begin{equation}\label{22} \langle P_{\omega} Ay, y\rangle + \langle y, P_{\omega}Ay\rangle + \langle i_{\omega}Ry, y\rangle = 0, \; y\in \mathcal{D}(A), \end{equation} (38) where $$R\in \mathcal{L}(L^2({\it{\Omega}}))$$ is a positive operator. Theorem 3.1 Suppose that $$B$$ is locally Lipschitz and $$P$$ is compact such that \begin{equation}\label{pp25} \langle P_{\omega}BS(t)z, S(t)z\rangle =0, \; \forall t\geq 0 \Longrightarrow \chi_{\omega} z=0 \end{equation} (39) then the control \begin{equation}\label{p27} u^*(t) = -\frac{\langle P_{\omega}By(t), y(t)\rangle}{\|y(t)\|^2} \end{equation} (40) is the unique solution of (37) and regionally weakly stabilizes system (1) on $$\omega$$. Proof. Let us define the function $$V(y)=\langle P_{\omega}y, y\rangle$$, $$ y\in L^2({\it{\Omega}})$$. Using (38), for all $$y_0\in\mathcal{D}(A)$$ and $$t\geq 0$$, we have \begin{equation}\label{deriv} \frac{{\rm{d}}V(y(t))}{{\rm{d}}t}= - 2\frac{\langle P_{\omega}By(t), y(t)\rangle^2}{\|y(t)\|^2} - \langle i_{\omega}Ry(t), y(t)\rangle. \end{equation} (41) Integrating this relation, we get \begin{equation}\label{p26} \int^{t}_0\frac{\langle P_{\omega}By(s), y(s)\rangle^2}{\|y(s)\|^2}{\rm{d}}s\leq \frac{1}{2}V(y_0), \; t\geq 0. \end{equation} (42) The solution $$y(.)$$ is continuous with respect to the initial condition $$y_0$$ (seePazy (1983)) and $$D(A)$$ is dense in $$L^2({\it{\Omega}})$$, then (42) holds for all $$y_0\in L^2({\it{\Omega}})$$ so $$J(u^*)$$ is finite for all $$y_0\in L^2({\it{\Omega}})$$. Let $$(t_k)$$ be a sequence of real numbers such that $$t_k\longrightarrow +\infty$$ as $$k\longrightarrow +\infty$$. By (16), $$y(t_k)$$ is bounded in $$L^2({\it{\Omega}})$$, then there exists a subsequence $$(t_ {\phi(k)})$$ of $$(t_k)$$ and $$z\in L^2({\it{\Omega}})$$ such that \begin{equation*} y(t_ {\phi(k)}) \rightharpoonup z\quad as\quad k\longrightarrow +\infty. \end{equation*} For all $$y_0\in L^2({\it{\Omega}})$$ and $$t\geq 0$$, we have \begin{equation*} \begin{array}{l@{\;}l@{\;}l} \langle P_{\omega}BS(t)y_0, S(t)y_0\rangle &=& \langle P_{\omega}BS(t)y_0 - P_{\omega}By(t), S(t)y_0\rangle \\\\ &&{}- \langle P_{\omega}By(t), y(t)- S(t)y_0\rangle + \langle P_{\omega}By(t), y(t)\rangle. \end{array} \end{equation*} Thus \begin{equation}\label{O13} \begin{array}{lll} |\langle P_{\omega}BS(t)y_0, S(t)y_0\rangle|\leq 2\delta^2\|P\|K\|y(t) - S(t)y_0\|\|y_0\| + |\langle P_{\omega}By(t), y(t)\rangle|. \end{array} \end{equation} (43) Moreover, we have \begin{equation*}\label{O14} \|y(t) - S(t)y_0\|\leq K\|y_0\|\int_{0}^{t}|\langle P_{\omega}By(s), y(s)\rangle|{\rm{d}}s. \end{equation*} Schwarz’s inequality yields \begin{equation}\label{O15} \|y(t)-S(t)y_0\|\leq K\sqrt{T\lambda(0)},\quad \forall t\in[0,T] \end{equation} (44) where $$\lambda(t) = \int_{t}^{ t + T}|\langle P_{\omega}By(s), y(s)\rangle|^2{\rm{d}}s$$. Integrating (43) over the interval $$[0, T]$$ and with (44), we obtain \begin{equation}\label{O17} \int_{0}^{T}|\langle P_{\omega}BS(s)y_0, S(s)y_0\rangle|{\rm{d}}s\leq (2\delta^2 K^2T\|P\|+1)\|y_0\|\sqrt{T\lambda(0)}. \end{equation} (45) Replacing $$y_0$$ by $$y(t_ {\phi(k)})$$ in (45), we get \begin{equation}\label{O18} \int_{0}^{T}|\langle P_{\omega}BS(s)y(t_ {\phi(k)}), S(s)y(t_ {\phi(k)})\rangle|{\rm{d}}s\leq (2\delta^2 K^2T\|P\|+1)\|y_0\|\sqrt{T\lambda(t_ {\phi(k)})}. \end{equation} (46) By (46), we get $$ \lim_{k\longrightarrow +\infty}\int_{0}^{T}\langle P_{\omega}BS(s)y(t_{\phi(k)}), S(s)y(t_{\phi(k)})\rangle{\rm{d}}s = 0.$$ Since $$P$$ is compact and $$S(s)$$ is continuous $$\forall s\geq 0$$, we have \begin{equation*} \lim\limits_{k\longrightarrow +\infty}\langle P_{\omega}BS(s)y(t_{\phi(k)}), S(s)y(t_{\phi(k)})\rangle = \langle P_{\omega}BS(s)z, S(s)z\rangle. \end{equation*} By dominated convergence theorem, we obtain \[\displaystyle\int_{0}^{T}|\langle P_{\omega}BS(s)z, S(s)z\rangle|{\rm{d}}s = 0\text{ and then }\langle P_{\omega}BS(s)z, S(s)z\rangle = 0,\quad \forall s\in[0,T].\] Using (39), we deduce that $$\chi_{\omega}y(t_{\phi(k)}) \rightharpoonup 0 \quad as\quad k\longrightarrow +\infty.$$ It follows that $$\chi_{\omega} y(t)$$ converges weakly to $$0$$ as $$t\longrightarrow +\infty$$, and system (1) is regionally weakly stabilizable on $$\omega$$. Now, let us prove that (40) is the unique solution of (37). Since $$P$$ is compact, it follows that $$V(y(t))\longrightarrow 0$$ as $$t\rightarrow +\infty$$. Let $$y_0\in \mathcal{D}(A)$$, formula (41) may be written as \begin{equation*} \frac{{\rm{d}}V(y(t))}{{\rm{d}}t}= \|y(t)\|^2\left(\left[\frac{\langle P_{\omega}By(t), y(t)\rangle}{\|y(t)\|^2}+u(t)\right]^2 - \frac{\langle P_{\omega}By(t), y(t)\rangle^2}{\|y(t)\|^2}- u^2(t)\right)-\langle i_{\omega}Ry(t), y(t)\rangle \end{equation*} integrating this relation, we obtain \begin{equation*} J(u) = V(y_0) + \int^{+\infty}_0\|y(s)\|^2\left[\frac{\langle P_{\omega}By(s), y(s)\rangle}{\|y(s)\|^2}+ u(s)\right]^2{\rm{d}}s. \end{equation*} Then \begin{equation*} J(u)\geq V(y_0), \; \forall u\in \mathcal{U}_{ad}. \end{equation*} For $$u=u^*$$, we get $$J(u^*)=V(y_0)$$. Let $$y_0\in L^2({\it{\Omega}})$$, and a sequence $$(y_{0k})\subset \mathcal{D}(A)$$ such that $$y_{0k} \longrightarrow y_0$$ as $$ k\longrightarrow +\infty$$, we have \begin{equation*} J(u) = V(y_{0k}) + \int^{+\infty}_0\|y_k(s)\|^2[\frac{\langle P_{\omega}By_k(s), y_k(s)\rangle}{\|y_k(s)\|^2}+ u(s)]^2 ds - \int^{+\infty}_0\langle i_{\omega}Ry_k(s), y_k(s)\rangle ds. \end{equation*} Thus $$J(u)\geq V(y_{0k})$$. We deduce that $$J(u)\geq V(y_{0})=J(u^*)$$, so (40) is the unique solution of problem (37). □ Proposition 3.2 Suppose that $$B$$ is locally Lipschitz and satisfies \begin{equation}\label{oss3} \displaystyle \int^{T}_{0} |\langle P_{\omega}BS(s)y, S(s)y\rangle|ds \geq \alpha \|\chi_{\omega}y\|^2_{L^2(\omega)}, \; ( T, \alpha >0) \end{equation} (47) then control (40) is the unique solution of (37) and regionally strongly stabilizes system (1) on $$\omega$$. Proof. For all $$y_0\in L^2({\it{\Omega}})$$, we have \begin{equation*} \begin{array}{lll} \langle P_{\omega}BS(t)y_0, S(t)y_0\rangle = \langle P_{\omega}BS(t)y_0 - P_{\omega}By(t), S(t)y_0\rangle + \langle P_{\omega}By(t), S(t)y_0\rangle. \end{array} \end{equation*} Using (15), we obtain the following relation \begin{equation*} \begin{array}{l@{\;}l@{\;}l} \langle P_{\omega}BS(t)y_0, S(t)y_0\rangle &=& \langle P_{\omega}BS(t)y_0 - P_{\omega}By(t), S(t)y_0\rangle - \langle P_{\omega}By(t), {\it{\Psi}}(t)\rangle\\\\ &&{} + \langle P_{\omega}By(t), y(t)\rangle, \end{array} \end{equation*} where $${\it{\Psi}}(t)=\int_{0}^{t} S(t-s)u(s) By(s){\rm{d}}s$$. Since $$\chi_{\omega}$$ is continuous, then there exists $$ \delta > 0$$ such that \begin{equation}\label{osk4} \begin{array}{lll} |\langle P_{\omega}BS(t)y_0, S(t)y_0\rangle| \leq \delta^2 K\|P\|\| {\it{\Psi}}(t)\|(\| S(t)y_0\| + \| y(t)\|) +| \langle P_{\omega}By(t), y(t)\rangle |. \end{array} \end{equation} (48) Then \begin{equation*} |\langle P_{\omega}BS(t)y_0, S(t)y_0\rangle| \leq 2\delta^2 K\|P\|\| {\it{\Psi}} (t)\|\| y_0 \| + |\langle P_{\omega}By(t), y(t)\rangle|, \forall t\in [0, T]. \end{equation*} Moreover, we have \begin{equation}\label{osk5} \| {\it{\Psi}}(t)\| \leq K\|y_0\|\sqrt T\left(\int_{0}^{T}|\langle P_{\omega}By(s), y(s)\rangle|^2{\rm{d}}s\right)^{\frac{1}{2}}\!. \end{equation} (49) Integrating (48) over the interval $$[0, T]$$ and using (49), we obtain \begin{equation*} \begin{array}{l@{\;}l@{\;}l} \displaystyle\int^T_0 |\langle P_{\omega}BS(s)y_0, S(s)y_0\rangle |{\rm{d}}s &\leq& \displaystyle 2\delta^2 T^\frac{3}{2}K\|P\|\|y_0\|^2 \left(\displaystyle\int_{0}^{T}|\langle P_{\omega}By(s), y(s)\rangle|^2{\rm{d}}s\right)^{\frac{1}{2}}\\\\ &&{} +\sqrt{T}\|y_0\|\left(\displaystyle\displaystyle\int^T_0 |\langle P_{\omega}By(s), y(s)\rangle|^2{\rm{d}}s\right)^{\frac{1}{2}}\!. \end{array} \end{equation*} Replacing $$y_0$$ by $$y(t)$$, we get \begin{equation}\label{osineq} \begin{array}{l@{\;}l@{\;}l} \displaystyle\int^T_0 |\langle P_{\omega}BS(s)y(t), S(s)y(t)\rangle |{\rm{d}}s & \leq& \displaystyle 2\delta^2 T^\frac{3}{2}K\|P\|\|y_0\| \left(\displaystyle\int_{t}^{t+T}|\langle P_{\omega}By(s), y(s)\rangle|^2{\rm{d}}s\right)^{\frac{1}{2}}\\\\ &&{} +\sqrt{T}\|y_0\|\left(\displaystyle\displaystyle\int^{t+T}_t |\langle P_{\omega}By(s), y(s)\rangle|^2{\rm{d}}s\right)^{\frac{1}{2}}\!. \end{array} \end{equation} (50) From (42) and (50), we deduce that \begin{equation}\label{oss7} \int^T_0 |\langle P_{\omega}BS(s)y(t), S(s)y(t)\rangle |{\rm{d}}s \longrightarrow 0,\; as\; t \longrightarrow +\infty. \end{equation} (51) From (47) and (51), we deduce that $$\|\chi_{\omega}y(t)\|_{L^2(\omega)} \longrightarrow 0, \: as\; t \longrightarrow +\infty$$, which completes the proof. □ Proposition 3.3 Assume that $$B$$ is locally Lipschitz such that \begin{equation}\label{p24} \displaystyle \int^{T}_{0} |\langle P_{\omega}BS(t)y, S(t)y\rangle|{\rm{d}}t \geq \delta \|\chi_{\omega}y\|_{L^2(\omega)}, \; (for \; some\; T, \delta >0) \end{equation} (52) and there exists $$\mu >0$$ such that \begin{equation}\label{ineq} \langle P_{\omega}By, y\rangle \leq \mu \langle i_{\omega}By, y\rangle \end{equation} (53) then control (40) is the unique solution of (37) and regionally exponentially stabilizes system (1) on $$\omega$$. Proof. Let us define the function $$F(y)=\langle P_{\omega}y, y\rangle$$, $$\forall y\in L^2({\it{\Omega}})$$. For all $$y_0\in\mathcal{D}(A)$$ and $$t\geq 0$$, using (38), we obtain \begin{equation*} \frac{{\rm{d}}F(y(t))}{{\rm{d}}t}= - 2\frac{\langle P_{\omega}By(t), y(t)\rangle^2}{\|y(t)\|^2} - \langle i_{\omega}Ry(t), y(t)\rangle. \end{equation*} Integrating this relation, we get \begin{equation}\label{sp26} \int^{t}_0\frac{\langle P_{\omega}By(s), y(s)\rangle^2}{\|y(s)\|^2}ds\leq \frac{1}{2}F(y_0), \; t\geq 0. \end{equation} (54) By density, we deduce that $$J(u^*)$$ is finite for all $$y_0\in L^2({\it{\Omega}})$$. For $$y_0\in L^2({\it{\Omega}})$$ and $$t\geq 0$$, we have \begin{equation*} \begin{array}{l@{\;}l@{\;}l} \langle P_{\omega}BS(t)y_0, S(t)y_0\rangle &=& \langle P_{\omega}BS(t)y_0 - P_{\omega}By(t), S(t)y_0\rangle - \langle P_{\omega}By(t), y(t)- S(t)y_0\rangle\\\\ &&+ \langle P_{\omega}By(t), y(t)\rangle. \end{array} \end{equation*} Since $$\chi_{\omega}$$ is continuous, there exists $$ \alpha > 0$$ such that \begin{equation}\label{xx13} \begin{array}{lll} |\langle P_{\omega}BS(t)y_0, S(t)y_0\rangle|\leq 2\alpha K\|P\|\|y(t) - S(t)y_0\|\|y_0\| + |\langle P_{\omega}By(t), y(t)\rangle|. \end{array} \end{equation} (55) Moreover, we have \begin{equation*}\label{xx14} \|y(t) - S(t)y_0\|\leq K\int_{0}^{T}\frac{|\langle P_{\omega}By(s), y(s)\rangle|}{\|y(s)\|}{\rm{d}}s. \end{equation*} Schwarz’s inequality yields \begin{equation}\label{xx15} \|y(t)-S(t)y_0\|\leq K\Big\{ T\int_{0}^{T}\frac{|\langle P_{\omega}By(s), y(s)\rangle|^2}{\|y(s)\|^2}{\rm{d}}s\Big\}^{\frac{1}{2}},\; \forall t\in [0,T]. \end{equation} (56) Using (16), we get \begin{equation*}\label{xx16} |\langle P_{\omega}By(t), y(t)\rangle|\leq \frac{|\langle P_{\omega}By(t), y(t)\rangle|}{\|y(t)\|}\|y_0\|,\; \forall t\in [0,T]. \end{equation*} Integrating (55) over the interval $$[0, T]$$ and taking into account (56), we obtain \begin{equation}\label{xx17} \displaystyle\int_{0}^{T}|\langle P_{\omega}BS(s)y_0, S(s)y_0\rangle|ds \leq (2\alpha K^2T\|P\|+1)\sqrt{T}\|y_0\|\sqrt{f(0)}, \end{equation} (57) where $$f(t)=\int_{t}^{t+T}\frac{|\langle P_{\omega}By(s), y(s)\rangle|^2}{\|y(s)\|^2}{\rm{d}}s$$. Replacing $$y_0$$ by $$y(t)$$ in (57), we get \begin{equation*}\label{xx18} \displaystyle\int_{0}^{T}|\langle P_{\omega}BS(s)y(t), S(s)y(t)\rangle|ds \leq (2\alpha\|P\|K^2T+1)\sqrt{T}\|y(t)\|\sqrt{f(t)}. \end{equation*} Using (52) and (16), we obtain $$\delta\|\chi_{\omega}y(t)\|_{L^2(\omega)}\leq \beta\sqrt{f(t)}$$ with $$\beta=(2\alpha K^2T\|P\|+1)\sqrt{T}\|y_0\|$$. Then \begin{equation}\label{xx19} \delta^2\|\chi_{\omega}y(t)\|^2_{L^2(\omega)}\leq \beta^2 \sqrt{f(t)}. \end{equation} (58) Using (53) and (11), we get $$\frac{d}{dt}\|\chi_{\omega}y(t)\|^2_{L^2(\omega)} \leq -\frac{2}{\mu^2}\frac{|\langle P_{\omega}By(t), y(t)\rangle|^2}{\|y(t)\|^2}.$$ Integrating this inequality from $$mT$$ to $$(m+1)T$$, $$(m\in\mathbb{N})$$, we obtain \begin{equation*} \|\chi_{\omega}y(mT)\|^2_{L^2(\omega)} - \|\chi_{\omega}y((m+1)T)\|^2_{L^2(\omega)} \geq \frac{2}{\mu^2} \int_{mT}^{(m+1)T}\frac{|\langle P_{\omega}By(s), y(s)\rangle|^2}{\|y(s)\|^2}{\rm{d}}s. \end{equation*} Using (58) and since $$\|\chi_{\omega}y(t)\|_{L^2(\omega)}$$ decreases, we deduce \begin{equation*} \left(1 + 2\left(\frac{\delta}{\mu\beta}\right)^2\right)\|\chi_{\omega}y((m+1)T)\|^2_{L^2(\omega)} \leq \|\chi_{\omega}y(mT)\|^2_{L^2(\omega)}. \end{equation*} Then \begin{equation*} \|\chi_{\omega}y((m+1)T)\|_{L^2(\omega)} \leq \sigma\|\chi_{\omega}y(mT)\|_{L^2(\omega)}, \end{equation*} where $$\sigma=\frac{1}{(1 + 2(\frac{\delta}{\mu\beta})^2)^{\frac{1}{2}}}$$. This implies that \begin{equation*} \|\chi_{\omega}y(mT)\|_{L^2(\omega)} \leq \sigma^m\|\chi_{\omega}y_0\|_{L^2(\omega)}. \end{equation*} Taking $$m=E(\frac{t}{T})$$ the integer part of $$\frac{t}{T}$$ and remarking that $$m\geq \frac{t}{T}-1$$, we obtain \begin{equation*} \|\chi_{\omega}y(t)\|_{L^2(\omega)} \leq \sigma^{\frac{t}{T}-1}\|\chi_{\omega}y_0\|_{L^2(\omega)}. \end{equation*} We deduce that \begin{equation}\label{pex} \|\chi_{\omega}y(t)\|_{L^2(\omega)} \leq Fe^{-\rho t}\|y_0\|, \; \forall t\geq 0, \end{equation} (59) where $$F = \alpha(1 + 2(\frac{\delta}{\mu\beta})^2)^{\frac{1}{2}}$$ and $$\rho = \frac{\ln(1 + 2(\frac{\delta}{\mu\beta})^2) }{2T}$$, then control (40) regionally exponentially stabilizes system (1) on $$\omega$$. It remains to show that (40) is the unique solution of (37). Remarking that $$F(y(t))\leq \alpha \|P\|\|\chi_{\omega}y(t)\|^2_{L^2(\omega)}$$, it follows from (59) that $$F(y(t))\longrightarrow 0$$ as $$t\rightarrow +\infty$$. Let $$y_0\in \mathcal{D}(A)$$, integrating the relation \begin{equation}\label{rel} \frac{{\rm{d}}F(y(t))}{{\rm{d}}t}= \|y(t)\|^2([\frac{\langle P_{\omega}By(t), y(t)\rangle}{\|y(t)\|^2}+u(t)]^2 - \frac{\langle P_{\omega}By(t), y(t)\rangle^2}{\|y(t)\|^4}- u^2(t))-\langle i_{\omega}Ry(t), y(t)\rangle \end{equation} (60) we have \begin{equation*} J(u) = F(y_0) + \int^{+\infty}_0\|y(s)\|^2[\frac{\langle P_{\omega}By(s), y(s)\rangle}{\|y(s)\|^2}+ u(s)]^2{\rm{d}}s \end{equation*} then $$J(u)\geq F(y_0)$$. Setting $$u=u^*$$, we obtain $$J(u^*)=F(y_0)$$. Let $$y_0\in L^2({\it{\Omega}})$$, and a sequence $$y_{0k}\subset \mathcal{D}(A)$$ such that $$y_{0k} \longrightarrow y_0$$ as $$ k\longrightarrow +\infty$$, we have \begin{equation*} J(u) = F(y_{0k}) + \int^{+\infty}_0\|y_k(s)\|^2[\frac{\langle P_{\omega}By_k(s), y_k(s)\rangle}{\|y_k(s)\|^2}+ u(s)]^2 {\rm{d}}s. \end{equation*} Thus $$J(u)\geq F(y_{0})=J(u^*)$$. Hence control (40) is the unique solution of the problem (37). □ In order to illustrate the previous results, we perform the following algorithm: Step 1: Initial data: initial condition $$y_0$$ and subregion $$\omega$$; Step 2: Solve equation (38) using Bartels-Stewart method given in Penzl (1998); Step 3: Apply the control given by (25) or (40); Step 4: Solve system (1) using Petrov–Galerkin method; 4. Simulation results Let $${\it{\Omega}} = ]0,1[$$, we consider the system \begin{equation}\label{expl1} \left\{\!\! \begin{array}{l@{\;}l@{\;}l} \displaystyle\frac{\partial y}{\partial t}(x, t) &=& 0.01{\it{\Delta}} y(x, t)+ u(t)\displaystyle\frac{y(x, t)}{1+\|y(x,t)\|}, \;\; {\it{\Omega}} \times ]0, +\infty[\\ y(0, t) &=& y(1, t) = 0, \hspace{4.3cm} ]0, +\infty[ \\ y(x, 0) &=& 2\pi \sin(2\pi x),\hspace{4.2cm} {\it{\Omega}} \end{array} \right. \end{equation} (61) where the state space is $$L^2({\it{\Omega}})$$. 1. Regional stabilizing control. Let $$\omega=]0, \frac{1}{2}[$$, we perform the above algorithm applying the control (25) to system (61), we obtain the following figures Figure 1 shows that system (61) is stabilized on the subregion $$\omega$$, with stabilization error equals to $$3.84\times 10^{-4}$$. Fig. 1. View largeDownload slide Evolution of the state. Fig. 1. View largeDownload slide Evolution of the state. Fig. 2. View largeDownload slide Evolution of the control. Fig. 2. View largeDownload slide Evolution of the control. For $$\omega=]0, 1[$$, we obtain Figure 3 shows that system (61) is stabilized on $${\it{\Omega}}$$, with stabilization error equals to $$9.83\times 10^{-4}$$. Fig. 3. View largeDownload slide Evolution of the state. Fig. 3. View largeDownload slide Evolution of the state. Fig. 4. View largeDownload slide Evolution of the control. Fig. 4. View largeDownload slide Evolution of the control. 2. Regional optimal stabilizing control. Consider system (61) and problem (37) with $$R=I$$ such that $$P_{\omega}$$ is the unique solution of the equation (38). Let $$\omega = ]0, \frac{1}{2}[$$, we perform the above algorithm applying the control (40) to system (61), we obtain the figures below Figure 5 shows how system (61) is stabilized by the control (40) on $$\omega$$ with a stabilization error equals to $$2.1$$$$10^{-4}$$ and the cost $$J(u^*)= 4.9\times 10^{-2}$$. Fig. 5. View largeDownload slide Evolution of the state. Fig. 5. View largeDownload slide Evolution of the state. Fig. 6. View largeDownload slide Evolution of the control. Fig. 6. View largeDownload slide Evolution of the control. For $$\omega={\it{\Omega}}$$, Figure 7 shows that system (61) is stabilized on $${\it{\Omega}}$$, with stabilization error equals to $$8.23\times 10^{-4}$$ and the cost $$J(u^*)= 9.75\times 10^{-2}$$. Fig. 7. View largeDownload slide Evolution of the state. Fig. 7. View largeDownload slide Evolution of the state. Fig. 8. View largeDownload slide Evolution of the control. Fig. 8. View largeDownload slide Evolution of the control. Table 1 shows that there exists a relation between the area of subregion $$\omega$$, the cost and the error of stabilization. Moreover, we remark that more the area of $$\omega$$ increases more the stabilization cost increases. $$\omega$$ $$]0,0.2[$$ $$]0,0.4[$$ $$]0,0.6[$$ $$]0,0.8[$$ $$]0,1[$$ Error 1.33 $$\times 10^{-4}$$ 1.73 $$\times 10^{-4}$$ 3.95$$\times 10^{-4}$$ 6.526 $$\times 10^{-4}$$ 8.23$$\times 10^{-4}$$ Cost 2.38 $$\times 10^{-2}$$ 3.84 $$\times 10^{-2}$$ 5.63 $$\times 10^{-2}$$ 7.56 $$\times 10^{-2}$$ 9.75 $$\times 10^{-2}$$ $$\omega$$ $$]0,0.2[$$ $$]0,0.4[$$ $$]0,0.6[$$ $$]0,0.8[$$ $$]0,1[$$ Error 1.33 $$\times 10^{-4}$$ 1.73 $$\times 10^{-4}$$ 3.95$$\times 10^{-4}$$ 6.526 $$\times 10^{-4}$$ 8.23$$\times 10^{-4}$$ Cost 2.38 $$\times 10^{-2}$$ 3.84 $$\times 10^{-2}$$ 5.63 $$\times 10^{-2}$$ 7.56 $$\times 10^{-2}$$ 9.75 $$\times 10^{-2}$$ $$\omega$$ $$]0,0.2[$$ $$]0,0.4[$$ $$]0,0.6[$$ $$]0,0.8[$$ $$]0,1[$$ Error 1.33 $$\times 10^{-4}$$ 1.73 $$\times 10^{-4}$$ 3.95$$\times 10^{-4}$$ 6.526 $$\times 10^{-4}$$ 8.23$$\times 10^{-4}$$ Cost 2.38 $$\times 10^{-2}$$ 3.84 $$\times 10^{-2}$$ 5.63 $$\times 10^{-2}$$ 7.56 $$\times 10^{-2}$$ 9.75 $$\times 10^{-2}$$ $$\omega$$ $$]0,0.2[$$ $$]0,0.4[$$ $$]0,0.6[$$ $$]0,0.8[$$ $$]0,1[$$ Error 1.33 $$\times 10^{-4}$$ 1.73 $$\times 10^{-4}$$ 3.95$$\times 10^{-4}$$ 6.526 $$\times 10^{-4}$$ 8.23$$\times 10^{-4}$$ Cost 2.38 $$\times 10^{-2}$$ 3.84 $$\times 10^{-2}$$ 5.63 $$\times 10^{-2}$$ 7.56 $$\times 10^{-2}$$ 9.75 $$\times 10^{-2}$$ 5. Conclusion In this article, sufficient conditions for regional exponential, strong and weak stabilization were given. Moreover, we gave a control which ensures regional stabilization minimizing an appropriate cost function. The obtained results are successfully illustrated by simulations. This work motivates new problems, this is the case where $$\omega$$ is a part of the boundary of the system evolution domain which is of a great interest. References Ball, J. M. & Slemrod, M. ( 1979 ) Feedback stabilization of distributed semilinear control systems. Appl. Math. Optim. , 5 , 169 – 179 . Google Scholar CrossRef Search ADS Bounit, H. & Hammouri, H. ( 1999 ) Feedback stabilization for a class of distributed semilinear control systems. Nonlinear Anal. , 37 , 953 – 969 . Google Scholar CrossRef Search ADS Mohler, R. R. ( 1973 ) Bilinear Control Processes: With Applications to Engineering, Ecology and Medicine . New York : Academic Press, Inc. Ouzahra, M. ( 2008 ) Strong stabilization with decay estimate of semilinear systems. Systems Control Lett. , 57 , 813 – 815 . Google Scholar CrossRef Search ADS Ouzahra, M. ( 2011 ) Exponential stabilization of distributed semilinear systems by optimal control. J. Math. Anal. Appl. , 380 , 117 – 123 . Google Scholar CrossRef Search ADS Pazy, A. ( 1983 ) Semigroups of Linear Operations to Partial Differential Equations . New York : Springer . Google Scholar CrossRef Search ADS Penzl, T. ( 1998 ) Numerical solution of generalized Lyapunov equations. Adv. Comp. Math. , 8 , 33 – 48 . Google Scholar CrossRef Search ADS Zerrik, E. & Ezzaki, L. ( 2016 ) Regional gradient stabilization of semilinear distributed systems. J. Dyn. Control Syst. , 23 , 405 – 420 . Google Scholar CrossRef Search ADS Zerrik, E. & Ezzaki, L. ( 2017 ) Output stabilization of distributed bilinear systems. Control Theory Technol. Zerrik, E. & Ouzahra, M. ( 2003 ) Regional stabilization for infinite-dimensional systems. Int. J. Cont. , 76 , 73 – 81 . Google Scholar CrossRef Search ADS Zerrik, E. & Ouzahra, M. ( 2005 ) Output stabilization for infinite bilinear systems. Int. J. Appl. Math. Comput. Sci. , 15 , 187 – 196 . Zerrik, E. & Ouzahra, M. ( 2007 ) Output stabilisation for distributed semilinear systems. IET Control Theory Appl. , 1 , 838 – 843 . Google Scholar CrossRef Search ADS © The authors 2017. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png IMA Journal of Mathematical Control and Information Oxford University Press

An output stabilization of infinite dimensional semilinear systems

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Oxford University Press
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© The authors 2017. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.
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0265-0754
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1471-6887
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10.1093/imamci/dnx038
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Abstract

Abstract This article considers stabilization of infinite dimensional semilinear systems on a subregion $$\omega$$ of the evolution domain $${\it{\Omega}}$$. Under sufficient conditions, we give controls which ensure exponential, strong and weak stabilization of such systems on $$\omega$$. Also, we determinate a control which allows stabilization on $$\omega$$ minimizing a given functional cost. The developed results are illustrated by many examples and simulations. 1. Introduction Semilinear systems is an important subclass of nonlinear systems and numerous real-world problems have a semilinear structure. They include applications in nuclear, thermal, chemical, social processes, etc., (Mohler, 1973). The stabilization of infinite dimensional semilinear systems has been the subject of many studies using different approaches. More precisely, on $${\it{\Omega}}\subset\mathbb{R}^n$$ an open bounded domain with regular boundary $$\partial {\it{\Omega}}$$, we consider the system: \begin{equation}\label{1} \left\{\!\!\!\begin{array}{r@{\;}c@{\;}l} \displaystyle\frac{{\rm{d}}y(t)}{{\rm{d}}t} &=& Ay(t)+u(t)By(t)\\[12pt] y(0) &=& y_0, \end{array} \right. \end{equation} (1) where the operator $$A$$ generates a semigroup of contractions $$(S(t))_{t\geq 0}$$ on the Hilbert space $$L^2({\it{\Omega}})$$ with inner product $$\langle ., . \rangle$$ and norm $$\|.\|$$, $$u(t)\in L^2(0, +\infty)$$ is a scalar valued control and $$B$$ is a nonlinear operator mapping $$L^2({\it{\Omega}})$$ into itself such that $$B(0)=0$$. The problem of stabilizing system (1) on the whole domain $${\it{\Omega}}$$ has been considered in many works: In Ball & Slemrod (1979), it was shown that the control \begin{equation}\label{gkey2} u(t)=-\langle By(t), y(t)\rangle \end{equation} (2) weakly stabilizes system (1) provided that B is sequentially weakly continuous (i.e. $$B\varphi_k\rightharpoonup B\varphi$$ if $$\varphi_k \rightharpoonup \varphi$$) and the following condition holds \begin{equation}\label{key1} \langle BS(t)z, S(t)z\rangle = 0, \; \forall t\geq 0 \Longrightarrow z = 0. \end{equation} (3) Also, the control (2) allows the strong stabilization of system (1) under the condition: \begin{equation}\label{assum} \displaystyle \int^{T}_{0} |\langle BS(s)y,S(s)y\rangle|{\rm{d}}s \geq \mu \|y\|^2, \; (for \; some\; T, \mu >0).\; (see\; \text{Ouzahra (2008)}) \end{equation} (4) Moreover, in Bounit & Hammouri (1999), one showed that under specific conditions, the control \begin{equation} u(t) = \displaystyle-\frac{\langle By(t), y(t)\rangle }{1 + |\langle By(t), y(t)\rangle|} \end{equation} (5) realizes the strong stabilization of system (1). On the other hand, in Ouzahra (2011), author showed that the control \begin{equation}\label{key3} u(t) = \left\{ \begin{array}{lcl} \displaystyle-\frac{\langle By(t), y(t)\rangle }{\| y(t)\|^2},\; y(t)\neq 0 \\[6pt] 0, \hspace{2cm } ~~~~~~~y(t)=0 \end{array} \right. \end{equation} (6) stabilizes exponentially system (1) when (4) holds. For $$\omega\subset{\it{\Omega}}$$ an open subregion and of non-null measure, regional stabilization consists in choosing a control that stabilizes system (1) only on $$\omega$$ and we say that system (1) is regionally weakly (respectively strongly, exponentially) stabilizable on $$\omega$$, if for any initial condition $$ y_0\in L^2({\it{\Omega}})$$ the corresponding solution $$y(t)$$ of (1) is unique and $$\chi_{\omega}y(t)$$ converges to $$0$$ weakly (respectively strongly, exponentially) as $$t \longrightarrow +\infty$$ with $$\chi_{\omega}$$ is the restriction operator to $$\omega$$ (Zerrik & Ouzahra, 2003). This notion makes sense for the usual concept of stabilization taking into account the spatial variable and then it becomes closer to real-world problems, where one wishes to stabilize system (1) only on a critical subregion $$\omega$$ of $${\it{\Omega}}$$. The question of stabilizing regionally a linear or bilinear distributed system on a subregion $$\omega\subset{\it{\Omega}}$$ was introduced and developed by Zerrik and co-workers using different standard stabilization techniques. Then the relationship between regional exponential stabilization and the existence of an appropriate solution of the Riccati equation was examined (Zerrik & Ouzahra, 2003); also the problem was considered by means of a decomposition of the state-space using spectral properties of the dynamic of the considered system (Zerrik & Ouzahra, 2005). In Zerrik & Ezzaki (2017), we gave sufficient conditions for regional exponential, strong and weak stabilization for infinite bilinear systems. Furthermore, the properties and characterizations of a class of controls which ensure regional strong and weak stabilization of the gradient of system (1) with various illustrating examples were given in Zerrik & Ezzaki (2016). In Zerrik & Ouzahra (2007), it was shown that if the assumption \begin{equation}\label{assump} \langle KS(t)z, S(t)z\rangle\langle BS(t)z, S(t)z\rangle = 0, \; \forall t\geq 0 \Longrightarrow \chi_{\omega}z = 0 \end{equation} (7) holds, where $$K$$ is a continuous operator mapping $$L^2({\it{\Omega}})$$ into itself, then, the control \begin{equation}\label{cont1} u(t)=-\langle Ky(t), y(t)\rangle \end{equation} (8) regionally weakly stabilizes system (1) on $$\omega$$. Moreover, the control \begin{equation*} u(t)=-\langle i_{\omega}Pi_{\omega}By(t), y(t)\rangle \end{equation*} regionally weakly stabilizes system (1) on $$\omega$$ and is the unique solution of the minimization problem \begin{equation}\label{pr} \left\{\!\!\! \begin{array}{r@{\;}c@{\;}l} \min q(u)&=& \displaystyle\int^{+\infty}_0\langle i_{\omega} Ry(t), y(t)\rangle{\rm{d}}t + \displaystyle\int^{+\infty}_0|\langle By(t), i_{\omega}Pi_{\omega}y(t)\rangle|^2{\rm{d}}t + \displaystyle\int^{+\infty}_0|u(t)|^2{\rm{d}}t\\[6pt] v\in \mathcal{U}_{ad} &=& \{u|y(t)\; \text{is a global solution and}\; J(u)<+\infty\}, \end{array} \right. \end{equation} (9) where $$i_{\omega} = \chi_{\omega}^*\chi_{\omega}$$ with $$\chi_{\omega}^*$$ is the adjoint operator, $$R\in \mathcal{L}(L^2({\it{\Omega}}))$$ is a positive operator and $$P$$ is a positive operator verifying the following equation \begin{equation*} \langle Ay, i_{\omega}Pi_{\omega}y\rangle + \langle i_{\omega}Pi_{\omega}y, Ay\rangle + \langle y, i_{\omega} Ry\rangle=0, \; y\in \mathcal{D}(A). \end{equation*} In this article, we study regional exponential and strong stabilization of system (1). Furthermore, we study regional weak stabilization using a different control than (8). Moreover, we give a control that stabilizes regionally system (1) on $$\omega$$ minimizing a functional cost. The plan of the article is as follows: Second section is devoted to regional exponential, strong and weak stabilization of system (1), with some illustrating examples. In the third section, we establish a control that stabilizes regionally system (1) minimizing a functional cost. Finally, the developed results are illustrated by simulations. 2. Regional stabilization Here we deal with regional exponential, strong and weak stabilization of system (1). $$\chi_{\omega} : L^2({\it{\Omega}}) \longrightarrow L^2(\omega)$$ the restriction operator to $$\omega$$. Let us denote $$i_{\omega} = \chi_{\omega}^*\chi_{\omega}$$ with $$\chi_{\omega}^*$$ is the adjoint operator. Assume that \begin{equation}\label{pos} \langle i_{\omega} By, y\rangle \langle By, y\rangle \geq 0,\quad \forall y\in L^2({\it{\Omega}}) \end{equation} (10) and \begin{equation}\label{diss} \langle i_{\omega} Ay, y\rangle \leq 0,\quad \forall y\in \mathcal{D}(A). \end{equation} (11) The following result gives sufficient conditions for regional exponential stabilization of system (1). Theorem 2.1 Suppose that $$B$$ is locally Lipschitz such that \begin{equation}\label{sassum3} \displaystyle \int^{T}_{0} |\langle i_{\omega}BS(s)y, S(s)y\rangle|{\rm{d}}s \geq \alpha \|\chi_{\omega}y\|_{L^2(\omega)}, \; (for \; some\; T, \alpha >0) \end{equation} (12) then the control \begin{equation}\label{sskey21} u(t)=\left\{ \begin{array}{lll} -\displaystyle\frac{\langle i_{\omega}By(t), y(t)\rangle }{\|y(t)\|^2},\; y(t)\neq 0 \\[6pt] 0, \quad\!\!\hspace{2.8cm}y(t)=0 \end{array} \right. \end{equation} (13) regionally exponentially stabilizes system (1) on $$\omega$$. Proof. First, let us show that the map $$h : y\mapsto \frac{\langle i_{\omega}By, y\rangle}{\|y\|^2}By$$ is locally Lipschitz. Since $$B$$ is locally Lipshitz, for each $$R> 0$$ there exists a constant $$K>0$$ such that \begin{equation}\label{sblip} \| By-Bz \| \leq K\| y-z\|,\; \forall y, z \in L^2({\it{\Omega}}) : 0<\| y\| \leq \| z\| \leq R. \end{equation} (14) Thus $$\begin{array}{lll} h(z)Bz - h(y)By = \displaystyle\frac{\|y\|^2(\langle i_{\omega}Bz, z\rangle Bz - \langle i_{\omega}By, y\rangle By)}{\|z\|^2\|y\|^2} -\displaystyle\frac{(\|z\|^2-\|y\|^2)\langle i_{\omega}By, y\rangle By}{\|z\|^2\|y\|^2}. \end{array}$$ Since $$\chi_{\omega}$$ is continuous, there exists $$\delta>0$$ such that $$\begin{array}{l@{\;}ll} \|h(z)Bz - h(y)By\| & \leq \displaystyle\frac{\|\langle i_{\omega}Bz, z\rangle Bz - \langle i_{\omega}By, y\rangle By\|}{\|z\|^2} +K^2\delta|\|z\|^2-\|y\|^2|.\displaystyle\frac{\|y\|}{\|z\|^2}\\\\ & \leq \displaystyle\frac{\|\langle i_{\omega}Bz, z\rangle (Bz-By) + (\langle i_{\omega}Bz, z-y \rangle }{\|z\|^2}\\\\ &\quad{} -\displaystyle\frac{\langle i_{\omega}By-i_{\omega}Bz, y\rangle)By\|}{\|z\|^2} +2K^2\delta\| z-y\|\\\\ &\leq K^2\delta\|z\|^2\|z-y\|+ K^2\delta\|y\|\|z-y\|(\|y\|+\|z\|) + 2K^2\delta\| z-y\|\\\\ &\leq 3K^2\delta\|z\|^2\|z-y\|+2K^2\delta\| z-y\|. \end{array}$$ We conclude that for all $$y, z \in L^2({\it{\Omega}})$$, we have $$ \|h(z)-h(y)\|\leq L\|z-y\| $$ where $$L=3K^2\delta R^2 + 2K^2\delta$$. It follows that system (1) has a unique global mild solution $$y(t)$$ defined on a maximal interval $$[0,t_{\max}[$$ (see Pazy (1983)) and given by the variation of constants formula \begin{equation}\label{sconst} y(t) = S(t)y_0 + \int_{0}^{t}S(t-s)u(s)By(s){\rm{d}}s. \end{equation} (15) Let us consider the nonlinear semigroup $$N(t)y_0:=y(t)$$, we have \begin{equation*} \frac{{\rm{d}}}{{\rm{d}}t}\| N(t)y_0\|^2 = 2\langle AN(t)y_0, N(t)y_0\rangle -2 \frac{\langle i_{\omega}BN(t)y_0, N(t)y_0\rangle}{1 + |\langle i_{\omega}BN(t)y_0, N(t)y_0\rangle|}\langle BN(t)y_0, N(t)y_0\rangle. \end{equation*} Since $$S(t)$$ is a semigroup of contractions, then \begin{equation*} \frac{{\rm{d}}}{{\rm{d}}t}\| N(t)y_0\|^2 \leq -2 \frac{\langle i_{\omega}BN(t)y_0, N(t)y_0\rangle}{1 + |\langle i_{\omega}BN(t)y_0, N(t)y_0\rangle|}\langle BN(t)y_0, N(t)y_0\rangle. \end{equation*} Integrating this inequality, we get \begin{equation*} \| N(t)y_0\|^2 \leq \| y_0\|^2 -2 \int_0^t \langle h(N(s)y_0), N(s)y_0\rangle ds, \; \forall t\in[0, t_{max}[. \end{equation*} Using (10), we obtain \begin{equation}\label{sbound1} \| N(t)y_0\| \leq \| y_0\|. \end{equation} (16) For all $$y_0\in L^2({\it{\Omega}})$$ and $$t\geq 0$$, we have \begin{equation*} \begin{array}{l@{\;}l@{\;}l} \langle i_{\omega}BS(t)y_0, S(t)y_0\rangle &=& \langle i_{\omega}BS(t)y_0 - i_{\omega}BN(t)y_0, S(t)y_0 \rangle \\\\ &&- \langle i_{\omega}BN(t)y_0, N(t)y_0- S(t)y_0\rangle\\\\ &&+ \langle i_{\omega}BN(t)y_0, N(t)y_0\rangle. \end{array} \end{equation*} Using (14) and the continuity of $$\chi_{\omega}$$, we obtain \begin{equation}\label{sx13} \begin{array}{l@{\;}l@{\;}l} |\langle i_{\omega}BS(t)y_0, S(t)y_0\rangle|&\leq& 2\delta K\|N(t)y_0 - S(t)y_0\|\|y_0\|\\\\ && + |\langle i_{\omega}BN(t)y_0, N(t)y_0\rangle|. \end{array} \end{equation} (17) Moreover, from (15) we have \begin{equation*}\label{sx14} \|N(t)y_0 - S(t)y_0\|\leq K\int_{0}^{t}\displaystyle\frac{|\langle i_{\omega}BN(s)y_0, N(s)y_0\rangle|}{\|N(s)y_0\|}{\rm{d}}s. \end{equation*} For a fixed $$T\in]0, t_{max}[$$, Schwarz’s inequality yields \begin{equation}\label{sx15} \|N(t)y_0-S(t)y_0\|\leq K\left\{ T\int_{0}^{T}\displaystyle\frac{|\langle i_{\omega}BN(s)y_0, N(s)y_0\rangle|^2}{\|N(s)y_0\|^2}{\rm{d}}s\right\}^{\frac{1}{2}},\; \forall t\in [0,T]. \end{equation} (18) Using (16), we get \begin{equation}\label{sx16} |\langle i_{\omega}BN(t)y_0, N(t)y_0\rangle|\leq \displaystyle\frac{|\langle i_{\omega}BN(t)y_0, N(t)y_0\rangle|}{\|N(t)y_0\|}\|y_0\|,\; \forall t\in [0,T]. \end{equation} (19) Integrating (17) over the interval $$[0, T]$$ and using (18) and (19), we obtain \begin{equation}\label{sx17} \begin{array}{l@{\;}l@{\;}l} \displaystyle\int_{0}^{T}|\langle i_{\omega}BS(s)y_0, S(s)y_0\rangle|{\rm{d}}s &\leq 2\delta K^2T^{\frac{3}{2}}\|y_0\| \left\{ \displaystyle\int_{0}^{T}\displaystyle\frac{|\langle i_{\omega}BN(s)y_0, N(s)y_0\rangle|^2}{\|N(s)y_0\|^2}{\rm{d}}s\right\}^{\frac{1}{2}}\\\\ &\quad{}+ \sqrt{T}\|y_0\|\left\{ \displaystyle\int_{0}^{T}\displaystyle\frac{|\langle i_{\omega}BN(s)y_0, N(s)y_0\rangle|^2}{\|N(s)y_0\|^2}{\rm{d}}s\right\}^{\frac{1}{2}}. \end{array} \end{equation} (20) Replacing $$y_0$$ by $$N(t)y_0$$ in (20), we get \begin{equation*}\label{sx18} \begin{array}{l@{\;}ll} \displaystyle\int_{0}^{T}|\langle i_{\omega}BS(s)N(t)y_0, S(s)N(t)y_0\rangle|{\rm{d}}s &\leq 2\delta K^2T^{\frac{3}{2}}\|N(t)y_0\|\\\\ &\quad\times\left\{ \displaystyle\int_{t}^{t+T}\displaystyle\frac{|\langle i_{\omega}BN(s)y_0, N(s)y_0\rangle|^2}{\|N(s)y_0\|^2}{\rm{d}}s\right\}^{\frac{1}{2}}\\\\ &\quad+ \sqrt{T}\|N(t)y_0\|\left\{ \displaystyle\int_{t}^{t+T}\displaystyle\frac{|\langle i_{\omega}BN(s)y_0, N(s)y_0\rangle|^2}{\|N(s)y_0\|^2}{\rm{d}}s\right\}^{\frac{1}{2}}. \end{array} \end{equation*} By (12) and (16), we obtain \begin{equation*} \alpha\|\chi_{\omega}N(t)y_0\|_{L^2(\omega)}\leq \beta\left\{ \int_{t}^{t+T}\displaystyle\frac{|\langle i_{\omega}BN(s)y_0, N(s)y_0\rangle|^2}{\|N(s)y_0\|^2}{\rm{d}}s\right\}^{\frac{1}{2}}, \end{equation*} where $$\beta=(2\delta K^2T + 1)\sqrt{T}\|y_0\|$$. Then \begin{equation}\label{sx19} \alpha^2\|\chi_{\omega}N(t)y_0\|^2_{L^2(\omega)}\leq \beta^2 \int_{t}^{t+T}\displaystyle\frac{|\langle i_{\omega}BN(s)y_0, N(s)y_0\rangle|^2}{\|N(s)y_0\|^2}{\rm{d}}s. \end{equation} (21) Using (11), we get \begin{equation}\label{sx20} \frac{{\rm{d}}}{{\rm{d}}t}\|\chi_{\omega}N(t)y_0\|^2_{L^2(\omega)}\leq -2\displaystyle\frac{|\langle i_{\omega}BN(t)y_0, N(t)y_0\rangle|^2}{\|N(t)y_0\|^2}. \end{equation} (22) Integrating (22) from $$mT$$ to $$(m+1)T$$, $$(m\in\mathbb{N})$$, we obtain \begin{equation*} \|\chi_{\omega}N(mT)y_0\|^2_{L^2(\omega)} - \|\chi_{\omega}N((m+1)T)y_0\|^2_{L^2(\omega)} \geq 2 \int_{mT}^{(m+1)T}\displaystyle\frac{|\langle i_{\omega}BN(s)y_0, N(s)y_0\rangle|^2}{\|N(s)y_0\|^2}{\rm{d}}s. \end{equation*} By (21) and (22), we deduce \begin{equation*} \left(1 + 2\left(\frac{\alpha}{\beta}\right)^2\right)\|\chi_{\omega}N((m+1)T)y_0\|^2_{L^2(\omega)} \leq \|\chi_{\omega}N(mT)y_0\|^2_{L^2(\omega)}. \end{equation*} Then $$\|\chi_{\omega}N((m+1)T)y_0\|_{L^2(\omega)} \leq \rho\|\chi_{\omega}N(mT)y_0\|_{L^2(\omega)}$$ where $$\rho=\frac{1}{(1 + 2(\frac{\alpha}{\beta})^2)^{\frac{1}{2}}}$$. By recurrence, we show that $$ \|\chi_{\omega}N(mT)y_0\|_{L^2(\omega)} \leq \rho^m\|\chi_{\omega}y_0\|_{L^2(\omega)}$$. Taking $$m=E(\frac{t}{T})$$ the integer part of $$\frac{t}{T}$$ and remarking that $$m\geq \frac{t}{T}-1$$, we get \begin{equation*} \|\chi_{\omega}N(t)y_0\|_{L^2(\omega)} \leq\rho^{\frac{t}{T}-1} \|\chi_{\omega}y_0\|_{L^2(\omega)}, \; \forall t\geq 0. \end{equation*} It follows that \begin{equation*} \|\chi_{\omega}N(t)y_0\|_{L^2(\omega)} \leq Me^{-\sigma t}\|y_0\|, \; \forall t\geq 0 \end{equation*} where $$M = \delta(1 + 2(\frac{\alpha}{\beta})^2)^{\frac{1}{2}}$$ and $$\sigma = \frac{\ln(1 + 2(\frac{\alpha}{\beta})^2) }{2T}$$, which shows the regional exponential stability of system (1) on $$\omega$$. □ Example 2.2 Let $${\it{\Omega}} = ]0,1[$$, we consider the wave equation defined by \begin{equation}\label{bexp23} \left\{\!\!\!\!\!\! \begin{array}{lll} &\displaystyle\frac{\partial^2 y}{\partial t^2}(x, t) = {\it{\Delta}} y(x, t) + u(t)\displaystyle\frac{\partial y}{\partial t}(x, t) \quad on \; {\it{\Omega}} \times ]0, +\infty[\\ & y(x, 0) = y_0, \; \displaystyle\frac{\partial y}{\partial t}(x, 0)=y_1 \hspace{2.8cm} on\; {\it{\Omega}}\\ & y(0,t) = y(1,t) = 0 \hspace{4.7cm} on\; [0, +\infty[. \end{array}\right. \end{equation} (23) This system has the form of (1) if we set $$A = \begin{pmatrix} 0 & I \\ {\it{\Delta}} & 0 \end{pmatrix}\; and\; B = \begin{pmatrix} 0 & 0 \\ 0 & I \end{pmatrix}\!. $$ Let $$\tilde{A}= -{\it{\Delta}}$$, with domain $$\mathcal{D}(\tilde{A})=H^2({\it{\Omega}})\cap H^1_0({\it{\Omega}})$$. Consider $$H = H^1_0({\it{\Omega}})\times L^2({\it{\Omega}})$$ the state space endowed with the inner product $$\langle (y_1, z_1), (y_2, z_2)\rangle = \langle \tilde{A}^\frac{1}{2}y_1, \tilde{A}^\frac{1}{2}y_2\rangle_{L^2({\it{\Omega}})} + \langle z_1, z_2\rangle_{L^2({\it{\Omega}})}$$. The eigenvalues of $$\tilde{A}$$ are $$\lambda_k = (k\pi)^2$$, corresponding to eigenfunctions $$\phi_k(x) = \sqrt{2}\sin(k\pi x),\; \forall k\,{\in}\,\mathbb{N}^*$$. We set $$y=(y_1, y_2)\in H$$ with $$y_1 = \sum_{k=1}^{\infty}a_k\phi_k$$ and $$y_2 = \sum_{k=1}^{\infty}\sqrt{\lambda_k}b_k\phi_k$$, where $$(a_k, b_k)\in \mathbb{R}^2$$$$\forall k\geq 1$$, and the semigroup is given by $$S(s)y = \sum_{k=1}^{\infty}\begin{pmatrix} a_k\cos(k\pi s) + b_k\sin(k\pi s)\\ b_k k\pi\cos(k\pi s) - a_k k\pi\sin(k\pi s) \end{pmatrix}\phi_k, \quad \forall s\geq 0. $$ For $$\omega = ]0,\frac{1}{2}[$$, we have \begin{equation*} \begin{array}{lll} \langle i_{\omega}BS(s)y, S(s)y\rangle = \displaystyle\sum_{k=1}^{\infty} \frac{(k\pi)^2}{2}\big(a_k^2\sin^2(k\pi s) - b_k a_k\sin(2k\pi s) + b_k^2\cos^2(k\pi s) \big) . \end{array} \end{equation*} Taking $$T=2$$, we get \begin{equation*} \begin{array}{lll} \displaystyle\int_{0}^{2} |\langle i_{\omega}BS(s)y, S(s)y\rangle|{\rm{d}}s = \displaystyle\sum_{k=1}^{\infty}\frac{(k\pi)^2}{2}(a_k^2 + b_k^2) \geq \alpha \|\chi_{\omega}y\|_{L^2(\omega)}, \end{array} \end{equation*} where $$\alpha = \frac{\pi}{2}\sqrt{\eta}$$, $$(\eta=\min\{(a_k^2+b_k^2)|k\in\mathbb{N}^*\})$$, thus (12) holds and the control \begin{equation*} u(t) = -\frac{\|\chi_{\omega}\partial_ty\|^2_{L^2(\omega)}}{\|y(t)\|^2_{H^1_0({\it{\Omega}})}+\|\partial_t y(t)\|^2} \end{equation*} regionally exponentially stabilizes system (23) on $$\omega$$. The next result gives sufficient conditions for regional strong stabilization. Theorem 2.3 Assume that $$B$$ is locally Lipschitz and satisfies \begin{equation}\label{ss3} \displaystyle \int^{T}_{0} |\langle i_{\omega}BS(s)y, S(s)y\rangle|{\rm{d}}s \geq \alpha \|\chi_{\omega}y\|^2_{L^2(\omega)}, \; ( T, \alpha >0) \end{equation} (24) then the control \begin{equation}\label{skey21} u(t)= -\frac{\langle i_{\omega}By(t), y(t)\rangle}{1 + |\langle i_{\omega}By(t), y(t)\rangle|} \end{equation} (25) regionally strongly stabilizes system (1) on $$\omega$$. Proof. By the same arguments using in the proof of Theorem 2.1, we can show that the map $$g : y\mapsto \frac{\langle i_{\omega}By, y\rangle}{1+|\langle i_{\omega}By, y\rangle|}By$$ is locally Lipschitz and we deduce that system (1) has a unique global mild solution $$y(t)$$ (see Pazy (1983)). It remains to show that $$\chi_{\omega} y(t) \longrightarrow 0\quad as\quad t\longrightarrow +\infty$$. For all $$y_0\in L^2({\it{\Omega}})$$, we have \begin{equation*} \begin{array}{lll} \langle i_{\omega}BS(t)y_0, S(t)y_0\rangle = \langle i_{\omega}BS(t)y_0 - i_{\omega}By(t), S(t)y_0\rangle + \langle i_{\omega}By(t), S(t)y_0\rangle. \end{array} \end{equation*} Using formula (15), we obtain \begin{equation*} \begin{array}{l@{\;}l@{\;}l} \langle i_{\omega}BS(t)y_0, S(t)y_0\rangle &=& \langle i_{\omega}BS(t)y_0 - i_{\omega}By(t), S(t)y_0\rangle - \langle i_{\omega}By(t), {\it{\Phi}}(t)\rangle\\\\ &&\quad{} + \langle i_{\omega}By(t), y(t)\rangle, \end{array} \end{equation*} where $${\it{\Phi}}(t)=\int_{0}^{t} S(t-s)u(s) By(s){\rm{d}}s$$. By (14) and the continuity of $$\chi_{\omega}$$, we deduce that \begin{equation}\label{sk4} \begin{array}{l@{\;}l@{\;}l} |\langle i_{\omega}BS(t)y_0, S(t)y_0\rangle| & \leq \delta K\| {\it{\Phi}}(t)\|(\| S(t)y_0\| + \| y(t)\|) +| \langle i_{\omega}By(t), y(t)\rangle |. \end{array} \end{equation} (26) Then \begin{equation*} |\langle i_{\omega}BS(t)y_0, S(t)y_0\rangle| \leq 2\delta K\| {\it{\Phi}} (t)\|\| y_0 \| + |\langle i_{\omega}By(t), y(t)\rangle|, \forall t\in [0, T]. \end{equation*} Moreover, Schwarz’s inequality gives \begin{equation}\label{sk5} \| {\it{\Phi}}(t)\| \leq K\|y_0\|\sqrt T\left(\int_{0}^{T}|\langle i_{\omega}By(s), y(s)\rangle|^2 {\rm{d}}s\right)^{\frac{1}{2}}\!. \end{equation} (27) Integrating (26) over the interval $$[0, T]$$ and using (27), we obtain \begin{equation*} \begin{array}{l@{\;}l@{\;}l} \displaystyle\int^T_0 |\langle i_{\omega}BS(s)y_0, S(s)y_0\rangle |{\rm{d}}s &\leq \displaystyle 2\delta T^\frac{3}{2}K\|y_0\|^2 \left(\displaystyle\int_{0}^{T}|\langle i_{\omega}By(s), y(s)\rangle|^2{\rm{d}}s\right)^{\frac{1}{2}}\\\\ &\quad +\sqrt{T}\|y_0\|\left(\displaystyle\displaystyle\int^T_0 |\langle i_{\omega}By(s), y(s)\rangle|^2{\rm{d}}s\right)^{\frac{1}{2}}\!. \end{array} \end{equation*} Replacing $$y_0$$ by $$y(t)$$, we get \begin{equation}\label{sineq} \begin{array}{l@{\;}ll} \displaystyle\int^T_0 |\langle i_{\omega}BS(s)y(t), S(s)y(t)\rangle |{\rm{d}}s & \leq \displaystyle 2\delta T^\frac{3}{2}K\|y_0\| \left(\displaystyle\int_{t}^{t+T}|\langle i_{\omega}By(s), y(s)\rangle|^2{\rm{d}}s\right)^{\frac{1}{2}}\\\\ &\quad +\sqrt{T}\|y_0\|\left(\displaystyle\displaystyle\int^{t+T}_t |\langle i_{\omega}By(s), y(s)\rangle|^2{\rm{d}}s\right)^{\frac{1}{2}}\!. \end{array} \end{equation} (28) Furthermore, using (11), we obtain \begin{equation*} \frac{d}{dt}\|\chi_{\omega}y(t)\|^2_{L^2(\omega)} \leq -2 \displaystyle\frac{|\langle i_{\omega}By(t), y(t)\rangle|^2}{1 + |\langle i_{\omega}By(t), y(t)\rangle|},\; \forall t\geq 0. \end{equation*} Integrating this inequality, we get \begin{equation*} \int_0^t |\langle i_{\omega}By(s), y(s)\rangle|^2 ds \leq \frac{L_{\|y_0\|}}{2}\|\chi_{\omega}y_0\|^2_{L^2(\omega)}, \; \forall t\geq 0 \end{equation*} with $$L_{\|y_0\|} = \sup_{\| y\| \leq \|y_0\|}(1 + |\langle i_{\omega}By, y\rangle|)$$. Hence \begin{equation}\label{sfini} \int_0^{+\infty} |\langle i_{\omega}By(s), y(s)\rangle|^2 ds < +\infty. \end{equation} (29) From (29) and (28), we deduce that \begin{equation}\label{ss7} \int^T_0 |\langle i_{\omega}BS(s)y(t), S(s)y(t)\rangle |{\rm{d}}s \longrightarrow 0,\; as\; t \longrightarrow +\infty. \end{equation} (30) It follows from (24) and (30) that $$\|\chi_{\omega}y(t)\|_{L^2(\omega)} \longrightarrow 0, \: as\; t \longrightarrow +\infty$$, which completes the proof. □ Example 2.4 On $${\it{\Omega}} = ]0,1[$$, we consider the following beam equation \begin{equation}\label{exp23} \left\{\!\! \begin{array}{l@{\;}l@{\;}l} \displaystyle\frac{\partial^2 y}{\partial t^2}(x, t) &=& -\displaystyle\frac{\partial^4 y}{\partial x^4}(x, t) + u(t)\displaystyle\frac{\partial y}{\partial t}(x, t) \quad on \; {\it{\Omega}} \times ]0, +\infty[\\[12pt] y(x, 0) &=& y_0,\; \displaystyle\frac{\partial y}{\partial t}(x, 0)=y_1 \hspace{3cm} on\; {\it{\Omega}}\\[12pt] y(\xi, t) &=& \displaystyle\frac{\partial^2 y}{\partial x^2}(\xi, t) = 0, \;\xi = 0, 1 \hspace{1.8cm} on\; ]0, +\infty[. \end{array}\right. \end{equation} (31) Let $$\tilde{A}= \frac{\partial^4 y}{\partial x^4}$$, with $$\mathcal{D}(\tilde{A}) = \{y\in L^2({\it{\Omega}})/\frac{\partial^4 y}{\partial x^4}\in L^2({\it{\Omega}}), y(\xi, t) = \frac{\partial^2 y}{\partial x^2}(\xi, t) = 0, \;\xi = 0, 1 \}$$. Setting $$H = (H^2({\it{\Omega}})\cap H^1_0({\it{\Omega}}))\times L^2({\it{\Omega}})$$ the state space endowed with the inner product $$\langle (y_1, z_1), (y_2, z_2)\rangle = \langle \tilde{A}^{\frac{1}{2}}y_1, \tilde{A}^{\frac{1}{2}}y_2\rangle_{L^2({\it{\Omega}})} + \langle z_1, z_2\rangle_{L^2({\it{\Omega}})}$$. The eigenvalues of $$\tilde{A}$$ are $$\lambda_j = (j\pi)^4$$, corresponding to eigenfunctions $$\varphi_j(x) = \sqrt{2}\sin((j\pi)^2 x),\; \forall j\,{\in}\,\mathbb{N}^*$$. System (31) has the form of (1) if we take $$A = \begin{pmatrix} 0 & I \\ -\tilde{A} & 0 \end{pmatrix}\; and\; B = \begin{pmatrix} 0 & 0 \\ 0 & I \end{pmatrix}. $$ $$B$$ satisfies condition (24), indeed, for $$y\in H$$, we have $y = \sum_{j=1}^{\infty}\begin{pmatrix} \alpha_j\\ \lambda_j^{\frac{1}{2}}\beta_j \end{pmatrix} \varphi_j$, where $$(\alpha_j, \beta_j)\in \mathbb{R}^2\; \forall j\geq 1$$ and the semigroup is given by $$S(s)y = \sum_{j=1}^{\infty}\begin{pmatrix} \alpha_j\cos(\lambda_j^{\frac{1}{2}} s) + \beta_j\sin(\lambda_j^{\frac{1}{2}}s)\\\\ \beta_j\lambda_j^{\frac{1}{2}}\cos(\lambda_j^{\frac{1}{2}} s) - \alpha_j\lambda_j^{\frac{1}{2}}\sin(\lambda_j^{\frac{1}{2}}s) \end{pmatrix}\varphi_j, \quad \forall s\geq 0. $$ For $$\omega = ]0,\frac{1}{2}[$$, we have \begin{equation*} \begin{array}{l@{\;}l@{\;}l} \langle i_{\omega}BS(s)y, S(s)y\rangle &=& \displaystyle\sum_{j=1}^{\infty} \frac{\lambda_j}{2}\big\{\alpha_j\sin(\lambda_j^{\frac{1}{2}} s) - \beta_j\cos(\lambda_j^{\frac{1}{2}}s)\big\}^2\\\\ &=& \displaystyle\sum_{j=1}^{\infty} \frac{\lambda_j}{2}\big\{\alpha_j^2\sin^2(j\pi s) + \beta_j^2\cos^2(j\pi s) - \alpha_j\beta_j\sin(2j\pi s)\big\} . \end{array} \end{equation*} Integrating this relation over the time interval $$[0, 2]$$, we obtain \begin{equation*} \begin{array}{lll} \displaystyle\int_{0}^{2} |\langle i_{\omega}BS(s)y, S(s)y\rangle|{\rm{d}}s = \displaystyle\sum_{j=1}^{\infty}\frac{\lambda_j}{2}(\alpha_j^2 + \beta_j^2) = \|\chi_{\omega}y\|^2 \end{array} \end{equation*} then (24) holds. We conclude that the control \begin{equation*} u(t) = -\frac{\| \chi_{\omega} \partial_t y(., t)\|^2_{L^2(\omega)}}{1+\| \chi_{\omega} \partial_t y(., t)\|^2_{L^2(\omega)}} \end{equation*} regionally strongly stabilizes system (31) on $$\omega$$. The following result gives sufficient conditions for regional weak stabilization. Theorem 2.5 Let $$B$$ be locally Lipschitz and weakly sequentially continuous such that \begin{equation}\label{s12} \langle i_{\omega}BS(t)z, S(t)z\rangle =0, \; \forall t\geq 0 \Longrightarrow \chi_{\omega} z=0 \end{equation} (32) then control (25) regionally weakly stabilizes system (1) on $$\omega$$. Proof. Consider the nonlinear semigroup $$N(t)y_0=y(t)$$ and let $$(t_k)$$ be a sequence of real numbers such that $$t_k\longrightarrow +\infty$$ as $$k\longrightarrow +\infty$$. From (16), $$N(t_ {k})y_0$$ is bounded in $$L^2({\it{\Omega}})$$, then there exists a subsequence $$(t_ {\phi(k)})$$ of $$(t_k)$$ and $$z\in L^2({\it{\Omega}})$$ such that $$N(t_ {\phi(k)})y_0 \rightharpoonup z\quad as\quad k\longrightarrow +\infty$$. For all $$y_0\in L^2({\it{\Omega}})$$ and $$t\geq 0$$, we have \begin{equation*} \begin{array}{l@{\;}l@{\;}l} \langle i_{\omega}BS(t)y_0, S(t)y_0\rangle &=& \langle i_{\omega}BN(t)y_0, N(t)y_0\rangle - \langle i_{\omega}BN(t)y_0, N(t)y_0 - S(t)y_0 \rangle \\\\ &&{}+ \langle i_{\omega}BS(t)y_0 - i_{\omega}BN(t)y_0, S(t)y_0\rangle. \\\\ \end{array} \end{equation*} Using (14) and the continuity of $$\chi_{\omega}$$, we get \begin{equation}\label{s13} \begin{array}{l@{\;}l@{\;}l} |\langle i_{\omega}BS(t)y_0, S(t)y_0\rangle|&\leq& |\langle i_{\omega}BN(t)y_0, N(t)y_0\rangle|\\\\ &&{} + 2\delta K\|N(t)y_0 - S(t)y_0\|\|y_0\|,\; (\text{for some}\; \delta > 0). \end{array} \end{equation} (33) From (15) and Schwarz’s inequality, we have \begin{equation}\label{s15} \|N(t)y_0-S(t)y_0\|\leq K\|y_0\|\sqrt{Tf(0)},\quad \forall t\in[0,T] \end{equation} (34) where $$f(t) = \int_{t}^{ t + T}|\langle i_{\omega}BN(s)y_0, N(s)y_0\rangle|^2{\rm{d}}s. $$ Integrating (33) over the interval $$[0, T]$$ and using (34), we obtain \begin{equation}\label{s17} \int_{0}^{T}|\langle i_{\omega}BS(s)y_0, S(s)y_0\rangle|{\rm{d}}s\leq \beta \|y_0\|\sqrt{f(0)} \end{equation} (35) with $$\beta = (2\delta K^2T\|y_0\|+1)\sqrt{T}$$. Replacing $$y_0$$ by $$N(t_ {\phi(k)})y_0$$ in (35), we get \begin{equation*}\label{s18} \int_{0}^{T}|\langle i_{\omega}BS(s)N(t_ {\phi(k)})y_0, S(s)N(t_ {\phi(k)})y_0\rangle|{\rm{d}}s\leq \beta\|y_0\|\sqrt{f(t_ {\phi(k)})}. \end{equation*} It follows that \begin{equation*} \int_{0}^{T}|\langle i_{\omega}BS(s)N(t_ {\phi(k)})y_0, S(s)N(t_ {\phi(k)})y_0\rangle|{\rm{d}}s \longrightarrow 0 \quad as \quad k\longrightarrow 0. \end{equation*} Since $$B$$ is weakly sequentially continuous and using the dominated convergence theorem, we deduce that $$\langle i_{\omega}BS(s)z, S(s)z\rangle = 0,\quad \forall s\in[0,T]$$. From (32), it follows that $$\chi_{\omega}N(t_{\phi(k)})y_0 \rightharpoonup 0 \quad as\quad k\longrightarrow +\infty$$, and then $$\forall \phi\in L^2({\it{\Omega}})$$, $$\langle \chi_{\omega}N(t_k)y_0, \phi\rangle \longrightarrow 0$$ as $$k\longrightarrow +\infty$$, hence $$\chi_{\omega} N(t)y_0 \rightharpoonup 0\quad as\quad t\longrightarrow +\infty.$$ In other words $$\chi_{\omega} y(t)$$ converges weakly to $$0$$ as $$t\longrightarrow +\infty$$, and then system (1) is regionally weakly stabilizable on $$\omega$$. □ Example 2.6 On $${\it{\Omega}} = ]0,1[$$, we consider the following equation \begin{equation}\label{2exp23} \left\{\!\! \begin{array}{l@{\;}l@{\;}l} \displaystyle\frac{\partial^2 y}{\partial t^2}(x, t) &=& -\displaystyle\frac{\partial^4 y}{\partial x^4}(x, t) + u(t)\displaystyle\frac{\partial y}{\partial t}(x, t) \quad on \; {\it{\Omega}} \times ]0, +\infty[\\ y(x, 0) &=& y_0, \; \displaystyle\frac{\partial y}{\partial t}(x, 0)=y_1 \hspace{2.7cm} on\; {\it{\Omega}}\\ y(\xi, t) &=& \displaystyle\frac{\partial^2 y}{\partial x^2}(\xi, t) = 0, \;\xi = 0, 1 \hspace{1.7cm} on\; ]0, +\infty[ \end{array}\!.\right. \end{equation} (36) The state space is $$H = (H^2({\it{\Omega}})\cap H^1_0({\it{\Omega}}))\times L^2({\it{\Omega}})$$, this system has the form of (1) if we set $$A = \begin{pmatrix} 0 & I \\ -{\it{\Delta}}^2 & 0 \end{pmatrix}\; and\; B = \begin{pmatrix} 0 & 0 \\ 0 & I \end{pmatrix}\!. $$ For $$\omega = ]0,\frac{1}{2}[$$, by the same arguments using in Example 2.4, we show that \begin{equation*} \begin{array}{lll} \displaystyle\int_{0}^{2} |\langle i_{\omega}BS(s)y, S(s)y\rangle|{\rm{d}}s = \|\chi_{\omega}y\|^2 \end{array}. \end{equation*} It follows that \begin{equation*} \langle i_{\omega}BS(t)y, S(t)y\rangle =0 \; \Longrightarrow \chi_{\omega} y=0. \end{equation*} Then, condition (32) holds. We conclude that the control \begin{equation*} u(t) = -\frac{\| \chi_{\omega} \partial_t y(., t)\|^2_{L^2(\omega)}}{1+\| \chi_{\omega} \partial_t y(., t)\|^2_{L^2(\omega)}} \end{equation*} regionally weakly stabilizes system (36) on $$\omega$$. 3. Regional stabilization problem This section deals with regional stabilization of system (1) by considering the following minimization problem \begin{equation}\label{p23} \left\{\!\! \begin{array}{r@{\;}c@{\;}l} \min J(u)&=& \displaystyle\int^{+\infty}_0\frac{\langle P_{\omega}By(t), y(t)\rangle^2}{\|y(t)\|^2}{\rm{d}}t + \displaystyle\int^{+\infty}_0\langle i_{\omega} Ry(t), y(t)\rangle{\rm{d}}t + \displaystyle\int^{+\infty}_0\|y(t)\|^2|u(t)|^2{\rm{d}}t\\[6pt] u\in \mathcal{U}_{ad} &=& \{u\in L^2(0,+\infty)|y(t)\; \text{is}~ \text{a}~ \text{global}~ \text{solution}~ \text{and}~ J(u)<+\infty\}, \end{array} \right. \end{equation} (37) where $$B$$ is bounded, $$A$$ satisfies (11) and $$P_{\omega}=i_{\omega}Pi_{\omega}$$ with $$P\in \mathcal{L}(L^2({\it{\Omega}}))$$ is a positive and bounded operator satisfying the equation \begin{equation}\label{22} \langle P_{\omega} Ay, y\rangle + \langle y, P_{\omega}Ay\rangle + \langle i_{\omega}Ry, y\rangle = 0, \; y\in \mathcal{D}(A), \end{equation} (38) where $$R\in \mathcal{L}(L^2({\it{\Omega}}))$$ is a positive operator. Theorem 3.1 Suppose that $$B$$ is locally Lipschitz and $$P$$ is compact such that \begin{equation}\label{pp25} \langle P_{\omega}BS(t)z, S(t)z\rangle =0, \; \forall t\geq 0 \Longrightarrow \chi_{\omega} z=0 \end{equation} (39) then the control \begin{equation}\label{p27} u^*(t) = -\frac{\langle P_{\omega}By(t), y(t)\rangle}{\|y(t)\|^2} \end{equation} (40) is the unique solution of (37) and regionally weakly stabilizes system (1) on $$\omega$$. Proof. Let us define the function $$V(y)=\langle P_{\omega}y, y\rangle$$, $$ y\in L^2({\it{\Omega}})$$. Using (38), for all $$y_0\in\mathcal{D}(A)$$ and $$t\geq 0$$, we have \begin{equation}\label{deriv} \frac{{\rm{d}}V(y(t))}{{\rm{d}}t}= - 2\frac{\langle P_{\omega}By(t), y(t)\rangle^2}{\|y(t)\|^2} - \langle i_{\omega}Ry(t), y(t)\rangle. \end{equation} (41) Integrating this relation, we get \begin{equation}\label{p26} \int^{t}_0\frac{\langle P_{\omega}By(s), y(s)\rangle^2}{\|y(s)\|^2}{\rm{d}}s\leq \frac{1}{2}V(y_0), \; t\geq 0. \end{equation} (42) The solution $$y(.)$$ is continuous with respect to the initial condition $$y_0$$ (seePazy (1983)) and $$D(A)$$ is dense in $$L^2({\it{\Omega}})$$, then (42) holds for all $$y_0\in L^2({\it{\Omega}})$$ so $$J(u^*)$$ is finite for all $$y_0\in L^2({\it{\Omega}})$$. Let $$(t_k)$$ be a sequence of real numbers such that $$t_k\longrightarrow +\infty$$ as $$k\longrightarrow +\infty$$. By (16), $$y(t_k)$$ is bounded in $$L^2({\it{\Omega}})$$, then there exists a subsequence $$(t_ {\phi(k)})$$ of $$(t_k)$$ and $$z\in L^2({\it{\Omega}})$$ such that \begin{equation*} y(t_ {\phi(k)}) \rightharpoonup z\quad as\quad k\longrightarrow +\infty. \end{equation*} For all $$y_0\in L^2({\it{\Omega}})$$ and $$t\geq 0$$, we have \begin{equation*} \begin{array}{l@{\;}l@{\;}l} \langle P_{\omega}BS(t)y_0, S(t)y_0\rangle &=& \langle P_{\omega}BS(t)y_0 - P_{\omega}By(t), S(t)y_0\rangle \\\\ &&{}- \langle P_{\omega}By(t), y(t)- S(t)y_0\rangle + \langle P_{\omega}By(t), y(t)\rangle. \end{array} \end{equation*} Thus \begin{equation}\label{O13} \begin{array}{lll} |\langle P_{\omega}BS(t)y_0, S(t)y_0\rangle|\leq 2\delta^2\|P\|K\|y(t) - S(t)y_0\|\|y_0\| + |\langle P_{\omega}By(t), y(t)\rangle|. \end{array} \end{equation} (43) Moreover, we have \begin{equation*}\label{O14} \|y(t) - S(t)y_0\|\leq K\|y_0\|\int_{0}^{t}|\langle P_{\omega}By(s), y(s)\rangle|{\rm{d}}s. \end{equation*} Schwarz’s inequality yields \begin{equation}\label{O15} \|y(t)-S(t)y_0\|\leq K\sqrt{T\lambda(0)},\quad \forall t\in[0,T] \end{equation} (44) where $$\lambda(t) = \int_{t}^{ t + T}|\langle P_{\omega}By(s), y(s)\rangle|^2{\rm{d}}s$$. Integrating (43) over the interval $$[0, T]$$ and with (44), we obtain \begin{equation}\label{O17} \int_{0}^{T}|\langle P_{\omega}BS(s)y_0, S(s)y_0\rangle|{\rm{d}}s\leq (2\delta^2 K^2T\|P\|+1)\|y_0\|\sqrt{T\lambda(0)}. \end{equation} (45) Replacing $$y_0$$ by $$y(t_ {\phi(k)})$$ in (45), we get \begin{equation}\label{O18} \int_{0}^{T}|\langle P_{\omega}BS(s)y(t_ {\phi(k)}), S(s)y(t_ {\phi(k)})\rangle|{\rm{d}}s\leq (2\delta^2 K^2T\|P\|+1)\|y_0\|\sqrt{T\lambda(t_ {\phi(k)})}. \end{equation} (46) By (46), we get $$ \lim_{k\longrightarrow +\infty}\int_{0}^{T}\langle P_{\omega}BS(s)y(t_{\phi(k)}), S(s)y(t_{\phi(k)})\rangle{\rm{d}}s = 0.$$ Since $$P$$ is compact and $$S(s)$$ is continuous $$\forall s\geq 0$$, we have \begin{equation*} \lim\limits_{k\longrightarrow +\infty}\langle P_{\omega}BS(s)y(t_{\phi(k)}), S(s)y(t_{\phi(k)})\rangle = \langle P_{\omega}BS(s)z, S(s)z\rangle. \end{equation*} By dominated convergence theorem, we obtain \[\displaystyle\int_{0}^{T}|\langle P_{\omega}BS(s)z, S(s)z\rangle|{\rm{d}}s = 0\text{ and then }\langle P_{\omega}BS(s)z, S(s)z\rangle = 0,\quad \forall s\in[0,T].\] Using (39), we deduce that $$\chi_{\omega}y(t_{\phi(k)}) \rightharpoonup 0 \quad as\quad k\longrightarrow +\infty.$$ It follows that $$\chi_{\omega} y(t)$$ converges weakly to $$0$$ as $$t\longrightarrow +\infty$$, and system (1) is regionally weakly stabilizable on $$\omega$$. Now, let us prove that (40) is the unique solution of (37). Since $$P$$ is compact, it follows that $$V(y(t))\longrightarrow 0$$ as $$t\rightarrow +\infty$$. Let $$y_0\in \mathcal{D}(A)$$, formula (41) may be written as \begin{equation*} \frac{{\rm{d}}V(y(t))}{{\rm{d}}t}= \|y(t)\|^2\left(\left[\frac{\langle P_{\omega}By(t), y(t)\rangle}{\|y(t)\|^2}+u(t)\right]^2 - \frac{\langle P_{\omega}By(t), y(t)\rangle^2}{\|y(t)\|^2}- u^2(t)\right)-\langle i_{\omega}Ry(t), y(t)\rangle \end{equation*} integrating this relation, we obtain \begin{equation*} J(u) = V(y_0) + \int^{+\infty}_0\|y(s)\|^2\left[\frac{\langle P_{\omega}By(s), y(s)\rangle}{\|y(s)\|^2}+ u(s)\right]^2{\rm{d}}s. \end{equation*} Then \begin{equation*} J(u)\geq V(y_0), \; \forall u\in \mathcal{U}_{ad}. \end{equation*} For $$u=u^*$$, we get $$J(u^*)=V(y_0)$$. Let $$y_0\in L^2({\it{\Omega}})$$, and a sequence $$(y_{0k})\subset \mathcal{D}(A)$$ such that $$y_{0k} \longrightarrow y_0$$ as $$ k\longrightarrow +\infty$$, we have \begin{equation*} J(u) = V(y_{0k}) + \int^{+\infty}_0\|y_k(s)\|^2[\frac{\langle P_{\omega}By_k(s), y_k(s)\rangle}{\|y_k(s)\|^2}+ u(s)]^2 ds - \int^{+\infty}_0\langle i_{\omega}Ry_k(s), y_k(s)\rangle ds. \end{equation*} Thus $$J(u)\geq V(y_{0k})$$. We deduce that $$J(u)\geq V(y_{0})=J(u^*)$$, so (40) is the unique solution of problem (37). □ Proposition 3.2 Suppose that $$B$$ is locally Lipschitz and satisfies \begin{equation}\label{oss3} \displaystyle \int^{T}_{0} |\langle P_{\omega}BS(s)y, S(s)y\rangle|ds \geq \alpha \|\chi_{\omega}y\|^2_{L^2(\omega)}, \; ( T, \alpha >0) \end{equation} (47) then control (40) is the unique solution of (37) and regionally strongly stabilizes system (1) on $$\omega$$. Proof. For all $$y_0\in L^2({\it{\Omega}})$$, we have \begin{equation*} \begin{array}{lll} \langle P_{\omega}BS(t)y_0, S(t)y_0\rangle = \langle P_{\omega}BS(t)y_0 - P_{\omega}By(t), S(t)y_0\rangle + \langle P_{\omega}By(t), S(t)y_0\rangle. \end{array} \end{equation*} Using (15), we obtain the following relation \begin{equation*} \begin{array}{l@{\;}l@{\;}l} \langle P_{\omega}BS(t)y_0, S(t)y_0\rangle &=& \langle P_{\omega}BS(t)y_0 - P_{\omega}By(t), S(t)y_0\rangle - \langle P_{\omega}By(t), {\it{\Psi}}(t)\rangle\\\\ &&{} + \langle P_{\omega}By(t), y(t)\rangle, \end{array} \end{equation*} where $${\it{\Psi}}(t)=\int_{0}^{t} S(t-s)u(s) By(s){\rm{d}}s$$. Since $$\chi_{\omega}$$ is continuous, then there exists $$ \delta > 0$$ such that \begin{equation}\label{osk4} \begin{array}{lll} |\langle P_{\omega}BS(t)y_0, S(t)y_0\rangle| \leq \delta^2 K\|P\|\| {\it{\Psi}}(t)\|(\| S(t)y_0\| + \| y(t)\|) +| \langle P_{\omega}By(t), y(t)\rangle |. \end{array} \end{equation} (48) Then \begin{equation*} |\langle P_{\omega}BS(t)y_0, S(t)y_0\rangle| \leq 2\delta^2 K\|P\|\| {\it{\Psi}} (t)\|\| y_0 \| + |\langle P_{\omega}By(t), y(t)\rangle|, \forall t\in [0, T]. \end{equation*} Moreover, we have \begin{equation}\label{osk5} \| {\it{\Psi}}(t)\| \leq K\|y_0\|\sqrt T\left(\int_{0}^{T}|\langle P_{\omega}By(s), y(s)\rangle|^2{\rm{d}}s\right)^{\frac{1}{2}}\!. \end{equation} (49) Integrating (48) over the interval $$[0, T]$$ and using (49), we obtain \begin{equation*} \begin{array}{l@{\;}l@{\;}l} \displaystyle\int^T_0 |\langle P_{\omega}BS(s)y_0, S(s)y_0\rangle |{\rm{d}}s &\leq& \displaystyle 2\delta^2 T^\frac{3}{2}K\|P\|\|y_0\|^2 \left(\displaystyle\int_{0}^{T}|\langle P_{\omega}By(s), y(s)\rangle|^2{\rm{d}}s\right)^{\frac{1}{2}}\\\\ &&{} +\sqrt{T}\|y_0\|\left(\displaystyle\displaystyle\int^T_0 |\langle P_{\omega}By(s), y(s)\rangle|^2{\rm{d}}s\right)^{\frac{1}{2}}\!. \end{array} \end{equation*} Replacing $$y_0$$ by $$y(t)$$, we get \begin{equation}\label{osineq} \begin{array}{l@{\;}l@{\;}l} \displaystyle\int^T_0 |\langle P_{\omega}BS(s)y(t), S(s)y(t)\rangle |{\rm{d}}s & \leq& \displaystyle 2\delta^2 T^\frac{3}{2}K\|P\|\|y_0\| \left(\displaystyle\int_{t}^{t+T}|\langle P_{\omega}By(s), y(s)\rangle|^2{\rm{d}}s\right)^{\frac{1}{2}}\\\\ &&{} +\sqrt{T}\|y_0\|\left(\displaystyle\displaystyle\int^{t+T}_t |\langle P_{\omega}By(s), y(s)\rangle|^2{\rm{d}}s\right)^{\frac{1}{2}}\!. \end{array} \end{equation} (50) From (42) and (50), we deduce that \begin{equation}\label{oss7} \int^T_0 |\langle P_{\omega}BS(s)y(t), S(s)y(t)\rangle |{\rm{d}}s \longrightarrow 0,\; as\; t \longrightarrow +\infty. \end{equation} (51) From (47) and (51), we deduce that $$\|\chi_{\omega}y(t)\|_{L^2(\omega)} \longrightarrow 0, \: as\; t \longrightarrow +\infty$$, which completes the proof. □ Proposition 3.3 Assume that $$B$$ is locally Lipschitz such that \begin{equation}\label{p24} \displaystyle \int^{T}_{0} |\langle P_{\omega}BS(t)y, S(t)y\rangle|{\rm{d}}t \geq \delta \|\chi_{\omega}y\|_{L^2(\omega)}, \; (for \; some\; T, \delta >0) \end{equation} (52) and there exists $$\mu >0$$ such that \begin{equation}\label{ineq} \langle P_{\omega}By, y\rangle \leq \mu \langle i_{\omega}By, y\rangle \end{equation} (53) then control (40) is the unique solution of (37) and regionally exponentially stabilizes system (1) on $$\omega$$. Proof. Let us define the function $$F(y)=\langle P_{\omega}y, y\rangle$$, $$\forall y\in L^2({\it{\Omega}})$$. For all $$y_0\in\mathcal{D}(A)$$ and $$t\geq 0$$, using (38), we obtain \begin{equation*} \frac{{\rm{d}}F(y(t))}{{\rm{d}}t}= - 2\frac{\langle P_{\omega}By(t), y(t)\rangle^2}{\|y(t)\|^2} - \langle i_{\omega}Ry(t), y(t)\rangle. \end{equation*} Integrating this relation, we get \begin{equation}\label{sp26} \int^{t}_0\frac{\langle P_{\omega}By(s), y(s)\rangle^2}{\|y(s)\|^2}ds\leq \frac{1}{2}F(y_0), \; t\geq 0. \end{equation} (54) By density, we deduce that $$J(u^*)$$ is finite for all $$y_0\in L^2({\it{\Omega}})$$. For $$y_0\in L^2({\it{\Omega}})$$ and $$t\geq 0$$, we have \begin{equation*} \begin{array}{l@{\;}l@{\;}l} \langle P_{\omega}BS(t)y_0, S(t)y_0\rangle &=& \langle P_{\omega}BS(t)y_0 - P_{\omega}By(t), S(t)y_0\rangle - \langle P_{\omega}By(t), y(t)- S(t)y_0\rangle\\\\ &&+ \langle P_{\omega}By(t), y(t)\rangle. \end{array} \end{equation*} Since $$\chi_{\omega}$$ is continuous, there exists $$ \alpha > 0$$ such that \begin{equation}\label{xx13} \begin{array}{lll} |\langle P_{\omega}BS(t)y_0, S(t)y_0\rangle|\leq 2\alpha K\|P\|\|y(t) - S(t)y_0\|\|y_0\| + |\langle P_{\omega}By(t), y(t)\rangle|. \end{array} \end{equation} (55) Moreover, we have \begin{equation*}\label{xx14} \|y(t) - S(t)y_0\|\leq K\int_{0}^{T}\frac{|\langle P_{\omega}By(s), y(s)\rangle|}{\|y(s)\|}{\rm{d}}s. \end{equation*} Schwarz’s inequality yields \begin{equation}\label{xx15} \|y(t)-S(t)y_0\|\leq K\Big\{ T\int_{0}^{T}\frac{|\langle P_{\omega}By(s), y(s)\rangle|^2}{\|y(s)\|^2}{\rm{d}}s\Big\}^{\frac{1}{2}},\; \forall t\in [0,T]. \end{equation} (56) Using (16), we get \begin{equation*}\label{xx16} |\langle P_{\omega}By(t), y(t)\rangle|\leq \frac{|\langle P_{\omega}By(t), y(t)\rangle|}{\|y(t)\|}\|y_0\|,\; \forall t\in [0,T]. \end{equation*} Integrating (55) over the interval $$[0, T]$$ and taking into account (56), we obtain \begin{equation}\label{xx17} \displaystyle\int_{0}^{T}|\langle P_{\omega}BS(s)y_0, S(s)y_0\rangle|ds \leq (2\alpha K^2T\|P\|+1)\sqrt{T}\|y_0\|\sqrt{f(0)}, \end{equation} (57) where $$f(t)=\int_{t}^{t+T}\frac{|\langle P_{\omega}By(s), y(s)\rangle|^2}{\|y(s)\|^2}{\rm{d}}s$$. Replacing $$y_0$$ by $$y(t)$$ in (57), we get \begin{equation*}\label{xx18} \displaystyle\int_{0}^{T}|\langle P_{\omega}BS(s)y(t), S(s)y(t)\rangle|ds \leq (2\alpha\|P\|K^2T+1)\sqrt{T}\|y(t)\|\sqrt{f(t)}. \end{equation*} Using (52) and (16), we obtain $$\delta\|\chi_{\omega}y(t)\|_{L^2(\omega)}\leq \beta\sqrt{f(t)}$$ with $$\beta=(2\alpha K^2T\|P\|+1)\sqrt{T}\|y_0\|$$. Then \begin{equation}\label{xx19} \delta^2\|\chi_{\omega}y(t)\|^2_{L^2(\omega)}\leq \beta^2 \sqrt{f(t)}. \end{equation} (58) Using (53) and (11), we get $$\frac{d}{dt}\|\chi_{\omega}y(t)\|^2_{L^2(\omega)} \leq -\frac{2}{\mu^2}\frac{|\langle P_{\omega}By(t), y(t)\rangle|^2}{\|y(t)\|^2}.$$ Integrating this inequality from $$mT$$ to $$(m+1)T$$, $$(m\in\mathbb{N})$$, we obtain \begin{equation*} \|\chi_{\omega}y(mT)\|^2_{L^2(\omega)} - \|\chi_{\omega}y((m+1)T)\|^2_{L^2(\omega)} \geq \frac{2}{\mu^2} \int_{mT}^{(m+1)T}\frac{|\langle P_{\omega}By(s), y(s)\rangle|^2}{\|y(s)\|^2}{\rm{d}}s. \end{equation*} Using (58) and since $$\|\chi_{\omega}y(t)\|_{L^2(\omega)}$$ decreases, we deduce \begin{equation*} \left(1 + 2\left(\frac{\delta}{\mu\beta}\right)^2\right)\|\chi_{\omega}y((m+1)T)\|^2_{L^2(\omega)} \leq \|\chi_{\omega}y(mT)\|^2_{L^2(\omega)}. \end{equation*} Then \begin{equation*} \|\chi_{\omega}y((m+1)T)\|_{L^2(\omega)} \leq \sigma\|\chi_{\omega}y(mT)\|_{L^2(\omega)}, \end{equation*} where $$\sigma=\frac{1}{(1 + 2(\frac{\delta}{\mu\beta})^2)^{\frac{1}{2}}}$$. This implies that \begin{equation*} \|\chi_{\omega}y(mT)\|_{L^2(\omega)} \leq \sigma^m\|\chi_{\omega}y_0\|_{L^2(\omega)}. \end{equation*} Taking $$m=E(\frac{t}{T})$$ the integer part of $$\frac{t}{T}$$ and remarking that $$m\geq \frac{t}{T}-1$$, we obtain \begin{equation*} \|\chi_{\omega}y(t)\|_{L^2(\omega)} \leq \sigma^{\frac{t}{T}-1}\|\chi_{\omega}y_0\|_{L^2(\omega)}. \end{equation*} We deduce that \begin{equation}\label{pex} \|\chi_{\omega}y(t)\|_{L^2(\omega)} \leq Fe^{-\rho t}\|y_0\|, \; \forall t\geq 0, \end{equation} (59) where $$F = \alpha(1 + 2(\frac{\delta}{\mu\beta})^2)^{\frac{1}{2}}$$ and $$\rho = \frac{\ln(1 + 2(\frac{\delta}{\mu\beta})^2) }{2T}$$, then control (40) regionally exponentially stabilizes system (1) on $$\omega$$. It remains to show that (40) is the unique solution of (37). Remarking that $$F(y(t))\leq \alpha \|P\|\|\chi_{\omega}y(t)\|^2_{L^2(\omega)}$$, it follows from (59) that $$F(y(t))\longrightarrow 0$$ as $$t\rightarrow +\infty$$. Let $$y_0\in \mathcal{D}(A)$$, integrating the relation \begin{equation}\label{rel} \frac{{\rm{d}}F(y(t))}{{\rm{d}}t}= \|y(t)\|^2([\frac{\langle P_{\omega}By(t), y(t)\rangle}{\|y(t)\|^2}+u(t)]^2 - \frac{\langle P_{\omega}By(t), y(t)\rangle^2}{\|y(t)\|^4}- u^2(t))-\langle i_{\omega}Ry(t), y(t)\rangle \end{equation} (60) we have \begin{equation*} J(u) = F(y_0) + \int^{+\infty}_0\|y(s)\|^2[\frac{\langle P_{\omega}By(s), y(s)\rangle}{\|y(s)\|^2}+ u(s)]^2{\rm{d}}s \end{equation*} then $$J(u)\geq F(y_0)$$. Setting $$u=u^*$$, we obtain $$J(u^*)=F(y_0)$$. Let $$y_0\in L^2({\it{\Omega}})$$, and a sequence $$y_{0k}\subset \mathcal{D}(A)$$ such that $$y_{0k} \longrightarrow y_0$$ as $$ k\longrightarrow +\infty$$, we have \begin{equation*} J(u) = F(y_{0k}) + \int^{+\infty}_0\|y_k(s)\|^2[\frac{\langle P_{\omega}By_k(s), y_k(s)\rangle}{\|y_k(s)\|^2}+ u(s)]^2 {\rm{d}}s. \end{equation*} Thus $$J(u)\geq F(y_{0})=J(u^*)$$. Hence control (40) is the unique solution of the problem (37). □ In order to illustrate the previous results, we perform the following algorithm: Step 1: Initial data: initial condition $$y_0$$ and subregion $$\omega$$; Step 2: Solve equation (38) using Bartels-Stewart method given in Penzl (1998); Step 3: Apply the control given by (25) or (40); Step 4: Solve system (1) using Petrov–Galerkin method; 4. Simulation results Let $${\it{\Omega}} = ]0,1[$$, we consider the system \begin{equation}\label{expl1} \left\{\!\! \begin{array}{l@{\;}l@{\;}l} \displaystyle\frac{\partial y}{\partial t}(x, t) &=& 0.01{\it{\Delta}} y(x, t)+ u(t)\displaystyle\frac{y(x, t)}{1+\|y(x,t)\|}, \;\; {\it{\Omega}} \times ]0, +\infty[\\ y(0, t) &=& y(1, t) = 0, \hspace{4.3cm} ]0, +\infty[ \\ y(x, 0) &=& 2\pi \sin(2\pi x),\hspace{4.2cm} {\it{\Omega}} \end{array} \right. \end{equation} (61) where the state space is $$L^2({\it{\Omega}})$$. 1. Regional stabilizing control. Let $$\omega=]0, \frac{1}{2}[$$, we perform the above algorithm applying the control (25) to system (61), we obtain the following figures Figure 1 shows that system (61) is stabilized on the subregion $$\omega$$, with stabilization error equals to $$3.84\times 10^{-4}$$. Fig. 1. View largeDownload slide Evolution of the state. Fig. 1. View largeDownload slide Evolution of the state. Fig. 2. View largeDownload slide Evolution of the control. Fig. 2. View largeDownload slide Evolution of the control. For $$\omega=]0, 1[$$, we obtain Figure 3 shows that system (61) is stabilized on $${\it{\Omega}}$$, with stabilization error equals to $$9.83\times 10^{-4}$$. Fig. 3. View largeDownload slide Evolution of the state. Fig. 3. View largeDownload slide Evolution of the state. Fig. 4. View largeDownload slide Evolution of the control. Fig. 4. View largeDownload slide Evolution of the control. 2. Regional optimal stabilizing control. Consider system (61) and problem (37) with $$R=I$$ such that $$P_{\omega}$$ is the unique solution of the equation (38). Let $$\omega = ]0, \frac{1}{2}[$$, we perform the above algorithm applying the control (40) to system (61), we obtain the figures below Figure 5 shows how system (61) is stabilized by the control (40) on $$\omega$$ with a stabilization error equals to $$2.1$$$$10^{-4}$$ and the cost $$J(u^*)= 4.9\times 10^{-2}$$. Fig. 5. View largeDownload slide Evolution of the state. Fig. 5. View largeDownload slide Evolution of the state. Fig. 6. View largeDownload slide Evolution of the control. Fig. 6. View largeDownload slide Evolution of the control. For $$\omega={\it{\Omega}}$$, Figure 7 shows that system (61) is stabilized on $${\it{\Omega}}$$, with stabilization error equals to $$8.23\times 10^{-4}$$ and the cost $$J(u^*)= 9.75\times 10^{-2}$$. Fig. 7. View largeDownload slide Evolution of the state. Fig. 7. View largeDownload slide Evolution of the state. Fig. 8. View largeDownload slide Evolution of the control. Fig. 8. View largeDownload slide Evolution of the control. Table 1 shows that there exists a relation between the area of subregion $$\omega$$, the cost and the error of stabilization. Moreover, we remark that more the area of $$\omega$$ increases more the stabilization cost increases. $$\omega$$ $$]0,0.2[$$ $$]0,0.4[$$ $$]0,0.6[$$ $$]0,0.8[$$ $$]0,1[$$ Error 1.33 $$\times 10^{-4}$$ 1.73 $$\times 10^{-4}$$ 3.95$$\times 10^{-4}$$ 6.526 $$\times 10^{-4}$$ 8.23$$\times 10^{-4}$$ Cost 2.38 $$\times 10^{-2}$$ 3.84 $$\times 10^{-2}$$ 5.63 $$\times 10^{-2}$$ 7.56 $$\times 10^{-2}$$ 9.75 $$\times 10^{-2}$$ $$\omega$$ $$]0,0.2[$$ $$]0,0.4[$$ $$]0,0.6[$$ $$]0,0.8[$$ $$]0,1[$$ Error 1.33 $$\times 10^{-4}$$ 1.73 $$\times 10^{-4}$$ 3.95$$\times 10^{-4}$$ 6.526 $$\times 10^{-4}$$ 8.23$$\times 10^{-4}$$ Cost 2.38 $$\times 10^{-2}$$ 3.84 $$\times 10^{-2}$$ 5.63 $$\times 10^{-2}$$ 7.56 $$\times 10^{-2}$$ 9.75 $$\times 10^{-2}$$ $$\omega$$ $$]0,0.2[$$ $$]0,0.4[$$ $$]0,0.6[$$ $$]0,0.8[$$ $$]0,1[$$ Error 1.33 $$\times 10^{-4}$$ 1.73 $$\times 10^{-4}$$ 3.95$$\times 10^{-4}$$ 6.526 $$\times 10^{-4}$$ 8.23$$\times 10^{-4}$$ Cost 2.38 $$\times 10^{-2}$$ 3.84 $$\times 10^{-2}$$ 5.63 $$\times 10^{-2}$$ 7.56 $$\times 10^{-2}$$ 9.75 $$\times 10^{-2}$$ $$\omega$$ $$]0,0.2[$$ $$]0,0.4[$$ $$]0,0.6[$$ $$]0,0.8[$$ $$]0,1[$$ Error 1.33 $$\times 10^{-4}$$ 1.73 $$\times 10^{-4}$$ 3.95$$\times 10^{-4}$$ 6.526 $$\times 10^{-4}$$ 8.23$$\times 10^{-4}$$ Cost 2.38 $$\times 10^{-2}$$ 3.84 $$\times 10^{-2}$$ 5.63 $$\times 10^{-2}$$ 7.56 $$\times 10^{-2}$$ 9.75 $$\times 10^{-2}$$ 5. Conclusion In this article, sufficient conditions for regional exponential, strong and weak stabilization were given. Moreover, we gave a control which ensures regional stabilization minimizing an appropriate cost function. The obtained results are successfully illustrated by simulations. This work motivates new problems, this is the case where $$\omega$$ is a part of the boundary of the system evolution domain which is of a great interest. References Ball, J. M. & Slemrod, M. ( 1979 ) Feedback stabilization of distributed semilinear control systems. Appl. Math. Optim. , 5 , 169 – 179 . Google Scholar CrossRef Search ADS Bounit, H. & Hammouri, H. ( 1999 ) Feedback stabilization for a class of distributed semilinear control systems. Nonlinear Anal. , 37 , 953 – 969 . Google Scholar CrossRef Search ADS Mohler, R. R. ( 1973 ) Bilinear Control Processes: With Applications to Engineering, Ecology and Medicine . New York : Academic Press, Inc. Ouzahra, M. ( 2008 ) Strong stabilization with decay estimate of semilinear systems. Systems Control Lett. , 57 , 813 – 815 . Google Scholar CrossRef Search ADS Ouzahra, M. ( 2011 ) Exponential stabilization of distributed semilinear systems by optimal control. J. Math. Anal. Appl. , 380 , 117 – 123 . Google Scholar CrossRef Search ADS Pazy, A. ( 1983 ) Semigroups of Linear Operations to Partial Differential Equations . New York : Springer . Google Scholar CrossRef Search ADS Penzl, T. ( 1998 ) Numerical solution of generalized Lyapunov equations. Adv. Comp. Math. , 8 , 33 – 48 . Google Scholar CrossRef Search ADS Zerrik, E. & Ezzaki, L. ( 2016 ) Regional gradient stabilization of semilinear distributed systems. J. Dyn. Control Syst. , 23 , 405 – 420 . Google Scholar CrossRef Search ADS Zerrik, E. & Ezzaki, L. ( 2017 ) Output stabilization of distributed bilinear systems. Control Theory Technol. Zerrik, E. & Ouzahra, M. ( 2003 ) Regional stabilization for infinite-dimensional systems. Int. J. Cont. , 76 , 73 – 81 . Google Scholar CrossRef Search ADS Zerrik, E. & Ouzahra, M. ( 2005 ) Output stabilization for infinite bilinear systems. Int. J. Appl. Math. Comput. Sci. , 15 , 187 – 196 . Zerrik, E. & Ouzahra, M. ( 2007 ) Output stabilisation for distributed semilinear systems. IET Control Theory Appl. , 1 , 838 – 843 . Google Scholar CrossRef Search ADS © The authors 2017. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

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IMA Journal of Mathematical Control and InformationOxford University Press

Published: Sep 6, 2017

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