/lp/ou_press/controllability-of-semilinear-stochastic-control-system-with-finite-mUZ02DS1Bd

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Abstract The objective of this paper is to present some sufficient conditions for approximate and exact controllability of semilinear stochastic control system with finite delay. Sufficient conditions for approximate controllability are obtained by separating the given semilinear system into two systems namely a semilinear deterministic system and a linear stochastic system using Schauder fixed point theorem. For obtaining the sufficient conditions for exact controllability of assumed system, Banach fixed point theorem is applied. Instead of a $$C_0$$-semigroup associated with the mild solution of the system we use the so-called fundamental solution. At the end, examples are given to illustrate the theory. 1. Introduction Controllability is one of the fundamental concepts in modern mathematical control theory. This is the qualitative property of control systems and is of particular importance in control theory. Many dynamical systems are such that the control does not affect the complete state of the dynamical system but only a part of it. Therefore, it is very important to determine whether or not control of the complete state of the dynamical system is possible. So, here the concept of complete controllability and approximate controllability arises. Roughly speaking, controllability generally means, that it is possible to steer dynamical system from an arbitrary final state using the set of admissible controls. It is well-known that controllability of deterministic equation is widely used in many fields of science and technology. But in many practical problems such as fluctuating stock prices or physical system subject to thermal fluctuations, population dynamics, etc, some randomness appear, so the system should be modelled in stochastic form. In deterministic setting, Kalman (1963) introduced the concept of controllability and obtained the results for linear deterministic systems. Naito (1987) established sufficient conditions for approximate controllability of deterministic semilinear control system dominated by the linear part using Schauder’s fixed point theorem. Wang (2005) extended the results of Naito (1987) and introduced the concept of fundamental solution to established sufficient conditions for delayed deterministic semilinear systems. Sukavanam & Tafesse (2011) established sufficient conditions for approximate controllability of a delayed semilinear control system with growing non-linearity using similar technique to Wang (2005). Jeong et al. (1999) introduced the sequence method and obtained some sufficient conditions for approximate controllability of semilinear retarded functional differential equations. In stochastic setting, Mahmudov & Denker (2000) and Mahmudov (2001) established results for controllability of linear stochastic systems. Klamka (2000a, 2007) obtained sufficient conditions for compete controllability of linear system with state delay and non-linear system using Schauder fixed point theorem. Shen & Sun (2012, 2013) established sufficient conditions for approximate controllability of abstract stochastic impulsive systems with multiple time-varying delays. The methods Shen & Sun choose are mainly Nussbaum fixed point theorem and stochastic analysis techniques combined with a strongly continuous semigroup. Shukla et al. (2015) obtained some sufficient conditions for approximate controllability of semilinear stochastic retarded system using Banach fixed point theorem. Balasubramaniam & Ntouyas (2006) obtained sufficient conditions for controllability of neutral stochastic functional differential inclusions with infinite delay in abstract with the help of Leray–Schuder non-linear alternative. Muthukumar & Rajivganthi (2014) obtained the results for approximate controllability of stochastic non-linear third-order dispersion equation in Hilbert spaces using fixed point theorems. Sukavanam & Kumar (2010) introduced the split technique and obtained sufficient conditions for stochastic controllability of an abstract first order semilinear control system using Schauder’s fixed point theorem. The aim of this paper is to investigate the approximate and exact controllability of semilinear stochastic systems with delay in state with some conditions such as Lipschitz continuity and strong continuity of the semigroup. Results are obtained using Schauder’s fixed point theorem, Banach fixed point theorem and concept of fundamental solution, which is strongly associated with semigroup. The rest of this paper is organized as follows: In Section 2, problem is formulated for study of approximate controllability. In Section 3, some basic definitions, assumptions, inequalities and operators of functional analysis are given. In Section 4, main results are discussed for approximate controllability. In Section 5, problem is formulated for study of exact controllability and some basic operators theory is discussed. In Section 6, main results are discussed for exact controllability. In Section 7, we give examples to testify the proposed results. 2. Approximate controllability of system Let $$X$$ and $$U$$ be the Hilbert spaces and $$Z=L_2[0,b;X]$$, $$Z_h=L_2[-h,b;X]$$, $$0<h<b$$ and $$Y=L_2[0,b;U]$$ be function spaces. $$\mathbb{R}^k$$ denotes $$k$$-dimensional real Euclidean space. Let $$({\it\Omega},\zeta,P)$$ be the probability space with a probability measure $$P$$ on $${\it\Omega}$$ and a filtration $$\{\zeta_t|t\in[0,b]\}$$ generated by a Wiener Process $$\{\omega(s):0\leq s\leq t\}$$. We consider the semilinear stochastic control system of the form: dx(t) = [Ax(t)+A1x(t−h)+Bu(t)+f(t,x(t−h))]dt+dω(t);t>0,x(t) =ξ(t),t∈[−h,0]. (2.1) where the state function $$x\in Z;$$$$\xi\in L_2[-h,0;X]$$; $$A:D(A)\subseteq X\rightarrow X$$ is a closed linear operator which generates a strongly continuous semigroup $$T(t)$$; $$A_1$$ is a bounded linear operator on $$X;$$$$B:Y\rightarrow Z$$ is a bounded linear operator; function $$f:[0,b]\times X\rightarrow X$$ is a non-linear operator such that, $$f$$ is measurable with respect to $$t$$ for all $$x\in Z$$ and continuous with respect to $$x$$ for almost all $$t\in[0,b]$$. Control $$u(t)$$ takes values in $$U$$ for each $$t\in[0,b]$$. By splitting the system (2.1), we get the following pair of coupled systems dy(t) = [Ay(t)+A1y(t−h)+Bv(t)+f(t,y(t−h)+z(t−h))]dt;0≤t≤b,y(t) =ψ(t),t∈[−h,0], (2.2) and dz(t) = [Az(t)+A1z(t−h)+Bw(t)]dt+dω(t);0≤t≤b,z(t) =ξ(t)−ψ(t),t∈[−h,0]. (2.3) The system represented by (2.3) is linear stochastic system with delay in state and for each realization $$z(t)$$ of system (2.3), the system given by (2.2) is a deterministic system. Thus the solution $$y(t)$$ of the semilinear system (2.2) depends on the solution $$z(t)$$ of linear stochastic system (2.3). The functions $$v$$ and $$w$$ are $$Y$$-valued control function, such that $$u=v+w$$. It can be easily seen that, the solution $$x(t)$$ of the semilinear stochastic system (2.1) is given by $$y(t)+z(t)$$ where $$y(t)$$ and $$z(t)$$ are the solutions of the systems (2.2) and (2.3), respectively. 3. Preliminaries Consider the linear delay system x′(t) =Ax(t)+A1x(t−h),t∈[0,b]x(t) =ξ(t).t∈[−h,0). (3.1) Let $$x_\xi (t)$$ be the unique solution of the system (3.1). Define the operator $$S(t)$$ on $$X$$ by S(t)ξ(0)={xξ(t),t∈[0,b]0,t∈[−h,0). Then, $$S(t)$$ is called the fundamental solution of (3.1) satisfying S(t) =T(t)+∫0tT(t−s)A1S(s−h)ds,t>0,S(0) =I,S(t)=0.for−h≤t<0. From Wang (2005), $$S(t)$$ is the unique solution of (3.1) and it can be easily shown that $$||S(t)||\leq M \exp(M(b-h)||A_1||)=M_1$$, where $$||T(t)||\leq M$$ (using Gronwall’s inequality). Now, we define the mild solution of the systems (2.1) as x(t)={S(t)ξ(0)+∫0tS(t−s){Bu(s)+f(s,x(s−h))}ds+∫0tS(t−s)dω(s),t>0ξ(t)−h≤t≤0 (3.2) the mild solution of the semilinear system (2.2), can be written as y(t)={S(t)ψ(0)+∫0tS(t−s){Bv(s)+f(s,y(s−h)+z(s−h))}ds,t>0ψ(t)−h≤t≤0 and the mild solution of the linear stochastic system (2.3), can be written as z(t)={S(t)(ξ(0)−ψ(0))+∫0tS(t−s)Bw(s)ds+∫0tS(t−s)dω(s),t>0ξ(t)−ψ(t)−h≤t≤0. Consider the linear system corresponding to the system (2.2), given by dp(t)dt =Ap(t)+A1p(t−h)+Br(t),t>0p(t) =ψ(t)t∈[−h,0]. (3.3) The mild solution of the above linear system is expressed as p(t)={S(t)ψ(0)+∫0tS(t−s)Br(s)dst>0ψ(t)−h≤t≤0. (3.4) For obtaining the sufficient conditions for the approximate controllability of control system (2.1) the following conditions are assumed. Throughout this paper $$D(A)$$, $$R(A)$$ and $$N_0(A)$$ denotes the domain, range and null space of operator $$A$$, respectively, $$(H_1)$$ For every $$p\in Z$$ there exists a $$q\in\overline{R(B)}$$ such that $$Lp=Lq$$ where the operator $$L:Z\rightarrow X$$ is defined as Lx=∫0bS(b−s)x(s)ds. $$(H_2)$$ The semigroup $$\{T(t), t\geq 0\}$$ generated by $$A$$ is compact on $$X$$ and there is a constant $$M\geq 0$$ such that $$||T(t)||\leq M$$. $$(H_3)$$$$f(t,x)$$ satisfies Lipschitz continuity on $$Z$$, i.e., ||f(t,x1)−f(t,x2)||≤lp||x1−x2||,lp>0 $$(H_4)$$$$f(t,x)$$ satisfies linear growth condition, that is, ||f(t,x)||≤a1+b1||x||, where $$a_1$$ and $$b_1$$ are constants. $$(H_5)$$$$M_1bb_1(1+c)<1$$, where the constants $$b$$ and $$b_1$$ appear in the above conditions. The constant $$c$$ is defined in Lemma 1. Let $$G:N_0^{\bot}(L)\rightarrow \overline{R(B)}$$ be an operator defined as follows Ga=a0, where $$a\in N_0^{\bot}(L)$$ and $$a_0$$ is the unique minimum norm element in the set $$\{a+N_0(L)\}\bigcap\overline{R(B)}\}$$ satisfying the following condition ||Ga||=||a0||=min[||e||:e∈{a+N0(L)}⋂R(B)¯}]. (3.5) The operator $$G$$ is well-defined, linear and continuous (see Naito, 1987, Lemma 1). From continuity of $$G$$, it follows that $$||Ga||\leq c||a||_Z$$, for some constant $$c\geq0$$. Since $$Z=N_0(L)+\overline{R(B)}$$ as is evident from condition $$(H_1)$$, any element $$z\in Z$$ can be expressed as z=n+q:n∈N0(L),q∈R(B)¯. Lemma 1 In Sukavanam & Kumar (2010), for $$z\in Z$$ and $$n\in N_0(L)$$; the following inequality holds ||n||Z≤(1+c)||z||Z, (3.6) where $$c$$ is such that $$||G||\leq c$$. Let us introduce some operators in the following way: $$K:Z\rightarrow Z$$ defined by (Kz)(t)=∫0tS(t−s)z(s)ds. Now, let $$M_0$$ be the subspace of $$Z_h$$ (see Sukavanam, 1993) such that M0={m∈Zh:m(t)=(Kn)(t),n∈N0(L)0≤t≤b,m(t)=0,−h≤t≤0. It can be noted that $$m(b)=0$$ for all $$m\in M_0$$. For each solution $$p(t)$$ of the system (3.3) with control $$r$$ and for each realization $$z(t)$$ of the system (2.3), define the random operator $$f_p:\overline{M_0}\rightarrow M_0$$ as fp={Kn,0<t<b,0,−h≤t≤0, (3.7) where $$n$$ is given by the unique decomposition Fh(p+z+m)=n+q:n∈N0(L),q∈R(B)¯, (3.8) where $$F_h:Z_h\rightarrow Z$$ given by (Fhx)(t)=f(t,x(t−h)),0≤t≤b. Note that if $$h=0$$, $$F_h$$ is the Nemytskii operator of $$f$$. It is easy to see that $$F_h$$ satisfies Lipschitz continuity $$(H_3)$$ and linear growth conditions $$(H_4)$$. Definition 3.1 The set given by $$K_b(f) =\{x(T)\in X :x\in Z_h\}$$ where $$x$$ is a mild solution of (2.1) corresponding to control $$u\in Y$$ is called Reachable set of the system (2.1). Definition 3.2 The system (2.1) is said to be approximately controllable if $$K_b(f)$$ is dense in $$X$$, means $$\overline{K_b(f)}=X$$. Definition 3.3 If, for $$t\geq t_0$$, (1) $$u(t)$$ and $$v(t)$$ are continuous and $$u(t)\geq0$$, $$v(t)\geq0$$; (2) $$ u(t)\leq K+\int_{t_0}^{t}u(s)v(s)ds$$, $$K>0$$; then u(t)≤Kexp(∫t0tv(s)ds),t≥t0. 4. Main results Lemma 2 Let assumption $$(H_2)$$ hold. Then $$S(t)$$ is compact operator for each $$t\in(0,b]$$. Proof. Define a sequence of operators $$S_n(t)$$ on $$[-h,b]$$ as S1(t) ={T(t),t∈(0,b]0t∈[−h,0], Sn+1(t) ={T(t)+∫0tT(t−s)A1Sn(s−h)dst∈(0,b]0,t∈[−h,0]. (4.1) From the compactness of $$T(t)$$ and boundedness of $$A_1$$ clearly $$\{S_n\}$$ is compact. Now we need to show that $$S_n(t)\rightarrow S(t)$$ in $$L(X)$$ for any $$t>0$$ as $$n\rightarrow \infty$$. Let $$||A_1||\leq K_1$$. Then it is easy to see from Sukavanam & Tafesse (2011) that for $$n>m$$ we have that ||Sn(t)−Sm(t)||≤∑i=mn−11i!Mi+1K1ibi. But right-hand side goes to zero as $$n,m\rightarrow \infty$$. Hence $$\{S_n\}$$ is a Cauchy sequence in $$L(X)$$ which is uniformly convergent, say $$S_n(t)\rightarrow S(t)$$ for every $$t\in(0,b]$$. Thus $$S(t)$$ is compact for $$t>0$$. □ Lemma 3 Suppose the linear control system without delay (when $$A_1=0$$ in (3.3)) is approximately controllable (see Naito, 1987). Then the linear control system with delay (3.3) is approximately controllable. Proof. Consider the system (3.3) with $$h\leq b$$. Since $$0<b<\infty$$ there exist a positive integer $$n$$ such that $$b\in((n-1)h,nh]$$. Suppose $$w_0=p_0, w_1, w_2,\ldots,w_{n-1} $$ are given in $$X$$. Then we want to prove that for any given $$\epsilon>0$$ and $$p_1$$, the mild solution $$p(t)$$ of (3.3) satisfies $$||p(b)-p_1||\leq \epsilon$$. Now consider the following control system without delay k′(t) =Ak(t)+Br(t),t∈(0,h]k(0) =p0=w0=ψ(0). (4.2) Let k1=w1−ph,ph=∫0hT(h−s)A1p(s−h)ds, which is known $$(p(t)=\psi(t)\;on\;[-h,0])$$. By assumption $$(H_1)$$ the system (4.2) is approximate controllable. Therefore there exists a control $$r_1$$ such that the mild solution of (4.2) k(t)=T(t)p0+∫0tT(t−s)Br1(s)ds,0<t≤h satisfies $$||k(h)-k_1||\leq \epsilon$$. Let p(t)=T(t)p0+∫0tT(t−s)Br1(s)ds+∫0tT(t−s)A1p(s−h)ds,0<t≤h. (The terms in the right-hand side are known.) Now ||p(h)−w1|| = ||k(h)+ph−w1||=||k(h)+w1−k1−w1|| = ||k(h)−k1||≤ϵ. Let us denote $$k(h)$$ by $$k_h$$. Consider k′(t) =Ak(t)+Br(t),t∈(h,2h]k(h) =kh. (4.3) Let k2=w2−p2h,p2h=∫02hT(2h−s)A1p(s−h)ds, which is known since $$p(s)$$ is known in the interval $$[0,h]$$ from the previous step. By a similar argument as in case of the system (4.2) there exists a control $$r_2$$ such that mild solution of (4.3) k(t)=T(t)p0+∫0tT(t−s)Br2(s)ds,h<t≤2h satisfies $$||k(2h)-k_2||\leq\epsilon$$. Let p(t)=T(t)p0+∫0tT(t−s)Br2(s)ds+∫0tT(t−s)A1p(s−h)ds,h<t≤2h. (The terms in the right-hand side are known.) Now ||p(2h)−w2|| = ||k(2h)+p2h−w2||=||k(2h)+w2−k2−w2|| = ||k(2h)−k2||≤ϵ. By proceeding in similar manner we have k′(t) =Ak(t)+Br(t),t∈((n−1)t,b]k((n−1)h) =k(n−1)h. (4.4) Let kn=p1−pb,pb=∫0bT(b−s)A1p(s−h)ds,t∈((n−1)h,b], which is known since $$p(s)$$ is known in the interval $$((n-2)h,(n-1)h]$$ from the previous step. By a similar argument as above there exists a control $$r_n$$ such that the mild solution of (4.4) k(t)=T(t)p0+∫0tT(t−s)Brnds,(n−1)h<t≤b satisfies $$||k(b)-k_n||\leq\epsilon$$. Let p(t)=T(t)p0+∫0tT(t−s)Brn(s)ds+∫0tT(t−s)A1p(s−h)ds,(n−1)h<t≤b. (The terms in the right-hand side are known.) Now ||p(b)−p1|| = ||k(b)+pb−p1||=||k(b)+p1−kn−p1|| = ||k(b)−kn||≤ϵ. Note that the controls $$r_1,r_2,\ldots,r_n$$ are known. Now we set the control $$r$$ as r(t) ={ri(t),t∈((i−1)h,ih],i=1,2,…,(n−1)0,t∈[0,b]∖((i−1)h,ih], r(t) ={rn(t),t∈((n−1)h,b],0,t∈[0,b]∖((n−1)h,b]. Hence the mild solution $$p(t)$$ of the system (4.4) can be written as p(t)=T(t)p0+∫0tT(t−s)Br(s)ds+∫0tT(t−s)A1p(s−h)ds,t∈[0,b], (4.5) which is the solution of the system (3.3) with $$||p(b)-p_1||\leq\epsilon$$. Therefore the proof is completed. □ For approximate controllability of (2.3) let us introduce some operators and lemmas. The mild solution of system (2.3) is z(t)={S(t)(ξ(0)−ψ(0))+∫0tS(t−s)Bw(s)ds+∫0tS(t−s)dω(s),t>0ξ(t)−ψ(t)−h≤t≤0. (4.6) Define the operator $$L_0^b:L_2[0,b;U]\rightarrow L_2[{\it {\Omega}},\zeta_t,X]$$, the controllability operator $${\it {\Pi}}_s^b:L_2[{\it {\Omega}},\zeta_t,X]\rightarrow L_2[{\it {\Omega}},\zeta_t,X]$$ associated with (4.8), and the controllability operator $${\it {\Gamma}}_s^b:X\rightarrow X$$ associated with the corresponding deterministic system of (4.6) as L0b = ∫0bS(b−s)Bw(s)ds,Πsb{.} = ∫sbS(b−t)BB∗S∗(b−t)E{.|ζt}dt,Γsb = ∫sbS(b−t)BB∗S∗(b−t)dt. It is easy to see that the operators $$L_0^b,{\it {\Pi}}_s^b,{\it {\Gamma}}_s^b$$ are linear bounded operators, and the adjoint $$(L_0^b)^*:L_2[{\it {\Omega}},\zeta_t,X]\rightarrow L_2[0,b;U]$$ of $$L_0^b$$ is defined by (L0b)∗ =B∗S∗(b−t)E{z|ζt},Π0b =L0b(L0b)∗. Before studying the approximate controllability of system (2.3), let us first investigate the relation between $${\it {\Pi}}_s^b$$ and $${\it {\Gamma}}_s^b$$, $$s\leq r<b$$ and resolvent operator $$R(\lambda,{\it {\Pi}}_s^b)=(\lambda I+{\it {\Pi}}_s^b)^{-1}$$ and $$R(\lambda,{\it {\Gamma}}_r^b)=(\lambda I+{\it {\Gamma}}_r^b)^{-1}$$, $$s\leq r<b$$ for $$\lambda>0$$, respectively. Lemma 4 For every $$z\in L_2[{\it {\Omega}},\zeta_t,X] $$there exists $$\varphi(.)\in L_2^\zeta(0,b;\mathbb{L}(\mathbb{R}^k,X))$$ such that (1) $$\mathbb{E}\{z|\zeta_t\}=\mathbb{E}\{z\}+\int_0^t\varphi(s)d\omega(s)$$, (2) $${\it {\Pi}}_s^bz={\it {\Gamma}}_s^b\mathbb{E}z+\int_s^b{\it {\Gamma}}_r^b\varphi(r)d\omega(r)$$, (3) $$R(\lambda,{\it {\Pi}}_s^b)z=R(\lambda,{\it {\Gamma}}_s^b)\mathbb{E}\{z|\zeta_t\}+\int_s^b{\it {\Gamma}}_r^b\varphi(r)d\omega(r)$$. Proof. The proof is straightforward adaption of the proof of Mahmudov (2001), Lemma (2.3). □ Theorem 4.1 The control system (2.3) is approximately controllable on $$[0,b]$$ if and only if one of the following conditions holds. (1) $${\it {\Pi}}_0^b>0.$$ (2) $$\lambda R(\lambda,{\it {\Pi}}_0^b)$$ converges to the zero operator as $$\lambda\rightarrow 0^+$$ in the strong operator topology. (3) $$\lambda R(\lambda,{\it {\Pi}}_0^b)$$ converges to the zero operator as $$\lambda\rightarrow 0^+$$ in the weak operator topology. Proof. The proof is straightforward adaption of the proof of Mahmudov & Denker (2000), Theorem 2. □ Lemma 5 Under the conditions $$(H_2), (H_4)$$ and $$(H_5)$$, the operator $$f_p$$ has a fixed point $$m_0\in M_0$$ for each realization $$z(t)$$ of the system (2.3). Proof. From the compactness of $$S(t)$$ the integral operator $$K$$ is compact and hence $$f_p$$ is compact for each $$p$$, (see Pazy, 1983). Now let $$B_{\tilde{r}}=\{v\in Z:||v||\leq \tilde{r},\;\tilde{r}>0\}$$ and suppose $$m\in B_{\tilde{r}}$$. Then using the inequality (3.6), equation (3.8) and the condition $$(H_4)$$, we get ||fp(m)||=Kn ≤ ∫0t||S(t−s)||||n(s)||ds ≤M1∫0t||n(s)||ds≤M1∫0t(1+c)||F(p+z+m)||ds ≤M1(1+c)∫0t(a1+b1||p+z+m||)ds ≤M1(1+c)∫0t(a1+b1||p+z||)ds+M1(1+c)∫0tb1||m||ds ≤M1(1+c)b(a1+b1||p+z||)+M1(1+c)bb1r~. Now let $$M_1b(1+c)(a_1+b_1||p+z||)+M_1(1+c)bb_1\tilde{r}<\tilde{r}$$. Then M1b(1+c)(a1+b1||p+z||)<r~−M1b(1+c)b1r~⇒r~>M1b(1+c)(a1+b1||p+z||)1−M1b(1+c)b1. Hence if $$M_1b(1+c)b_1<1$$, for a large $$\tilde{r}$$ , $$f_p$$ maps $$B_{\tilde{r}}$$ in to itself. Then by using Schauder’s fixed point theorem $$f_p$$ has a fixed point $$m_0$$, i.e., fp(m0)=m0=Kn. (4.7) The approximate controllability of the semilinear system (2.2) is proved in following manner using the above lemma. □ Lemma 6 For each realization $$z(t)$$ of the system (2.3), the semilinear control system (2.2) is approximate controllable under the conditions $$(H_1)\text{--}(H_4)$$. Proof. From the equation (3.8), we have Fh(p+z+m)=n+q. Operating $$K$$ on both the sides at $$m=m_0$$ (fixed point of $$f_p$$) and using (3.7) , we get KFh(p+z+m0) =Kn+Kq =m0+Kq. Adding $$p$$ on both sides, we get p+KFh(p+z+m0)=p+m0+Kq. Let $$p+m_0=y^*$$, then the above equation is equivalent to p+KFh(y∗+z)=y∗+Kq. Since, from the equation (3.4) p=S(t)ψ(0)+KBr we have S(t)ψ(0)+KBr+KFh(y∗+z)=y∗+KqS(t)ψ(0)+K(Br−q)+KFh(y∗+z)=y∗. Thus, it follows that $$y^*(t)$$ is a solution of the semilinear system dy∗(t)dt =Ay∗(t)+A1y∗(t−h)+f(t,y∗(t−h)+z(t−h))+Br(t)−q(t),y∗(0) =ψ(0), (4.8) with control $$(Br-q)$$. Moreover, since $$y^*(t)=p(t)+m_0(t)$$, it follows that y∗(b)=p(b)+m0(b), as $$m_0(b)=0$$ it follows that y∗(b)=p(b)∈Kb(0)(reachablesetoflinearsystem(3.3)). (4.9) Since $$q\in\overline{R(B)}$$ implies that for any given $$\epsilon_1>0$$, there exists $$v_1\in Y$$ such that $$||q-Bv_1||\leq \epsilon_1$$. Now consider the equation dy(t)dt =Ay(t)+A1y(t−h)+f(t,y(t−h)+z(t−h))+B(r(t)−v1(t)),y(0) =ψ(0). (4.10) Let $$y(t)$$ be the solution of the system (4.10), corresponding to control $$v=r-v_1$$. Then $$||y^*(b)-y(b)||$$ can be made arbitrary small by choosing a suitable $$\epsilon_1$$, which implies that the reachable set of the system (4.10) is dense in the reachable set of the system (4.8), which in turn is dense in $$X$$. This proves that the system (2.2) is approximately controllable. □ Remark 4.1 If $$A_1=0$$ and $$h=0$$ then the $$C_0$$ semigroup $$T(t)$$ is used in place of fundamental solution $$S(t)$$ and the system (2.1) becomes dx(t) = [Ax(t)+Bu(t)+f(t,x(t))]dt+dω(t),t>0,x(0) =ξ(0)=x0. (4.11) Thus the main result of Sukavanam & Kumar (2010) follows as corollary to Lemma 6. Corollary 4.1 Suppose $$A_1=0$$ and $$h=0$$. Under conditions $$(H_1)\text{--}(H_4)$$ the semilinear stochastic control system (4.12) is approximately controllable if the constants $$b_1$$ and $$c$$ satisfies the condition $$M_1bb_1(1+c)<1$$. 5. Exact controllability of system In this section we discussed exact controllability of assumed system (5.1). For understanding the problem we adopt the following notations: (i) $$({\it\Omega},\digamma,P)$$: The triple $$({\it\Omega},\digamma,P)$$ is probability space of the $$n$$-dimensional Wiener process $$\omega$$. (ii) $$\{\digamma _t|t\in[0,T]\}$$: The filtration generated by $$\{\omega(s):0\leq s\leq t\}$$, here $$\omega$$ is Wiener Process. (iii) $$L_2({\it\Omega},\digamma_T,\mathbb{R}^n)$$: The Hilbert space of all $$\digamma_T$$-measurable square integrable random variables with values in $$\mathbb{R}^n$$. (iv) $$L_2^\digamma([0,T],\mathbb{R}^n)$$: The Hilbert space of all square-integrable and $$\digamma_t$$-measurable processes with values in $$\mathbb{R}^n$$. (v) $$H_2$$: The Banach space of all square integrable and $$\digamma_t$$-adapted processes $$\varphi(t)$$ with norm ||φ||2=supt∈[0,T]E||φ(t)||2,whereEistheexpectedvalue. (vi) $$\mathbb{L}(X,Y)$$: The space of all linear bounded operators from a Banach space $$X$$ into a Banach space $$Y$$. (vii) $$U_{\rm ad}=L_2^\digamma([0,T],\mathbb{R}^m)$$: The set of admissible controls. Consider the following stochastic system with delay in state term: dx(t) = [A0x(t)+A1x(t−h)+B0u(t)+f(t,x(t−h))]dt +σ(t,x(t−h))dω(t),fort∈(0,T] (5.1) x(t) =ψ(t),fort∈[−h,0),x(0)=x0. (5.2) where the state $$x(t)\in L_2({\it\Omega},\digamma_t,R^n)=X$$ and the control $$u(t)\in R^m=U$$, $$ A_0 $$ and $$A_1$$ are an $$n\times n$$ constant matrices, $$B_0$$ is an $$n\times m$$ constant matrix. $$f:[0,T]\times R^n\rightarrow R^{n}$$ and $$\sigma :[0,T]\times R^n\rightarrow R^{n\times n}$$ are non-linear functions. $$\omega$$ is a $$n$$-dimensional Wiener process and $$h>0$$ is a constant point delay. It is well-known from Shen & Sun (2012) and Balasubramaniam & Ntouyas (2006) that for a given initial condition (5.2), any admissible control $$u\in U_{\rm ad}$$ and suitable non-linear functions $$f(t,x(t))$$ and $$\sigma(t,x(t))$$ for $$t\in[0,T]$$ there exists a unique solution $$x(t;x_0,u)\in L_2({\it\Omega},\digamma_T,R^n)$$ of the semilinear stochastic system (5.1) which can be represented as follows: x(t;x0,u)={x(t;x0,0)+∫0tF(t−s)(B0u(s)+f(s,x(s−h)))ds+∫0tF(t−s)σ(s,x(s−h))dω(s)fort≥0ψ(t)fort<0, (5.3) where $$F(t)$$ is the $$n\times n$$ matrix for the delayed state equation (5.1), which satisfies the matrix integral equation. F(t)=I+∫0tF(s)A0ds+∫0t−hF(s)A1ds (5.4) for $$t>0$$, with the initial conditions F(0)=I,F(t)=exp(A0t)fort∈[0,h),F(t)=0fort<0. Moreover, for $$t>0$$, $$x(t;x_0,0)$$ is given by x(t;x0,0)=exp(A0t)x0+∫−h0F(t−s−h)A1x0(s)ds or, equivalently x(t;x0,0) =exp(A0t)x0+∫0hF(t−s)A1x0(s−h)ds. (5.5) Now, for a given final time $$T>h$$, taking into account the form of the integral solution $$x(t;x_0,u)$$, let us introduce the following operators and sets. Define the bounded linear operator $$L_T\in L_2([0,T],R^m)\rightarrow L_2({\it\Omega},\digamma_T,R^n)$$ by LTu=∫0hexp(A0(T−s))B0u(s)ds+∫hTF(T−s)B0u(s)ds. Its adjoint bounded linear operator $$L_T^*\in L_2({\it\Omega},\digamma_T,R^n)\rightarrow L_2([0,T],R^m) $$ has the following form: LT∗z={(B0∗exp(A0∗(T−t))+B0∗F∗(T−t))E{z|ϝT}fort∈[h,T]B0∗exp(A0∗(T−t))E{z|ϝT}fort∈[0,h). Define the set of all the states reachable in the final time $$T$$ from a given initial state $$x_0\in L_2([-h,0],R^n)$$, using a set of admissible controls,as follows RT(Uad)={x(T;x0,u)∈L2(Ω,ϝT,Rn):u∈Uad}. Now, we introduce the linear controllability operator $${\it\Pi}_0^T\in L(L_2({\it\Omega},\digamma_T,R^n)\rightarrow L_2({\it\Omega},\digamma_T,R^n))$$, which is strongly associated with the control operator $$L_T$$ and is given the following equality: Π0T{.} =LTLT∗{.} ={∫0Texp(A0(T−t))B0B0∗exp(A0∗(T−t))E{.|ϝt}dtforT≤h∫hTF(T−t)B0B0∗F∗(T−t)E{.|ϝt}+∫0hexp(A0(T−t))B0B0∗exp(A0∗(T−t))E{.|ϝt}dtforT>h. Let us recall that the $$n\times n$$ deterministic controllability matrix is given by Γ0T =LTLT∗ ={∫0Texp(A0(T−t))B0B0∗exp(A0∗(T−t))dtforT≤h∫hTF(T−t)B0B0∗F∗(T−t)dt+∫0hexp(A0(T−t))B0B0∗exp(A0∗(T−t))dtforT>h. In the proofs of the main results we shall also use the deterministic controllability operator $${\it\Gamma}_s^T$$ depending on time $$s\in [0,T]$$. It is defined as: ΓsT =LT(s)LT∗(s) ={∫sTexp(A0(T−t))B0B0∗exp(A0∗(T−t))dtforT≤h∫hTF(T−t)B0B0∗F∗(T−t)+∫shexp(A0(T−t))B0B0∗exp(A0∗(T−t))forT>h. Now we define some definitions and results which will be used in further section: Definition 5.1 Let $$G:[0,T]\times R^n\rightarrow R^{n\times n}$$ be a strongly measurable mapping such that $$\int_{0}^{T}E||G(t)||^p dt<\infty$$. Then E||∫0tG(s)dω(s)||p≤LG∫0tE||G(s)||pds, (5.6) for all $$t\in[0,T]$$ and $$p\geq 2$$, where $$L_G$$ is the constant involving $$p$$ and $$T$$. Definition 5.2 The stochastic dynamic system (5.1) is said to be exactly controllable on $$[0,T]$$ if RT(Uad)=L2(Ω,ϝT,Rn), that is, if all the points in $$L_2({\it\Omega},\digamma_T,R^n)$$ can be exactly reached at time $$T$$ from any arbitrary initial condition $$x_0\in L_2^F([-h,0],L_2({\it\Omega},F_T,R^n))$$. Definition 5.3 A control system is said to be exact controllable in the interval $$I=[0,T]$$ if for every initial state $$x_0$$ and desired final state $$x_1$$, there exists a control $$u(t)$$ such that the solution $$x(t)$$ of the system corresponding to this control u satisfies $$x(T)=x_1$$. Remark 5.1 For dynamical system (5.1) it is possible to define many different concepts of controllability (complete, approximate). Using this admissible controls, Klamka (2000b, 2001a,,b, 2002) obtained complete controllability with constrained admissible controls of non-linear systems. It is generally assumed that the control values are in a convex and closed cone with vertex at zero, or in a cone with non-empty interior. Klamka obtained sufficient conditions for constrained exact local controllability using the generalized open mapping theorem. Let $$U_0\subset U$$ be a closed convex cone with non-empty interior. The set of admissible controls for the system (5.1) is given by $$U_{\rm ad}=L_\infty([0,T],U_0)$$ (for more detail see Klamka, 2000b, 2001a,b, 2002). In this article some sufficient conditions for exact controllability with unconstrained admissible controls of system (5.1) is obtained. Unconstrained admissible control for the system (5.1) in this paper is defined in notation (vii). From equation (5.4) we have ||F(t)||=||I+∫0tF(s)A0ds+∫0t−hF(s)A1ds|| using Gronwall’s inequality ||F(t)||≤exp(t(||A0||+||A1||)) (5.7) let $$l_1=\max (||F(t)||^2) \quad in \quad t\in[0,T]$$. From equation (5.5) we have E||x(t;x0,0)||2 =E||exp(A0t)x0+∫0hF(t−s)A1x0(s−h)ds||2 ≤2(l1||x0||2+||A1||2||ψ(t)||2l1). (5.8) 6. Main results Lemma 7 Assume that the operator $$({\it\Pi}_0^T)$$ is invertible. Then for arbitrary final state $$x_T\in L_2({\it\Omega},\digamma_T,R^n)$$ the control defined as: u(t)={B0∗F∗(T−t))E{(Π0T)−1p(x)|ϝT}fort∈[h,T]B0∗exp(A0∗(T−t))E{(Π0T)−1p(x)|ϝT}fort∈[0,h), (6.1) where $$p(x)=x_T-x(T;x_0,0)-\int_{0}^{T}F(T-s)(f(s,x(s-h))ds+\sigma (s,x(s-h))d\omega(s))$$ transfers the system (5.1) from $$x_0\in R^n$$ to $$x_T$$ at time $$T$$ and x(t;x0,u) =x(t;x0,0)+Π0t[F∗(T−t)(Π0T)−1p(r)] +∫0tF(t−s)f(s,x(s−h))ds+∫0tF(t−s)σ(s,x(s−h))dω(s) (6.2) provided the solution of (6.2) exists. Proof By substituting (6.1) in (5.3) and using definition of $${\it\Pi}_0^T$$,we can easily obtain the following For $$T<h$$ x(t;x0,u) =x(t;x0,0)+∫0tF(t−s)(B0B0∗exp(A0∗(T−s))E{(Π0T)−1p(r)|ϝT}+f(s,x(s−h)))ds +∫0tF(t−s)σ(s,x(s−h))dω(s) =x(t;x0,0)+∫0texp(A0(T−s))(B0B0∗exp(A0∗(T−t))exp(A0∗(t−s)) ×E{(Π0T)−1p(r)|ϝT}+f(s,x(s−h)))ds+∫0tF(t−s)σ(s,x(s−h))dω(s) =x(t;x0,0)+Π0t[F∗(T−t)(Π0T)−1p(r)] +∫0tF(t−s)f(s,x(s−h))ds+∫0tF(t−s)σ(s,x(s−h))dω(s). In the same manner for $$T\geq h$$ x(t;x0,u) =x(t;x0,0)+(∫0hF(t−s)(B0B0∗exp(A0∗(T−s))) +∫htF(t−s)(B0B0∗F∗(T−s)))E{(Π0T)−1p(r)|ϝT}ds +∫0tF(t−s)(f(s,x(s−h))ds+σ(s,x(s−h))dω(s)) =x(t;x0,0)+(∫0hexp(A0(T−s))(B0B0∗exp(A0∗(T−t))exp(A0∗(t−s))) +∫htF(t−s)(B0B0∗F∗(T−t)F∗(t−s)))E{(Π0T)−1p(r)|ϝT}ds +∫0tF(t−s)(f(s,x(s−h))ds+σ(s,x(s−h))dω(s)) =x(t;x0,0)+Π0t[F∗(T−t)(Π0T)−1p(r)] +∫0tF(t−s)f(s,x(s−h))ds+∫0tF(t−s)σ(s,x(s−h))dω(s). Put $$t=T$$ and value of $$p(x)$$ in (6.2) we get x(T;x0,u) =x(T;x0,0)+Π0T[F∗(T−T)(Π0T)−1(xT−x(T;x0,0) −∫0TF(T−s)(f(s,x(s−h)))ds+σ(s,x(s−h))dω(s))] +∫0TF(T−s)f(s,x(s−h))ds+∫0TF(T−s)σ(s,x(s−h))dω(s)x(T;x0,u) =xT. □ Remark 6.1 In Theorem 6.1 sufficient condition are given for the existence and uniqueness of solution of system $$(5.3))$$. Now let us assume the following conditions for obtaining the controllability results $$(A1)$$$$(f,\sigma)$$ satisfies the Lipschitz condition with respect to $$x$$, i.e., ||f(t,x1)−f(t,x2)||2≤L1||x1−x2||2,||σ(t,x1)−σ(t,x2)||2≤L2||x1−x2||2. $$(A2)$$$$(f,\sigma)$$ is continuous on $$[0,T]\times R^n$$ and satisfies ||f(t,x)||2≤L3(||x||2+1),||σ(t,x)||2≤L4(||x||2+1)$. $$(A3)$$ The linear system corresponding to (5.1) is completely controllable. Lemma 8 (Oksendal, 2003) For every $$z\in L_2({\it\Omega},\digamma_T,R^n)$$, there exists a process $$\varphi(.)\in L_2([0,T],R^{n\times n})$$ such that z =Ez+∫0Tφ(s)dω(s)Π0Tz =Γ0TEz+∫0TΓsTφ(s)dω(s). Moreover, E||Π0Tz||2 ≤ME||E{z|ϝT}||2 ≤ME||z||2,z∈L2(Ω,ϝT,Rn). Note that if the Assumption $$(A3)$$ holds, then for some $$\gamma>0$$ E⟨Π0Tz,z⟩≥γE||z||2,forallz∈L2(Ω,ϝT,Rn) and consequently E||(Π0T)−1||2≤1γ=l4(let). Define the operator $$\mathbf{S}$$ for (6.2) for $$t\in[-h,T]$$ as follows S(x)(t){ψ(t)fort∈[−h,0]x(t;x0,0)+Π0t[F∗(T−t)((Π0T)−1×(xT−x(T;x0,0)−∫0TF(T−r)f(r,x(r−h))dr−∫0TF(T−r)σ(r,x(r−h))dω(r))]+∫0tF(t−s)f(s,x(s−h))ds+∫0tF(t−s)σ(s,x(s−h))dω(s)fort∈[0,T]. From Lemma 7, the control $$u(t)$$ transfer the system (6.2) from the initial state $$x_0$$ to the final state $$x_T$$ provided that the operator $$\textbf{S}$$ has a fixed point. So, if the operator $$\textbf{S}$$ has a fixed point then the system (5.1) is completely controllable. Now for convenience, let us introduce the notation l1=max||F(t)||2:t∈[0,T],l2=||B0||2,l3=E||xT||2,M=max||ΓsT||2:s∈[0,T]. Theorem 6.1 Assume that the conditions $$(A1)$$, $$(A2)$$ and $$(A3)$$ hold.In addition if the inequality 4l1(2Ml1l4+1)(L1T+L2Lσ)T<1 (6.3) holds, then the system (5.1) is completely controllable. Proof As mentioned above, to prove the complete controllability it is enough to show that $$\textbf{S}$$ has a fixed point in $$H_2$$. To do this, we use the Banach contraction mapping principle. To apply the contraction mapping principle, first we show that $$\textbf{S}$$ maps $$H_2$$ into itself. Now by Lemma 8 and equations (5.7) and (5.8) we have E||(Sx)(t)||2 =E||ψ(t)+x(t;x0,0)+Π0t[F∗(T−t)×(Π0T)−1(xT−x(T;x0,0) −∫0TF(T−r)f(r,x(r−h))dr−∫0TF(T−r)σ(r,x(r−h))dω(r))] +∫0tF(t−s)f(s,x(s−h))ds+∫0tF(t−s)σ(s,x(s−h))dω(s)||2 ≤B1+B2(∫0t(TE||f(r,x(r−h))||2+LσE||σ(r,x(r−h))||2)dr), where $$B_1>0$$ and $$B_2>0$$ are suitable constants. It follows from the above and the condition $$(A2)$$ that there exists $$C_1>0$$ such that E||(Sx)(t)||2 ≤C1(1+∫−hTE||x(v)||2dv) ≤C1(1+(T+h)sup−h≤v≤TE||x(v)||2) for all $$t\in [-h,T]$$. Therefore $$\textbf{S}$$ maps $$H_2$$ into itself. Secondly, we show that $$\textbf{S}$$ is a contraction mapping. indeed E||(Sx1)(t)−(Sx2)(t)||2 =E||Π0t[F∗(T−t)(Π0T)−1×(∫0TF(T−s)(f(s,x2(s−h))−f(s,x1(s−h)))ds +∫0TF(T−s)(σ(s,x2(s−h))−σ(s,x1(s−h)))dω(s))] +∫0tF(t−s)(f(s,x1(s−h))−f(s,x2(s−h)))ds +∫0tF(t−s)(σ(s,x1(s−h))−σ(s,x2(s−h)))dω(s)||2 ≤4Ml12l4(2T∫0TE||f(s,x1(s−h))−f(s,x2(s−h))||2ds +2Lσ∫0TE||σ(s,x1(s−h))−σ(s,x2(s−h))||2ds) +4l1(T∫0tE||f(s,x1(s−h))−f(s,x2(s−h))||2ds +Lσ∫0tE||σ(s,x1(s−h))−σ(s,x2(s−h))||2ds) ≤4Ml12l4(2L1T+2L2Lσ)∫0TE||x1(s−h)−x2(s−h)||2ds +4l1(L1T+L2Lσ)∫0tE||x1(s−h)−x2(s−h))||2ds ≤4l1(2Ml1l4+1)(L1T+L2Lσ)∫−hT−hE||x1(v)−x2(v)||2dv. It results that supt∈[−h,T]E||(Sx1)(t)−(Sx2)(t)||2≤4l1(2Ml1l4+1)(L1T+L2Lσ)Tsupt∈[−h,T]E||x1(t)−x2(t)||2. Therefore $$\textbf{S}$$ is a contraction mapping if the inequality (6.3) holds. Then the mapping $$\textbf{S}$$ has a unique fixed point $$x(\cdot)$$ in $$H_2$$ which is the solution of the equation (5.1). Thus the system (6.3) is exactly controllable. The theorem is proved. □ 7. Examples Example 1 Consider the stochastic control system with delay governed by the semilinear heat equation ∂y(t,x) = [∂2y(t,x)∂x2+y(t−h,x)+Bu(t,x)+f(t,y(t−h,x))]∂t+∂ω(t)for0<t<τ,0<x<πwithconditionsy(t,0)=y(t,π)=0,0≤t≤τy(t,x) =ξ(t,x),−h≤t≤0,0≤x≤π. (7.1) The system (7.1) can be written in the abstract form (2.1), by setting $$X=L_2(0,\pi)$$ and $$A=\frac{d^2}{dx^2}$$, with domain consisting of all $$y\in X$$ with $$(\frac{d^2y}{dx^2})\in X$$ and $$y(0)=0=y(\pi)$$. Take $$\phi_(x)=(2/\pi)^{1/2}\sin(nx),\; 0\leq x\leq \pi,\;n=1,2,3,\ldots$$, then $$\{\phi_n(x)\}$$ is an orthonormal basis for $$X$$ and $$\phi_n$$ ia an eigenfunction corresponding to the eigenvalue $$\lambda_n=-n^2$$ of the operator $$A$$, $$n=1,2,3,\ldots$$. Then the $$C_0$$-semigroup $$T(t)$$ generated by $$A$$ has $$e^{\lambda_nt}$$ as the eigenvalues and $$\phi_n$$ as their corresponding eigenfunctions. Define an infinite dimensional space $$U$$ by U={u:u=∑n=2∞unϕnwith∑n=2∞un2<∞}. The norm defined by ||u||U=(∑n=2∞un2)1/2$$\xi(t,x)$$ is known function. Let $$B$$ be a continuous linear operator from $$U$$ to $$X$$ defined as Bu=2u2ϕ1+∑n=2∞unϕn,u=∑n=2∞unϕn∈U. The non-linear operator $$f$$ is assumed to satisfy conditions $$(H_3)$$ and $$(H_4)$$. The system (7.1), can be associated with two control systems under the initial and boundary conditions, as given below ∂y(t,x)∂t =∂2y(t,x)∂x2+y(t−h,x) +Bv(t,x)+f(t,y(t−h,x)+z(t−h,x))t∈[0,b]x∈[0,π],y(t,x) =ξ(t,x),−h≤t≤0,0≤x≤π, (7.2) ∂z(t,x) = [∂2z(t,x)∂x2+z(t−h,x)+Bw(t)]∂t+∂ω(t). (7.3) The system (7.3) is a linear stochastic system and for each realization $$z(t)$$ of the system (7.3), the system (7.2) is a deterministic system. From Lemma 6 and using the conditions $$(H_1)\text{--}(H_4)$$, it is clear that for each realization $$z(t)$$ of the system (7.3), the system (7.2) is approximately controllable. The linear system corresponding to (7.2) is approximately controllable from Lemma 3. Example 2 Consider a two-dimensional semilinear stochastic system with delay in state dx(t)=[A0x(t)+A1x(t−h)+B0u(t)+f(t,x(t−h))]dt+σ(t,x(t−h))dω(t);t∈[0,T] (7.4) with initial condition (5.2). $$\omega(t)$$ is a one-dimensional Wiener process and A0 =[−11−1−1],A1=[−1110],B0=[1001]f(t,x(t−h)) =1a[sinx(t−h)x(t−h)],σ(t,x(t−h))=1b[x(t−h)00cosx(t−h)]. Here $$f(t,x(t-h))$$ and $$\sigma(t,x(t-h))$$ are satisfying conditions $$(A1)$$ and $$(A2)$$. For $$x=(x_1,x_2)$$ with the initial value $$x_0$$ and final point $$x_T\in R^2$$. For this system the controllability matrix is Γst=12[1−exp(−2(t−s))001−exp(−2(t−s))]. If we take Euclidean norm then ||A0||=2,||A1||=3,||B0||=2,||Γst||=1−exp(−2(t−s))2>0∀0≤s<t. We can see that conditions of Theorem 6.1 for any time $$T$$ are satisfied. So system (7.4) is exactly controllable. Example 3 Suppose the system dx(t) = (u(t)+x(t−1))dt+dω(t)t∈(0,1/2],x(t) =0−1≤t≤0, (7.5) on splitting the system in deterministic and stochastic systems we have dy(t) = (v(t)+y(t−1)+z(t−1))dty(t) =0−1≤t≤0. (7.6) and stochastic system dz(t) =w(t)dt+dω(t)z(t) =0−1≤t≤0 (7.7) The system represented by (7.7) is linear stochastic system with delay in state and for each realization $$z(t)$$ of system (7.7), the system given by (7.6) is a deterministic system. Thus the solution $$y(t)$$ of the semilinear system (7.6) depends on the solution $$z(t)$$ of linear stochastic system (7.6). The functions $$v$$ and $$w$$ are $$Y$$-valued control function, such that $$u=v+w$$. For time interval $$[0,1/2]$$ dy(t)=0+v(t)dt. Therefore at final time $$t=1/2$$ y(1/2)=∫01/2v(t)dt=10. So for control $$v(t)=20$$, the system is controllable. Hence Provided conditions are only sufficient not necessary. Because here non-linear term did not satisfy Lipschitz condition (assumption $$(H_3)$$). 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IMA Journal of Mathematical Control and Information – Oxford University Press

**Published: ** Nov 10, 2016

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