Abstract In this article, sufficient conditions have been obtained for the controllability of retarded semilinear fractional system with non-local conditions using the compactness of the non-linear function. The Nussbaum fixed point theorem combined with the theory of strongly continuous semigroup and fractional calculus are the main tools used in this problem. Finally, some examples have been provided to illustrate the application of the obtained results. 1. Introduction The theory of fractional differential equations is an important branch of the theory of differential equations, which is used in the modelling of many physical phenomena in various fields of engineering, physics, economics and science and has many applications in viscoelasticity, electrochemistry, control, porous media, etc., see Debnath (2003) and the references therein. These fractional differential equations are considered as alternative models to non-linear differential equations which induced extensive research work in various theoretical and applied fields, see Podlubny (1999). It is well-known that the concept of controllability plays an important role in control theory and engineering. Controllability of the deterministic linear and non-linear control systems are well-developed when the state space is infinite-dimensional. There are different types of controllability like complete controllability, approximate controllability, boundary controllability, constrained controllability, etc. Bragdi & Hazi (2010) discussed the existence and controllability of an evolution fractional semilinear integrodifferential system and Matar (2010) proved the controllability of fractional semilinear mixed integrodifferential systems with non-local conditions. Wang et al. (2010) discussed the existence of mild solution and optimal control of fractional non-local integrodifferential equations using Banach contraction mapping principle and Krasnosel’skii’s fixed point theorem via Gronwall’s inequality. Ahmed (2010) established sufficient conditions for boundary controllability of non-linear fractional integro differential systems in Banach spaces using Schauder’s fixed point theorem. Wang & Zhou (2011) proved the existence of mild solutions for semilinear fractional evolution equations and optimal controls in the q-norm. They introduced a suitable $$q$$-mild solution of the semilinear fractional evolution equation. Zhou & Jiao (2010) discussed the non-local Cauchy problem for the fractional equations in an arbitrary Banach space in which the mild solution is introduced based on the probability density function and semigroup theory. Sakthivel et al. (2011) considered a class of control systems governed by the semilinear fractional differential equations in Hilbert spaces. They gave a set of sufficient conditions for approximate controllability of semilinear fractional differential systems under the assumption that the associated linear system is approximate controllable. Sakthivel et al. (2012) established sufficient conditions for the controllability of non-linear fractional neutral evolution systems by using a fixed point analysis approach under the natural assumption that the associated linear control system is exact controllable. The constrained controllability problems for infinite-dimensional linear and non-linear dynamical systems in the differential state equations has been studied by Klamka (1995a,b, 1996a,b). In Klamka (1996b), obtained sufficient conditions for constrained exact local controllability using the generalized open mapping theorem. It is generally assumed that the values of controls are in a convex and closed cone with vertex at zero or in a cone with non-empty interior. Now, in the last few decades, there has been an expanding interest in the problems involving retarded systems. Retarded Systems are the systems having retarded arguments. Many real life problems that have in the past, sometimes been modelled by initial value problems for differential equations actually involve a significant memory effect that can be represented in a more refined model, using a differential equation incorporating retarded or delayed arguments (arguments that ‘lag behind’ the current value). Therefore it becomes necessary and important to consider retarded systems as these systems have found many applications in mathematical physics, biology and finance. Shukla et al. (2015) studied the approximate controllability of semilinear retarded stochastic system with non-local conditions. Klamka (1995a) investigated the constrained controllability for retarded dynamical systems described by abstract differential equations with unbounded control operator. On the other hand, Byszewski & Lakshmikantham (1991) introduced non-local conditions into the initial value problems and argued that the corresponding models more accurately describe the phenomena since more information was taken into account at the onset of the experiment, thereby reducing the ill effects incurred by a single initial measurement. Non-local initial conditions have various applications in fields such as blood flow problems, chemical engineering, thermoelasticity, underground water flow, population dynamics, etc. For more details on non-local conditions, see Wang & Wei (2010) and the references therein. Arora & Sukavanam (2015) obtained the sufficient conditions for the approximate controllability of second order semilinear stochastic system with non-local conditions using Sadovskii’s fixed point theorem. Most of the aforementioned works require the assumption of compactness of semigroup generated by the linear part of these systems. Also they have used uniform boundedness of the non-linear function. But in this article, we have removed these assumptions. The main results in the paper are motivated by the work of Chang et al. (2009). They derived a set of novel sufficient conditions for the controllability of a class of first order semilinear differential systems with non-local initial conditions by using non-compactness technique and Sadovskii’s fixed point theorem. Motivated by this consideration, in the present problem, it has been shown that for a compact non-linear function, the fractional order retarded semilinear system with non-local conditions is controllable on $$J=[0,b]$$. Here, we discuss the controllability of the system by dropping the compactness of the semigroup and uniform boundedness of non-linear function, which are main assumptions of earlier work (Ahmed, 2010; Sakthivel et al., 2011). A new set of sufficient conditions are established for controllability of the system using Nussbaum fixed point theorem. 2. Preliminaries Let $$V, \hat{V}$$ be two Banach spaces and $$Z=L_2([0,b];V)$$. Let $$C$$ be the Banach space of all continuous functions from $$[-h,0]$$ to $$V$$ with the supremum norm |x|=supt∈[−h,0]{||x(t)||}. In this paper, we examine the controllability of the following fractional order retarded semilinear system with non-local conditions of the form: cDtαx(t)=Ax(t)+Bu(t)+f(t,xt);t∈J=[0,b]x(t)=ψ(t);t∈[−h,0] and x(0)=ψ(0)=x0+p(x)}, (2.1) where $$0<\alpha<1$$; $$^cD_t^\alpha$$ denotes the Caputo fractional derivative operator of order $$\alpha$$; state $$x(.)$$ takes values in Banach space V and the control function u(.) is given in $$L_2(J, \hat{V})$$, the Banach space of admissible control functions. Let $$A:D(A)\subset V\rightarrow V$$ generates a strongly continuous semigroup $$\{T_0(t),\;t\in\mathbb{R}\}$$. $$B:L_2([0,b];\hat{V})\rightarrow L_2([0,b],V)$$ is a bounded linear operator. $$x_t\in L^2([-h,0],V)$$ and is defined as $$x_t(s)=\{x(t+s)|-h\leq s\leq 0|\}$$ and $$\psi=\{\psi(s)|-h\leq s\leq 0\}\in L^2([-h,0],V)$$. Moreover, the function $$f:J\times C([-h,0];V)\rightarrow V$$ is a compact non-linear function and satisfy caratheodory conditions (that is measurable with respect to $$t$$ for all $$x\in V$$ and continuous with respect to $$x$$ for almost all $$t\in J$$). The function $$p:C[J,V]\rightarrow V$$ is a non-linear function and $$x_0$$ is an element of $$V.$$ By using some constructive control function, we transfer the controllability problem for fractional order retarded system into a fixed point problem for an appropriate non-linear operator in a function space. Using Nussbaum fixed point theorem, we guarantee the existence of a fixed point of this operator and study the controllability of the system (2.1). In particular, in this paper we derive results on exact controllability of fractional order retarded semilinear system with non-local conditions by assuming the compactness of the non-linear function. First, let us recall some basic definitions and lemmas which are used throughout this paper and then a set of hypothesis $$(H_1)$$–$$(H_5)$$ has been presented to establish the controllability result. A measurable function $$x:J\rightarrow V$$ is Bochner integrable if and only if $$||x||$$ is Lebesgue integrable. For the properties of the Bochner integral, see Yosida (1980). Let $$L_1[J;V]$$ denotes the Banach space of functions $$x:J\rightarrow V$$, which are Bochner integrable with the norm |x|L1=∫0b||x(t)||dt. Definition 2.1 (Miller & Ross, 1993; Balachandran & Park, 2009) A real function $$f(t)$$ is said to be in the space $$C_{\alpha}$$, $$\alpha\in\mathbb{R}$$ if there exists a real number $$q>\alpha$$, such that $$f(t)=t^{q}g(t)$$, where $$g\in C[0,\infty)$$ and it is said to be in the space $$C_\alpha^m$$ if $$f^{(m)}\in C_\alpha$$, $$m\in \mathbb{N}$$. Definition 2.2 (Miller & Ross, 1993; Balachandran & Park, 2009) The fractional integral of order $$\alpha\in \mathbb{R}^{+}$$ with the lower limit $$0$$ for a function $$f$$ is defined as Iαf(t)=1Γ(α)∫0tf(s)(t−s)1−αds,t>0,α>0 provided the right-hand side is point-wise defined on $$[0,\infty)$$, where $$\Gamma(.)$$ is gamma function. Definition 2.3 (Miller & Ross, 1993; Balachandran & Park, 2009) The Riemann–Liouville derivative of order $$\alpha\in \mathbb{R}^{+}$$ with lower limit zero for a function $$f:[0,\infty)\rightarrow \mathbb{R}$$ can be written as LDαf(t)=1Γ(n−α)dndtn∫0tf(s)(t−s)α+1−nds,t>0,n−1<α<n. Definition 2.4 The Caputo derivative of order $$\alpha\in \mathbb{R}^{+}$$ for a function $$f:[0,\infty)\rightarrow \mathbb{R}$$ can be written as cDαf(t)=LDα(f(t)−f(0)),t>0,0<α<1. Remark 2.1 (i) If $$f(t)\in C^n[0,\infty)$$, then cDαf(t)=1Γ(n−α)∫0t(t−s)n−α−1fn(s)ds=In−αfn(s),t>0,n−1≤α<n. (ii) The Caputo derivative of a constant is equal to zero. (iii) If $$f$$ is an abstract function with values in $$V$$, then integrals which appear in Definitions 2.2 and 2.3 are taken in Bochner’s sense. More details about the theory of fractional differential equations can be seen in Miller & Ross (1993), Podlubny (1999) and Kilbas et al. (2006). 3. Existence of mild solution In this section, we define the mild solution of the system (2.1) and then the sufficient conditions for the controllability of the system (2.1) have been established. For this, first we find the mild solution of the non-local Cauchy problem described by the following differential equation dαx(t)dtα=Ax(t)+f(t,x(t));t∈J=[0,b],α∈(0,1),x(0)=x0+p(x).} (3.1) According to the Definitions (2.1)–(2.4), the above Cauchy problem (3.1) can be rewritten in the equivalent form of the integral equation x(t)=x0+1Γ(α)∫0t(t−s)α−1[Ax(s)+f(s,x(s))]ds, (3.2) provided the integral (3.2) exists. Let γ(μ)=∫0∞e−μsx(s)ds and ω(μ)=∫0∞e−μsf(s,x(s))ds,μ>0. (3.3) Now taking the Laplace transformation on both sides of the equation (3.2), we have γ(μ)=1μ[x0+p(x)]+1μαAγ(α)+1μαω(μ)⇒(μαI−A)γ(μ)=μα−1[x0+p(x)]+ω(μ)⇒γ(μ)=μα−1(μαI−A)−1[x0+p(x)]+(μαI−A)−1ω(μ)=μα−1∫0∞e−μαsT0(s)[x0+p(x)]ds+∫0∞e−μαsT0(s)ω(μ)ds. (3.4) Consider the one-dimensional stable probability density function, see Mainardi et al. (2003), ψα(ξ)=1π∑n=1∞(−1)n−1ξ−αn−1Γ(nα+1)n!sin(nπα),ξ∈(0,∞), (3.5) whose Laplace transformation is given by ∫0∞e−μξψα(ξ)dξ=e−μα,whereα∈(0,1). (3.6) Let I1=μα−1∫0∞e−μαsT0(s)[x0+p(x)]ds=μα−1∫0∞αtα−1e−(μt)αT0(tα)[x0+p(x)]dt=−1μ∫0∞ddt(e−(μt)α)T0(tα)[x0+p(x)]dt. Using (3.6), we get I1=∫0∞∫0∞ξe−(μtξ)ψα(ξ)T0(tα)[x0+p(x)]dξdt=∫0∞e−(μt)[∫0∞ψα(ξ)T0(tαξα)[x0+p(x)]dξ]dt. (3.7) Let I2=∫0∞e−μαtT0(t)ω(μ)dt. Using (3.3), we get I2=∫0∞e−μαtT0(t)∫0∞e−μsf(s,x(s))dsdt=∫0∞αtα−1e−(μt)αT0(tα)∫0∞e−μsf(s,x(s))dsdt. From (3.6), we get I2=∫0∞∫0∞∫0∞αtα−1e−(μtξ)T0(tα)e−μsf(s,x(s))dξdsdt=α∫0∞∫0∞∫0∞(tξ)α−1e−(μt)ψα(ξ)T0(tαξα)e−μsf(s,x(s))dξdsdt=α∫0∞e−(μt)[∫0t∫0∞(t−sξ)α−1T0((t−s)αξα)ψα(ξ)f(s,x(s))dξds]dt. (3.8) From equations (3.4), (3.7) and (3.8), we have γ(μ)=∫0∞e−(μt)[∫0∞T0(tαξα)ψα(ξ)[x0+p(x)]dξ+α∫0t∫0∞(t−sξ)α−1T0((t−s)αξα)ψα(ξ)f(s,x(s))dξds]dt. (3.9) Taking inverse Laplace transformation on both sides, we have x(t)=∫0∞T0(tαξα)ψα(ξ)[x0+p(x)]dξ+α∫0t∫0∞(t−sξ)α−1T0((t−s)αξα)ψα(ξ)f(s,x(s))dξds=∫0∞ζα(ξ)T0(tαξ)[x0+p(x)]dξ+α∫0t∫0∞ξ(t−s)α−1ζα(ξ)T0((t−s)αξ)f(s,x(s))dξds, (3.10) where $$\zeta_\alpha(\xi)=\frac{1}{\alpha}\xi^{-1-\frac{1}{\alpha}}\psi_\alpha(\xi^{-\frac{1}{\alpha}})$$ is the probability density function defined on $$(0,\infty)$$ and ∫0∞ξζα(ξ)dξ=∫0∞1ξαψα(ξ)dξ=1Γ(1+α). (3.11) Due to above expansions, we give the following definition of the mild solution of system (2.1): Definition 3.1 The mild solution of the system (2.1) for a given control $$u\in L_2[J;\hat{V}]$$ can be written as x(t)={Tα^(t)[x0+p(x)]+∫0t(t−s)α−1Tα(t−s)[Bu(s)+f(s,xs)]dsfort>0ψ(t)fort∈[−h,0], (3.12) where $$\widehat{T_\alpha}(t)=\int_{0}^{\infty}\zeta_\alpha(\xi)T_0(t^\alpha \xi)d\xi$$ and $$T_\alpha(t)=\alpha\int_{0}^{\infty}\xi \zeta_\alpha(\xi)T_0(t^\alpha \xi)d\xi$$. Definition 3.2 The system (2.1) is said to be controllable over the time interval $$J$$, if for any given final state $$x_F\in V$$, there exists a control function $$u\in L_2[J;\hat{V}]$$ such that the corresponding mild solution $$x(t)$$ of the system (2.1) satisfies $$x(b)=x_F$$. Lemma 3.1 (Wang & Zhou, 2011) For any fixed $$t\geq 0$$, the operators $$\widehat{T_\alpha}(t)$$ and $$T_\alpha(t)$$ are linear and bounded, that is, for any $$x\in V$$, $$||\widehat{T_\alpha}(t)x||\leq M||x||$$ and $$||T_\alpha(t)x||\leq \dfrac{M \alpha}{\Gamma(1+\alpha)}||x||$$, where M is a constant such that $$||T_0(t)||\leq M$$ for all $$t\geq 0$$. To prove the controllability of the system (2.1), the following conditions are assumed: $$(H_1)$$$$A$$ is the infinitesimal generator of a strongly continuous semigroup $$\{T_0(t): t\geq 0\}$$ of bounded linear operators on $$V$$. $$(H_2)$$The linear operator $$W:L_2[J;\hat{V}]\rightarrow V$$ defined by Wu=1Γ(α)∫0b(b−s)α−1T0(b−s)Bu(s)ds, has an invertible operator $$\widetilde{W}$$ defined on $$L_2[J;\hat{V}]/\;{\rm ker}\;W$$ and there exists a positive constant $$K>0$$, such that $$|B\widetilde{W}^{-1}|\leq K$$. For more detail of $$\widetilde{W}$$, see Quinn & Carmichael (1985). $$(H_3)$$The function $$f:J\times V\rightarrow V$$ satisfy Lipschitz condition, i.e., there exists a function $$L_f\in L_1[J;\mathbb{R}^{+}]$$, such that ||f(t,xt)−f(t,yt)||≤Lf(t)||xt−yt||;forallxt,yt∈L2[−h,0;V]. $$(H_4)$$There exists a constant $$L_p>0$$, such that ||p(x)−p(y)||≤Lp|x−y|;x,y∈C[J;V]. $$(H_5)$$$$M\bigg( L_p +\dfrac{b^{1+\alpha}}{\Gamma(1+\alpha)}|L_f|_{L^1}(b+h)\bigg) \bigg(1+\dfrac{M K b^{1+\alpha}}{\Gamma(1+\alpha)}\bigg)<1$$ The Nussbaum fixed point theorem as given below will be used in next section to establish our results. Theorem 3.1 (Nussbaum, 1969) Let $$N$$ be a closed, bounded and convex subset of a Banach space $$\mathbb{H}$$. Let $$\it {\Phi}_1$$, $$\it {\Phi}_2$$ be continuous mappings from $$N$$ into $$\mathbb{H}$$ such that (1) $$(\it {\Phi}_1+\it {\Phi}_2)N\subset N$$, (2) $$||\it {\Phi}_1 x-\it {\Phi}_1 y||\leq k||x-y||$$ for all $$x,y\in N$$ where $$0\leq k<1$$ is a constant, (3) $$\overline{\it {\Phi}_2[N]}$$ is compact. Then the operator $$\it {\Phi}_1+\it {\Phi}_2$$ has a fixed point in $$N$$. 4. Controllability result In this section, we derive the main result concerning the mild solution of the system (2.1) under the assumptions $$(H_1)$$–$$(H_5)$$ of $$f, p$$ and $$T_0(t)$$. Theorem 4.1 Under the assumptions $$(H_1)$$–$$(H_5)$$, the fractional order semilinear system (2.1) is controllable in the time interval $$J$$. Proof Using condition $$(H_2)$$, for an arbitrary function $$x(\cdot)$$, define the control u(t)=W^−1[xF−Tα^(b)[x0+p(x)]−∫0b(b−s)α−1Tα(b−s)f(s,xs)ds](t). (4.1) This control is substituted into the equation (3.12) to define an operator $$\it {\Phi}$$ by (Φx)(t)=Tα^(t)(x0+p(x))+∫0t(t−s)α−1Tα(t−s)BW^−1×[xF−Tα^(b)(x0+p(x))−∫0b(b−s1)α−1Tα(b−s1)f(s1,xs1)ds1]ds+∫0t(t−s)α−1Tα(t−s)f(s,xs)ds. (4.2) Clearly $$(\it {\Phi} x_b)=x_F$$, which means that the control u steers the system from the initial state $$x(0)$$ to $$x_F$$ in time b, provided we can obtain a fixed point of the non-linear operator $$\it {\Phi}$$. For that we define, (Φ1x)(t)=Tα^(t)(x0+p(x))+∫0t(t−s)α−1Tα(t−s)×BW^−1[xF−Tα^(b)[x0+p(x)]−∫0b(b−s1)α−1Tα(b−s1)f(s1,xs1)ds1]ds and (Φ2x)(t)=∫0t(t−s)α−1Tα(t−s)f(s,xs)ds. Define the set Bλ={x|x∈C[J,V],|x|≤λ for each t∈J}, then for each $$\lambda$$, $$B_\lambda$$ is obviously a bounded, closed and convex set in $$C[J,V]$$. From the definition, we have Φx=(Φ1+Φ2)x. □ Step-1: Now, we claim that there exists a positive number $$\lambda$$ such that $$\it {\Phi}(B_\lambda)\subseteq B_\lambda$$. If it is not true, then for each positive number q, there is a function $$x^{\lambda}\in B_\lambda$$ but $$\it {\Phi} (x^{\lambda})$$ does not belong to $$B_\lambda$$, that is $$||(\it {\Phi} x^{\lambda})(t)||>\lambda$$. On the other hand, from assumptions $$(H_1)$$–$$(H_4)$$ and Lemma 3.1, we have, λ<||(Φxλ)(t)||=||Tα^(t)(x0+p(xλ))+∫0t(t−s)α−1Tα(t−s)BW~−1×[xF−Tα^(t)(x0+p(xλ))−∫0b(b−s1)α−1Tα(b−s1)f(s1,xs1λ)ds1]ds+∫0t(t−s)α−1Tα(t−s)f(s,xsλ)ds||≤M[||x0||+Lp|xλ|+||p(0)||]+MαΓ(1+α)Kbαα∫0t[||xF+M{||x0||+Lp|xλ|+||p(0)||}]ds+(Mα)2(Γ(1+α))2K(bα)2(α)2∫0t∫0b||f(s1,xs1λ)−f(s1,0)+f(s1,0)||ds1ds+MαΓ(1+α)bαα∫0t||f(s,xsλ)−f(s,0)+f(s,0)||ds≤M[||x0||+Lp|xλ|+||p(0)||]+MKbαΓ(1+α)b[||xF+M{||x0||+Lp|xλ|+||p(0)||}]+M2Kb2α(Γ(1+α))2b[∫0bLf(s1)||xs1λ||ds1+∫0b||f(s1,0)||ds1]+MbαΓ(1+α)[∫0tLf(s)||xsλ||ds+∫0t||f(s,0)||ds]≤M[||x0||+Lp|xλ|+||p(0)||]+MKb1+αΓ(1+α)[||xF+M{||x0||+Lp|xλ|+||p(0)||}]+M2Kb1+2α(Γ(1+α))2|Lf|L1b(b+h)|xλ|+M2Kb1+2α(Γ(1+α))2∫0b||f(s1,0)||ds1+MbαΓ(1+α)|Lf|L1b(b+h)|xλ|+MbαΓ(1+α)∫0b||f(s,0)||ds≤M[||x0||+||p(0)||+Kb1+αΓ(1+α){||xF||+M||x0||+M||p(0)||}]+[MLp(1+MKb1+αΓ(1+α))+Mb1+αΓ(1+α)|Lf|L1(b+h)(1+MKb1+αΓ(1+α))]|xλ|+MbαΓ(1+α)(1+MKb1+αΓ(1+α))∫0b||f(s,0)||ds. Since ∫0b||xs||ds≤∫0b∫−h0||x(s+r)||drds=∫0b∫s−hs||x(v)||dvds, where s+r=v=∫0b∫−hb||x(v)||dvds≤b(b+h)supt∈[−h,b]||x(v)||≤b(b+h)|x|. Thus we get λ<||(Φxλ)(t)||≤M[||x0||+||p(0)||+Kb1+αΓ(1+α){||xF||+M||x0||+M||p(0)||}]+MbαΓ(1+α)(1+MKb1+αΓ(1+α))∫0b||f(s,0)||ds+M(Lp+b1+αΓ(1+α)|Lf|L1(b+h))(1+MKb1+αΓ(1+α))λ. Dividing both sides by $$\lambda$$ and taking the limit as $$\lambda\rightarrow \infty$$, we get M(Lp+b1+αΓ(1+α)|Lf|L1(b+h))(1+MKb1+αΓ(1+α))>1. This contradicts the condition $$(H_5)$$. Hence for some positive number $$\it {\Phi}(B_\lambda)\subseteq B_\lambda$$. Step-2: In this step, we show that $$\it {\Phi}_1$$ is a contraction mapping on $$B_\lambda$$. For it, let $$x, y\in B_\lambda$$, then for each $$t\in J$$, we have ||(Φ1x)(t)−(Φ1y)(t)||=||Tα^(t)[p(x)−p(y)]−∫0t(t−s)α−1Tα(t−s)BW~−1×[Tα^(b)[p(x)−p(y)]+∫0b(b−s1)α−1Tα(b−s1){f(s1,xs1)−f(s1,ys1)}ds1]ds||≤MLp|x−y|+MKbαΓ(1+α)[bMLp|x−y|+MbαΓ(1+α)∫0bLf(s1)||xs1−ys1||ds1]≤MLp|x−y|+MKbαΓ(1+α)[MbLp|x−y|+MbαΓ(1+α)|Lf|L1b(b+h)|x−y|]≤M[Lp+MKb1+αΓ(1+α)Lp+MKb2(1+α)(Γ(1+α))2(b+h)|Lf|L1]|x−y|. Since ∫0b||xs−ys||ds≤∫0b∫−h0||x(s+r)−y(s+r)||drds=∫0b∫s−hs||x(v)−y(v)||dvds, where s+r=v=∫0b∫−hb||x(v)−y(v)||dvds≤b(b+h)supt∈[−h,b]||x(v)−y(v)||≤b(b+h)|x−y|. So, we have ||(Φ1x)(t)−(Φ1y)(t)||≤M[Lp+MKb1+αΓ(1+α)Lp+MKb2(1+α)(Γ(1+α))2(b+h)|Lf|L1]|x−y|. Taking supremum over $$t\in [0,b]$$, |Φ1x−Φ1y|≤M[Lp+MKb1+αΓ(1+α)Lp+MKb2(1+α)(Γ(1+α))2(b+h)|Lf|L1]|x−y|. By the condition $$(H_5)$$, M[Lp+MKb1+αΓ(1+α)Lp+MKb2(1+α)(Γ(1+α))2(b+h)|Lf|L1]<1. Hence $$\it {\Phi}_1$$ is a contraction mapping on $$B_\lambda$$. Step-3: Now we prove that $$\it {\Phi}_2$$ is a continuous mapping on $$C[J,V]$$. For it, let $${x_n}$$ be a sequence in $$C[J,V]$$, which converges to $$x$$. From the continuity of $$f$$, f(s,xns)→f(s,xs). Using Lebesgue’s Convergence Theorem, ||Φ2xn−Φ2x||C[J,V]=||∫0t(t−s)α−1Tα(t−s){f(s,xns))−f(s,xs)}ds||≤MbαΓ(1+α)∫0t||f(s,xns))−f(s,xs)||ds,$$\rightarrow 0 \mbox{ as } n\rightarrow \infty$$. Thus we obtain $$\it {\Phi}_2 x_n\rightarrow \it {\Phi}_2 x$$ in $$C[J,V]$$. Step-4: Now, we claim that $$\it {\Phi}_2$$ maps bounded sets into equicontinuous sets of $$C[J,V]$$. For it, let $$\epsilon>0$$ be given and $$t\in J$$. Then for any $$x\in B_\lambda$$, we have ||(Φ2x)(t+ϵ)−(Φ2x)(t)||=||∫0t+ϵ(t+ϵ−s)α−1Tα(t+ϵ−s)f(s,xs)ds−∫0t(t−s)α−1Tα(t−s)f(s,xs)ds||≤MbαΓ(1+α)∫tt+ϵ||f(s,xs)−f(s,0)+f(s,0)||ds+∫0t{(t+ϵ−s)α−1−(t−s)α−1}×||{Tα(t+ϵ−s)−Tα(t−s)}×f(s,xs)||ds≤MbαΓ(1+α)[∫tt+ϵLf(s)||xs||ds+∫tt+ϵ||f(s,0)||ds]λ+[(t+ϵ)α−tα−ϵαα]×∫0t||{Tα(t+ϵ−s)−Tα(t−s)}×f(s,xs)||ds≤MbαΓ(1+α)[|Lf|L1b(b+h)|x|+∫tt+ϵ||f(s,0)||ds]+[(t+ϵ)α−tα−ϵαα]×∫0t||{Tα(t+ϵ−s)−Tα(t−s)}×f(s,xs)||ds. Since $$f$$ is compact, therefore $$||\{T_\alpha(t+\epsilon-s)-T_\alpha(t-s)\}\times f(s,x_s)||$$ tends to zero(as $$\epsilon\rightarrow 0$$) uniformly for $$s\in [-h,b]$$ and $$x\in B_\lambda$$. This implies that, for any $$\epsilon_1>0$$, there exists $$\delta>0$$ such that ||{Tα(t+ϵ−s)−Tα(t−s)}×f(s,xs)||≤ϵ1 for $$0\leq \epsilon<\delta$$ and $$x\in B_\lambda$$. So, we get ||(Φ2x)(t+ϵ)−(Φ2x)(t)||≤MbαΓ(1+α)[|Lf|L1.b(b+h)λ+∫tt+ϵ||f(s,0)||ds]λ+bϵ1[(t+ϵ)α−tα−ϵαα] for $$0\leq \epsilon <\delta$$ and for all $$x\in B_\lambda$$. Therefore $$\it {\Phi}_2(B_\lambda)\subset C[J,V]$$ is equicontinuous. Step-5: Finally, we show that $$Y=\{\it {\Phi}_2 x(t):x\in B_\lambda\}$$ is precompact in $$V$$. Let $$t>0$$ be fixed and $$\{\it {\Phi}_2 x_n(t):x_n\in B_\lambda\}$$ be a bounded sequence in $$B_\lambda$$. Since $$x_n\in B_\lambda$$, $$\{x_n\}$$ is a bounded sequence in $$C[J,V]$$. Thus for any $$t^*\in J$$, the sequence $$\{x_n(t^*)\}$$ is bounded in $$B_\lambda$$. Since $$f$$ is compact, then it has a convergent subsequence such that f(t∗,xnt∗)→f(t∗,xt∗), or ||f(t∗,xnt∗)−f(t∗,xt∗)||→0 as n→∞. Using bounded convergence theorem, it can be seen that (Φ2xn)(t)→(Φ2x)(t) in Bλ. This proves that $$\it {\Phi}_2$$ is compact operator. Thus all the conditions of Nussbaum fixed point theorem are satisfied. Hence by Nussbaum fixed point theorem, $$\it {\Phi}_1+\it {\Phi}_2$$ has a fixed point $$x(.)$$ in $$B_\lambda$$, which is a mild solution of the system (2.1). This shows that the system (2.1) is controllable on $$J$$. 5. Example Example 2.1 Consider the following retarded fractional differential equation with non-local conditions ∂α∂tαz(t,y)=∂2∂y2z(t,y)+Bu(t,y)+p(t,zt);t∈J=[0,b],0<y<πz(t,0)=z(t,π)=0,0≤t≤bz(t,y)=ψ(y),−h≤t<0,0≤y≤πz(0,y)+∑k=1mckz(tk,y)=z0(y),0≤y≤π}, (5.1) where $$0<\alpha<1$$ and $$c_k>0$$. Let $$V=\hat{V}=L_2[0,\pi]$$. $$B$$ is a bounded linear operator from a Hilbert space $$\hat{V}$$ into $$V$$, $$x_t\in L^2([-h,0],V)$$ and is defined as $$x_t(s)=\{x(t+s)|-h\leq s\leq 0\}$$ and $$\psi=\{\psi(s)|-h\leq s\leq 0\}$$. $$u(t)$$ is a feedback control and $$p:J\times V\rightarrow V$$ is a non-linear function. Define $$A:D(A)\subseteq V\rightarrow V$$ by Aw=w″,w∈D(A), where $$D(A)=\{w\in V: w,w_y \mbox{ are absolutely continuous }, w_{yy}\in V, w(0)=w(\pi)=0\},$$ then $$A$$ has spectral representation Aw=∑n=1+∞−n2<w,wn>wn,w∈D(A), where $$w_n(s)=(2/\pi)^{1/2}\sin ns$$, $$n=1,2,3,\ldots$$ is the orthogonal set of eigenfunctions of $$A$$. Further it can be shown that $$A$$ is the infinitesimal generator of a strongly continuous semigroup $$\{T_0(t):t\in \mathbb{R}\}$$ defined on $$V$$ which is given by T0(t)w=∑n=1+∞exp(−n2t)<w,wn>wn,w∈V. Let $$f:J\times V\rightarrow V$$ be defined by f(t,xt)(y)=p(t,zt(y)),(t,zt)∈J×V,y∈[0,π]. The function $$g:C(J,V)\rightarrow V$$ is defined as g(z)(y)=∑k=1mckz(tk,y) for $$0<t_k<b$$ and $$y\in [0,\pi]$$. Under the above assumptions, the system (5.1) is converted into the abstract form of (2.1). Assume that the operator $$W$$ defined as (Wμ)(y)=1Γ(α)∑n=1∞∫0b(b−s)α−1exp(−n2(b−s))(μ(t,y),wn)wnds, has a bounded invertible operator $$\widetilde{W}$$ defined on $$L_2[J;U]/{\rm Ker} W$$. Thus the system (5.1) is controllable on $$J$$ by theorem 4.1 for a class of non-linear functions satisfying the conditions $$(H_2)$$–$$(H_5)$$. Example 2 Consider the electric circuit shown in Fig. 1 with given resistances $$R_1,R_2, R_3$$ inductances $$L_1,L_2$$ and a non-linear device $$N$$ (for example, diode, non-linear resistor, etc.) connected to a source voltage $$u(t)$$. Let the non-linear device produce a voltage $$k(i_2(t))$$, where $$k$$ is a non-linear function of $$i_2$$ and satisfies Lipsctitz condition. Apply Kirchhoff’s law in closed loop (I) (Kaczorek, 2011), we get u(t)=i1(t)R1+(i1(t)−i2(t))R3+L1dαi1(t)dtα⇒dαi1(t)dtα=−(R1+R3)L1i1(t)+R3L1i2(t)+u(t)L1. (5.2) Apply Kirchhoff’s law in closed loop (II), we get 0=i2(t)R2+L2i2(t)dtα+k(i2(t))−(i1(t)−i2(t))R3⇒dαi2(t)dtα=R3L2i1(t)−(R2+R3)L2i2(t)−k(i2(t))L2. (5.3) Writing equations (5.2) and (5.3) in the matrix form dαdtα[i1(t)i2(t)]=[−(R1+R3)L1R3L1R3L2−(R2+R3)L2][i1(t)i2(t)]+[1L10]u(t)+[0−k(i2(t))L2]. Denoting $$x(t)=[i_1(t),i_2(t)]^{T}$$, the above system can be written as dαx(t)dtα=Ax(t)+Bu(t)+k(t,x(t)), (5.4) where $$A=\left[\begin{array}{@{}rr@{}} -\dfrac{(R_1+R_3)}{L_1} & \dfrac{R_3}{L_1} \\ \dfrac{R_3}{L_2} & -\dfrac{(R_2+R_3)}{L_2} \end{array}\right]$$, $$B=\left[\begin{array}{@{}rr@{}} \dfrac{1}{L_1} \\ 0 \end{array}\right]$$ and $$k(t,x(t))=\left[\begin{array}{@{}rr@{}} 0 \\ -\dfrac{k(i_2(t))}{L_2} \end{array}\right].$$ It can be easily seen that the linear system corresponding to (5.4) is controllable as the rank of the matrix $$[B,AB]$$ is $$2$$, see Das (2011) and Monje et al. (2010). So, the controllability Grammian matrix is non-singular. We can define the control function for the semilinear system using the inverse of the controllability Grammian matrix. Also the non-linear function satisfies the Lipsctitz condition, therefore the semilinear fractional system (5.4) is controllable using Theorem 4.1 for $$0<\alpha<1$$. Fig. 1. View largeDownload slide Electrical control system. Fig. 1. View largeDownload slide Electrical control system. Now, we give an example of a system which does not satisfy the assumptions of Theorem 4.1 but is still controllable Example 3 Consider the system cDtαx(t)=(u(t)+x(t−1),0<t≤12x(t)=0 for −1≤t≤0}. (5.5) Let us take $$\alpha=\frac{1}{2}$$. Here we have $$f(t,x(t))=\sqrt{x(t)}$$, which does not satisfy Lipschitz condition but the system (5.5) is controllable, as for the time interval $$t=[0,\frac{1}{2}]$$, the system (5.5) reduces to cDtαx(t)=u(t). The solution of the above system is x(t)=1Γ(α)∫0t(t−s)α−1u(s)ds,=1Γ(12)∫0t(t−s)−1/2u(s)ds. Now, at the final time $$t=\frac{1}{2}$$ and for control $$u(t)=20$$, we get $$x(\frac{1}{2})=20\sqrt{\frac{2}{\pi}}=15.9577$$. Thus for control $$u(t)=20$$, the system (5.5) is controllable. Thus, from the above example, we conclude that the conditions $$(H_1)$$–$$(H_5)$$ are the sufficient conditions for the controllability of the system (2.1) but not necessary. 6. Conclusion In this problem, an approach has been given for controllability of fractional order retarded semilinear system with non-local conditions using the compactness of non-linear function. In this approach, we used general mild solution of the system which is based on the probability density function and semigroup. In this theory, we avoid the compactness of $$C_0$$-semigroup and uniform boundedness of non-linear function. The use of developed theory has been demonstrated by controlled diffusion. By adapting the techniques and ideas established in this paper, one can prove the controllability of fractional control systems with impulses. 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IMA Journal of Mathematical Control and Information – Oxford University Press
Published: Sep 21, 2018
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