Approximate controllability of some nonlinear systems in Banach spaces

The problems of controllability of infinite dimensional nonlinear (fractional) systems were studied widely by many authors; see [1–6] and the references therein. The approximate controllability of nonlinear systems when the semigroup $S(t)$, $t>0$, generated by A is compact has been studied by many authors. The results of Zhou [6] and Naito [7] give sufficient conditions on B with finite dimensional range or necessary and sufficient conditions based on more strict assumptions on B. Li and Yong in [8] studied the same problem assuming the approximate controllability of the associated linear system under arbitrary perturbation in $L_{\mathrm{\infty }}(I,L(X))$. Bian [9] investigated the approximate controllability for a class of semilinear systems. For abstract nonlinear systems, Carmichael and Quinn [10] used the Banach fixed-point theorem to obtain a local exact controllability in the case of nonlinearities with small Lipschitz constants. Zhang [11] studied the local exact controllability of semilinear evolution systems. Naito [7] and Seidman [12] used Schauder’s fixed-point theorem to prove invariance of the reachable set under nonlinear perturbations. Other related abstract results were given by Lasiecka and Triggiani [13].

In recent years, controllability problems for various types of nonlinear fractional dynamical systems in infinite dimensional spaces have been considered in many publications. An extensive list of these publications focused on the complete and approximate controllability of the fractional dynamical systems can be found (see [1–5, 7, 9–47]). A pioneering work has been reported by Bashirov and Mahmudov [17], Dauer and Mahmudov [28] and Mahmudov [31]. Sakthivel et al. [40] studied the approximate controllability of nonlinear deterministic and stochastic evolution systems with unbounded delay in abstract spaces. Klamka [23–26] derived a set of sufficient conditions for constrained local controllability near the origin for semilinear dynamical control systems. Wang and Zhou [3] investigated the complete controllability of fractional evolution systems without assuming the compactness of characteristic solution operators. Sukavanam and Kumar [47] obtained a new set of sufficient conditions for the approximate controllability of a class of semilinear delay control systems of fractional order by using the contraction principle and Schauder’s fixed-point theorem.

and define the following sets:

in the strong operator topology as $\epsilon \to 0^{+}$, see [30]. Thus, the control (3) transfers system (2) from $y_0\in Y$ to a small neighborhood of an arbitrary point $h\in X$ if and only if $\overline{QR(L,0)}=X$.

It is clear that the fixed points of the nonlinear operator $T^{\epsilon }$ are the solutions of nonlinear control system (5) and vice versa.

To the best of our knowledge, the approximate controllability problem for semilinear abstract systems in Banach spaces has not been investigated yet. Motivated by this consideration, in this paper we study the approximate controllability of semilinear abstract systems in Banach spaces. The approximate controllability of (1) is derived under the compactness assumption of the linear operator involved. We prove that the approximate controllability of linear system (2) implies the approximate controllability of semilinear system (1) under some assumptions. On the other hand, it is known that if the operator L is compact, then $ImQLB\ne X$, that is, linear system (2) is not exactly controllable. Thus the analogue of this result is not true for exact controllability, that is why we investigate just the approximate controllability. Notice that a similar result for semilinear equations in Hilbert spaces was obtained by Dauer and Mahmudov [27].

In Section 2 an abstract result concerning the approximate controllability of semilinear system (1) is obtained. It is proven that the controllability of (2) implies the controllability of (1). Finally, these abstract results are applied to the approximate controllability of semilinear fractional integrodifferential equations. These equations serve as an abstract formulation of a fractional partial integrodifferential equation arising in various applications such as viscoelasticity, heat equations and many other physical phenomena.

is bijective, demicontinuous, i.e., continuous from X with a strong topology into $X^{\ast }$ with weak topology and strictly monotonic. Moreover, $J^{-1}:X^{\ast }\to X$ is also a duality mapping.

for all $z_1^{\ast },z_2^{\ast }\in X^{\ast }$. It is easy to see that Γ is linear and continuous. Γ is nonnegative if $〈z^{\ast },\mathrm{\Gamma }{z}^{\ast }〉\ge 0$ for all $z^{\ast }\in X^{\ast }$.

Lemma 1 [31]

Theorem 2 [31]

We impose the following assumptions:

(A1) $F:Y\times V\to Y$ is continuous and there exists $C>0$ such that $\parallel F(y,v)\parallel \le C$ for all $(y,v)\in Y\times V$.

(A2) $L:Y\to Y$ is compact.

(A3) For all $h\in X$, $z_{\epsilon }(h)=\epsilon {(\epsilon I+\mathrm{\Gamma }J)}^{-1}(h)$ strongly converges to zero as $\epsilon \to 0^{+}$.

Note that the condition (A3) holds if and only if $\overline{Im(QLB)}=\overline{QR(L,0)}=X$, i.e., system (2) is approximately controllable.

Theorem 4 Assume (A1)-(A3) hold. Then semilinear system (1) is approximate controllability.

Proof Step 1. Show that the operator $T^{\epsilon }$ has a fixed point in $Y\times V$ for all $\epsilon >0$. For our convenience, let us introduce the following notation:

Thus we proved that $T^{\epsilon }$ maps $B_{\epsilon }=\{(z,w)\in Y\times V:\parallel (z,w)\parallel \le r(\epsilon )\}$ into itself. On the other hand, the operator $T^{\epsilon }$ is continuous and $T^{\epsilon }(B_{\epsilon })$ is relatively compact. By Schauder’s fixed-point theorem, for all $\epsilon >0$, $T^{\epsilon }$ has a fixed point in the ball $B_{\epsilon }$.

for some ${\overline{z}}_0\in Z$. Applying $J({\overline{z}}_0)$ to equation (9) and taking the limit, we obtain

since Γ is positive. So, $J(z_{\epsilon })\rightharpoonup 0$ as $\epsilon \to 0^{+}$. Now, applying $J(z_{\epsilon })$ to equation (9), dividing through by ε and taking the limit, we obtain

Thus ${lim}_{\epsilon \to 0^{+}}\parallel Q{y}_{\ast }^{\epsilon }-h\parallel =0$, consequently $\overline{QR(L,F)}=X$. The theorem is proved. □

where the state variable x takes values in a separable reflexive Banach space X; ${}^{c}D^{\alpha }$ is the Caputo fractional derivative of order $\frac{1}{2}<\alpha <1$; A is the infinitesimal generator of a $C_0$ semigroup $S(t)$ of bounded operators on X; the control function u is given in $L_2([0,b],U)$, U is a Hilbert space; B is a bounded linear operator from U into X, $\mathrm{\Delta }=\{(t,s):0\le s\le t\le T\}$ and $g:\mathrm{\Delta }\times X\to X$, $f:I\times X\times X\to X$ are continuous bounded functions and $x_0\in X$.

provided the right-hand side is pointwise defined on $[0,\mathrm{\infty })$, where γ is the gamma function.

For basic facts about fractional integrals and fractional derivatives, one can refer to [49].

where

and ${\eta }_{\alpha }$ is a probability density function defined on $(0,\mathrm{\infty })$, that is, ${\eta }_{\alpha }(\theta )\ge 0$, $\theta \in (0,\mathrm{\infty })$ and ${\int }_0^{\mathrm{\infty }}{\eta }_{\alpha }(\theta )d\theta =1$.

Lemma 9 [34]

For any fixed $t\ge 0$, the operators ${\hat{S}}_{\alpha }(t)$ and $S_{\alpha }(t)$ are linear compact and bounded operators, i.e., for any $x\in X$, $\parallel {\hat{S}}_{\alpha }(t)x\parallel \le M\parallel x\parallel$ and $\parallel S_{\alpha }(t)x\parallel \le \frac{M}{\mathrm{\Gamma }(\alpha )}\parallel x\parallel$.

Definition 10 A solution $x(\cdot ;x_0,u)\in C([0,b],X)$ is said to be a mild solution of (10) if for any $u\in L_2([0,b],U)$ and the integral equation (11) is satisfied.

Let $x_b(x_0;u)$ be the state value of (10) at terminal time b corresponding to the control u and the initial value $x_0$. Introduce the set $\mathrm{\Re }(b,x_0)=\{x_b(x_0;u)(0):u\in L_2([0,b],U)\}$, which is called the reachable set of system (10) at terminal time b, its closure in X is denoted by $\overline{\mathrm{\Re }(b,x_0)}=X$.

Definition 11 System (10) is said to be approximately controllable on J if $\overline{\mathrm{\Re }(b,x_0)}=X$, that is, given an arbitrary $ϵ>0$, it is possible to steer from the point $x_0$ to within a distance ϵ from all points in the state space X at time b.

where $B^{\ast }$ denotes the adjoint of B and $S_{\alpha }^{\ast }$ is the adjoint of $S_{\alpha }$. It is straightforward that the operator ${\mathrm{\Gamma }}_0^b$ is a linear bounded operator. By Theorem 2, linear fractional control system (12) is approximately controllable on $[0,b]$ if and only if for any $h\in X$, $z_{\epsilon }(h)=\epsilon {(\epsilon I+{\mathrm{\Gamma }}_0^{b}J)}^{-1}(h)$ converges strongly to zero as $\epsilon \to 0^{+}$.

is compact from $L^p([0,b],X)$ into $C([0,b],X)$.

Consequently, $L_{\alpha }^{\eta ,\delta }\to L_{\alpha }$ in the operator norm so that $L_{\alpha }$ is compact and (i) follows immediately.

To prove (ii), note that, for $0\le t\le t+h\le b$ and ${\parallel f\parallel }_{L^p}\le 1$, we have

Applying the Hölder inequality, we obtain

It is clear that $I_1,I_2\to 0$ as $h\to 0$. On the other hand, the compactness of $S(t)$, $t>0$ (and consequently $S_{\alpha }(t)$), implies the continuity of $S_{\alpha }(t)$, $t>0$, in the uniform operator topology. Then, by the Lebesque dominated convergence theorem, $I_3\to 0$ as $h\to 0$. Thus the proof of (ii), and therefore the proof of the proposition, is complete. □

Theorem 13 Suppose $S(t)$, $t>0$, is compact and $\frac{1}{2}<\alpha \le 1$. Then system (10) is approximately controllable on $[0,b]$ if the corresponding linear system is approximately controllable on $[0,b]$.

Proof Let $Y=L_2([0,b],X)$, $V=L_2([0,b],U)$, and $y_0=S_{\alpha }(\cdot )x_0\in Y$. Define the linear operators Q, L, $L_1$ and the nonlinear operator F by

for $y\in Y$, $v\in V$. It is easy to see that by Proposition 12 all the conditions of Theorem 4 are satisfied and (10) is approximately controllable. This completes the proof. □

where $e_n(\theta )=\sqrt{2/\pi }sinn\theta$, $0\le x\le \pi$, $n=1,2,\dots$ . It is known that A generates a compact semigroup $S(t)$, $t>0$, in X and is given by

Now if $(b,e_n)\ne 0$ for all n, then $(z,e_n)=0$ for all n and $z=0$. Therefore, the associated linear system is approximately controllable provided that ${\int }_0^{\pi }b(\theta )e_n(\theta )d\theta \ne 0$ for $n=1,2,3,\dots$ . Because of the compactness of the semigroup $S(t)$ (and consequently ${\hat{S}}_{\alpha }(t)$, $S_{\alpha }(t)$) generated by A, the associated linear system of (13) is not completely controllable but it is approximately controllable. Hence, according to Theorem 13, system (13) will be approximately controllable on $[0,b]$.

In this paper, abstract results concerning the approximate controllability of semilinear evolution systems in a separable reflexive Banach space are obtained. An approximate controllability result for semilinear systems is obtained by means of Schauder’s fixed-point theorem under the compactness assumption. It is also proven that the controllability of the linear system implies the controllability of the associated semilinear system. Then the obtained results are applied to derive sufficient conditions for the approximate controllability of the semilinear fractional integrodifferential equations in Banach spaces. Upon making some appropriate assumptions, by employing the ideas and techniques as in this paper, one can establish the approximate controllability results for a wide class of fractional deterministic and stochastic differential equations.