Controllability results for fractional semilinear delay control systems

Abstract

In this article, we have presented the controllability relationship between the semilinear control system of fractional order (1, 2] with delay and that of the semilinear control system without delay. Suppose X and U be Hilbert spaces which are separable and $Z=L_2[0,b;X],\;Z_h=L_2[-h,b;X],\;0\le h\le b$ and $Y=L_2[0,b;U]$ be the function spaces. Let the semilinear control system of fractional order with delay as

$$\begin{aligned} ^CD_\tau ^\alpha z(\tau )= & {} Az(\tau )+Bv(\tau )+g(\tau ,z(\tau -h)),\;0\le \tau \le b;\\ z_0(\theta )= & {} \phi (\theta ),\;\;\;\; \theta \in [-h,0]\\ z'(0)= & {} z_0. \end{aligned}$$

where $1<\alpha \le 2$, fractional Caputo derivative is denoted as $^CD_{\tau }^\alpha$, time constant b is positive and finite.$A:D(A)\subseteq X\rightarrow X$ is a operator which is linear and closed having densed domain X and A is the infinitesimal generator of solution operator $\{C_\alpha (\tau )\}_{\tau \ge 0}$. The control function is denoted by $v(\tau )$ and defined as $v:[0,b]\rightarrow U$. The continuous state variable $z(\tau )\in Z$, $\phi \in L_2[-h,0;X]$. The operator $B:Y\rightarrow Z$ is linear and bounded. The function $g:[0,b]\times X\rightarrow V$ is purely nonlinear and satisfies Lipschitz continuity. We assumed that the fractional semilinear system without delay is approximate/exact controllable and by imposing some conditions on the range of the nonlinear term, we obtained the controllability results of the fractional semilinear system with delay. Approximate controllability of proposed problem is discussed under three different sets of assumptions. Exact controllability of proposed problem is also discussed. Finally an example is given to understand the theoretical results in better manner.