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

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.

在本文中，我们介绍了具有分数阶的分数阶（1，2）的半线性控制系统与没有延迟的半线性控制系统之间的可控制性关系。假设X和U是可分离的希尔伯特空间，并且\（Z = L_2 [0，b; X]，\; Z_h = L_2 [-h，b; X]，\; 0 \ le h \ le b \）和\（Y = L_2 [0，b; U] \）为将具有延迟的分数阶半线性控制系统设为

其中\（1 <\ alpha \ le 2 \），分数Caputo导数表示为\（^ CD _ {\ tau} ^ \ alpha \），时间常数b为正且有限。\（A：D（A）\ subseteq X \ rightarrow X \）是线性且封闭的，具有密集的域X的算子，而A是解算子\（\ {C_ \ alpha（\ tau）\}的无穷小生成器_ {\ tau \ ge 0} \）。控制函数由\（v（\ tau）\）表示，并定义为\（v：[0，b] \ rightarrow U \）。连续状态变量\（z（\ tau）\ in Z \），\（\ phi \ in L_2 [-h，0; X] \）。运算符\（B：Y \ rightarrow Z \）是线性且有界的。函数\（g：[0，b] \ x X \ rightarrow V \）完全是非线性的，并且满足Lipschitz连续性。我们假设没有延迟的分数半线性系统是近似/精确可控的，并且通过在非线性项的范围上施加一些条件，我们获得了具有延迟的分数半线性系统的可控制性结果。在三种不同的假设下讨论了所提出问题的近似可控性。还讨论了所提出问题的精确可控性。最后给出一个例子，以更好地理解理论结果。