Exact Null Controllability of Non-Autonomous Conformable Fractional Semi-Linear Systems with Nonlocal Conditions
Dev Prakash Jha, Raju K. George
TL;DR
The paper investigates exact null controllability for non-autonomous conformable fractional semi-linear evolution systems with nonlocal initial conditions in Hilbert spaces, formulating the problem in terms of a fractional evolution operator $\Psi_\alpha(t,s)$. It establishes a framework where the linearized controllability problem, via the operators $L^{t_2}_{\zeta}$ and $N^{t_2}_{\zeta}$, yields a bounded inverse $H_{\alpha}=(L_{\zeta})^{-1}N^{t_2}_{\zeta}$ and a feedback-like control law $u(t) = -H_{\alpha}(x_0-g(x),F)(t)$, which is then extended to the nonlinear setting through a mild-solution representation. Existence of mild solutions is shown using Schauder's fixed-point theorem on a fixed-point map $Q$ under growth/boundedness conditions on the nonlinear terms and a smallness inequality $M^2L+\|B\|\|H_\alpha\|NM(LM+\gamma)+NM\gamma<1$, ensuring exact null controllability on $[\zeta,t_2]$. An illustrative application to a fractional-order PDE with nonlocal boundary conditions confirms the practicality of the results and demonstrates exact null controllability under explicit choices of the nonlinear terms and parameters.
Abstract
We study the exact null controllability of a class of non-autonomous conformable fractional semi-linear evolution systems with nonlocal initial conditions in Hilbert spaces. The analysis is carried out within the framework of conformable fractional calculus and linear evolution operator theory. Under suitable assumptions, we establish the existence of mild solutions and provide sufficient conditions for exact null controllability. Notably, the nonlocal term is allowed to be continuous without requiring compactness or Lipschitz-type conditions. An example is included to illustrate the applicability of the main results.