Controllability problems of a neutral integro-differential equation with memory
Sumit Arora, Akambadath Nandakumaran
TL;DR
The paper tackles approximate controllability of a semilinear neutral integro-differential equation with memory in Banach spaces, formalized as $\frac{d}{dt}\left[w(t)+\int_{-\infty}^t G(t-s)w(s)\,ds\right]=A w(t)+\int_0^t N(t-s)w(s)\,ds+B u(t)+f(t,w_t)$. It develops a resolvent-family framework for the linear part, formulates a linear-quadratic regulator to obtain an explicit optimal control achieving approximate controllability, and then uses a Schauder fixed-point argument to secure a mild solution for the semilinear system, establishing approximate controllability under suitable conditions. The results are extended to general Banach spaces under a Lipschitz-type nonlinear term, and the theory is illustrated via an application to heat conduction with fading memory, where the PDE is recast in an abstract evolution form and shown to be approximately controllable. A key contribution is proving controllability results without resorting to fractional power theory, relying instead on resolvent-operator techniques and phase-space methods in reflexive Banach spaces with uniformly convex duals. Overall, the work provides a rigorous, general framework for controllability of memory-bearing neutral systems and a concrete PDE example demonstrating practical relevance.
Abstract
The current study addresses the control problems posed by a semilinear neutral integro-differential equation with memory. The primary objectives of this study are to investigate the existence of a mild solution and approximate controllability of both linear and semilinear control systems in Banach spaces. To accomplish this, we begin by introducing the concept of a resolvent family associated with the homogeneous neutral integro-differential equation without memory. In the process, we establish some important properties of the resolvent family. Subsequently, we develop approximate controllability results for a linear control problem by constructing a linear-quadratic regulator problem. This involves establishing the existence of an optimal pair and determining the expression of the optimal control that produces the approximate controllability of the linear system. Furthermore, we deduce sufficient conditions for the existence of a mild solution and approximate controllability of a semilinear system in a reflexive Banach space with a uniformly convex dual. Additionally, we delve into the discussion of approximate controllability for a semilinear problem in general Banach space, assuming a Lipschitz type condition on the nonlinear term. Finally, we implement our findings to examine the approximate controllability of certain partial differential equations, demonstrating their practical relevance.