Controllability problem of an evolution equation with singular memory
Sumit Arora, Rodrigo Ponce
TL;DR
The paper addresses the controllability of evolution equations with a singular memory kernel $κ(t)=α e^{-β t} t^{ν-1}/Γ(ν)$ by developing an $( ext{α}, ext{β}, u)$-resolvent family for a sectorial operator $A$, establishing well-posedness and mild solutions for the linear and semilinear problems. It then formulates and solves a linear-quadratic regulator to achieve approximate controllability for the linear system and provides sufficient conditions for approximate controllability of the semilinear system in super-reflexive Banach spaces, with extensions to general Banach spaces. The methodology hinges on controllability operators, memory Gramians, and a feedback optimal control expressed via the resolvent and duality mappings. An application to the heat equation with memory demonstrates the practical relevance and adaptability of the framework to parabolic problems with nonlocal memory effects.
Abstract
This work addresses control problems governed by a semilinear evolution equation with singular memory kernel $κ(t)=αe^{-βt}\frac{t^{ν-1}}{Γ(ν)}$, where $α>0, β\ge 0$, and $0<ν<1$. We examine the existence of a mild solution and the approximate controllability of both linear and semilinear control systems. To this end, we introduce the concept of a resolvent family associated with the linear evolution equation with memory and develop some of its essential properties. Subsequently, we consider a linear-quadratic regulator problem to determine the optimal control that yields approximate controllability for the linear control system. Furthermore, we derive sufficient conditions for the existence of a mild solution and the approximate controllability of a semilinear system in a super-reflexive Banach space. Additionally, we present an approximate controllability result within the framework of a general Banach space. Finally, we apply our theoretical findings to investigate the approximate controllability of the heat equation with singular memory.