2025/04/24 by Arora, Sumit, Ponce, Rodrigo
#34H05 #45J05 #93B05 #FOS: Mathematics #Optimization and Control (math.OC)
paper · doi:10.48550/arxiv.2504.17566
This work addresses control problems governed by a semilinear evolution equation with singular memory kernel κ(t)=αe-βt\fractν-1Γ(ν), where α>0, β≥ 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.