Non Controllability to Rest of the Two-Dimensional Distributed System Governed by the Integrodifferential Equation

TL;DR

This paper investigates the controllability of a two-dimensional distributed system governed by the Gurtin-Pipkin integrodifferential equation, demonstrating that certain memory kernels prevent finite-time control to equilibrium.

Contribution

It establishes non-controllability results for systems with specific memory kernels, advancing understanding of control limitations in integrodifferential systems.

Findings

If the Laplace transform of the memory kernel has a non-zero root, the system cannot reach equilibrium in finite time.

The controllability depends critically on the properties of the memory kernel.

The results apply to systems with twice continuously differentiable kernels.

Abstract

The paper deals with controllability problem for a distributed system governed by the two-dimensional Gurtin-Pipkin equation. We consider a system with compactly supported distributed control and show that if the memory kernel is a twice continuously differentiable function, such that its Laplace transformation has at least one non-zero root, then the system cannot be driven to the equilibrium in a finite time.