Uniform asymptotic stability of a PDE's system arising from a flexible robotics model

TL;DR

This paper establishes the uniform asymptotic stability of a fourth-order PDE model from flexible robotics using semigroup theory, providing new results on solution existence, stability conditions, and attractors.

Contribution

It introduces novel theoretical results on the stability and attractors of PDEs with memory terms, specifically applied to a flexible robotics model.

Findings

Proved existence and uniqueness of solutions for the PDE.

Derived sufficient conditions for uniform asymptotic stability.

Established the existence of attractors for the system.

Abstract

In this paper we investigate the asymptotic stability of a fourth-order PDE with a fading memory forcing term and boundary conditions arising from a flexible robotics model. We carry on our study by using an abstract formulation of the problem based on the $C_0$-semigroup. To achieve our objective, we first provide new results on the existence, uniqueness, continuous dependence on initial data of either mild and strong solutions for semilinear integro-differential equations in Banach spaces. Then, we also find sufficient conditions for the uniform asymptotic stability of solutions and for the existence of attactors. As an application of these abstract results, we can ensure existence, uniqueness and continuous dependence on initial data for the solutions of the boundary value problem under investigation and, finally, we prove the uniform asymptotic stability of solutions and the existence of attactors under suitable conditions on the nonlinear term.