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

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.