Uniform Asymptotic Stability of a PDE'S System Arising From a Flexible Robotics Model

ABSTRACT In this paper, we investigate the uniform asymptotic stability of a fourth‐order partial differential equation with a fading memory forcing term and boundary conditions arising from a flexible robotics model. To achieve this goal, the model is reformulated in an abstract framework using the ‐semigroup theory. First, we present results on the existence, uniqueness, and continuous dependence on initial data for both mild and strong solutions of a class of semilinear integro‐differential equations in Banach spaces. Within this abstract framework, we also derive new sufficient conditions for the uniform asymptotic stability of solutions and the existence of attractors. As a consequence of these theoretical results, we establish the existence, uniqueness, and continuous dependence on initial data for the solutions of the boundary value problem associated to the model. Moreover, under suitable assumptions on the nonlinear term, we guarantee the uniform asymptotic stability of solutions and the existence of attractors.