Lyapunov Stability Analysis in Integro-Differential Systems with Damping Memory Terms
Description
This work presents an analytical study of Lyapunov stability theory applied to integro-differential equations and dynamical systems with dissipative memory terms. Initially, we investigate criteria for the existence and uniqueness of strong and weak solutions using the theory of linear and nonlinear operator semigroups, combined with Dafermos's history space formulation. Subsequently, we formalize the concepts of Lyapunov stability, asymptotic stability, and exponential stability for systems whose evolution depends on their entire past history. We rigorously address the construction of an adapted Lyapunov functional (or Lyapunov-Krasovskii functional), which incorporates the energy stored in the memory variable. We analyze the conditions imposed on the memory kernel g(t), highlighting the effects of the transition between kernels exhibiting exponential decay and those with slow decay (polynomial or sub-exponential). Finally, we conduct a comparative study of dissipation mechanisms and state dynamics in systems with and without memory.