Stability in probability and inverse optimal control of evolution systems driven by Levy processes

This paper first develops a Lyapunov-type theorem to study global well-posedness (existence and uniqueness of the strong variational solution) and asymptotic stability in probability of nonlinear stochastic evolution systems (SESs) driven by a special class of Levy processes, which consist of Wiener and compensated Poisson processes. This theorem is then utilized to develop an approach to solve an inverse optimal stabilization problem for SESs driven by Levy processes. The inverse optimal control design achieves global well-posedness and global asymptotic stability of the closed-loop system, and minimizes a meaningful cost functional that penalizes both states and control. The approach does not require to solve a Hamilton-Jacobi-Bellman equation (HJBE). An optimal stabilization of the evolution of the frequency of a certain genetic character from the population is included to illustrate the theoretical developments.