Optimal Impulsive Control and Stabilization of Dynamic Systems Based on Quasi-Variational Inequalities

In this paper, we investigate the optimal control problem regarding a class of dynamic systems, aiming to address the challenge of simultaneously ensuring cost minimization and system asymptotic stability. The theoretical framework proposed in this paper integrates the value function concept from optimal control theory with Lyapunov stability theory. By setting the impulse cost at any finite time to be strictly positive, we exclude Zeno behavior, and a set of sufficient conditions is established that simultaneously guarantees system asymptotic stability and cost minimization based on Quasi-Variational Inequalities (QVIs). To address the challenge of solving the Hamilton–Jacobi–Bellman (HJB) equation in high-dimensional nonlinear systems, we employ an inverse optimal control framework to synthesize the strategy and its corresponding cost function. Finally, we validate the feasibility of our method by applying the theoretical results obtained to three numerical examples.