Inverse optimal incremental control of nonlinear jump-diffusion systems

In this study, we address the challenge of solving the inverse optimal incremental control problem for nonlinear jump-diffusion systems by proposing an innovative inverse optimal incremental controller framework. A pivotal aspect of our approach lies in the novel utilization of an auxiliary incremental controller as a cornerstone for constructing the inverse optimal controller. This design not only ensures that the resultant controller is optimal in the sense of minimizing a meaningful cost functional but also imparts upon the closed-loop jump-diffusion system the property of incremental global $K_{\infty }$-exponential stability. This dual capability of achieving optimality and robust stability underscores the significance and novelty of our proposed controller design. Leveraging our inverse incremental controller design, we derive a comprehensive set of conditions that guarantee the inverse incremental $H_{\infty }$ control of nonlinear jump-diffusion systems. Simultaneously, we develop a methodology for estimating the incremental Hamilton-Jacobi inequality (iHJI), which serves as a cornerstone for validating the controller's performance. We present two illustrative engineering examples, showcasing the practical implications and robustness of our approach.