Differential games, continuous Lyapunov functions, and stabilisation of non-linear dynamical systems

• Published in: IET control theory & applications 2017-10, Vol.11 (15), p.2486-2496
• Main Author: L'Afflitto, Andrea
• Format: Article
• Language: English
• Subjects: asymptotic stability; closed loop systems; closed‐loop asymptotic stability; closed‐loop system; continuous Lyapunov function; cost function; differential games; game theory; infinite time horizon; inverse optimality framework; Lyapunov methods; nonlinear dynamical systems; nonlinear nonquadratic cost functions; optimal robust control problems; partial differential equations; polynomial form; Research Article; robust control; stabilisation; steady‐state Hamilton‐Jacobi‐Isaacs equation; two‐player zero‐sum differential game problem; viscosity solutions
• ISSN: 1751-8644; 1751-8652; 1751-8652
• DOI: 10.1049/iet-cta.2017.0271

In this study, the authors address the two-player zero-sum differential game problem for non-linear dynamical systems with non-linear-non-quadratic cost functions over the infinite time horizon. The pursuer's goal is to minimise the cost function and guarantee asymptotic stability of the closed-loop system, whereas the evader's goal is to maximise the cost function. Closed-loop asymptotic stability is certified by continuous Lyapunov functions that are viscosity solutions of the steady-state Hamilton–Jacobi–Isaacs equation for the controlled system. Since it is difficult to find viscosity solutions of partial differential equations for numerous problems of practical interest, they extend an inverse optimality framework to provide explicit closed-form solutions of differential game problems, which involve affine in the controls dynamical systems with quadratic cost functions and linear dynamical systems with Lagrangians in polynomial form. The authors' framework allows also to solve optimal robust control problems involving non-linear dynamical systems with non-linear-non-quadratic cost functionals and provides a generalisation of the mixed-norm $\mathcal {H}_2 / \mathcal {H}_{\infty }$H2/H∞ optimal robust control framework. Two numerical examples illustrate the applicability of theoretical results provided.