@article{https://doi.org/10.1049/iet-cta.2017.0271,
author = {LAfflitto, Andrea},
title = {Differential games, continuous {L}yapunov functions, and stabilisation of non-linear dynamical systems},
journal = {IET Control Theory \& Applications},
volume = {11},
number = {15},
pages = {2486-2496},
keywords = {game theory, Lyapunov methods, nonlinear dynamical systems, robust control, closed loop systems, asymptotic stability, partial differential equations, differential games, continuous Lyapunov function, stabilisation, nonlinear dynamical systems, two-player zero-sum differential game problem, nonlinear nonquadratic cost functions, infinite time horizon, cost function, closed-loop system, closed-loop asymptotic stability, steady-state Hamilton-Jacobi-Isaacs equation, viscosity solutions, partial differential equations, inverse optimality framework, polynomial form, optimal robust control problems},
doi = {https://doi.org/10.1049/iet-cta.2017.0271},
url = {https://ietresearch.onlinelibrary.wiley.com/doi/abs/10.1049/iet-cta.2017.0271},
eprint = {https://ietresearch.onlinelibrary.wiley.com/doi/pdf/10.1049/iet-cta.2017.0271},
abstract = {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 optimal robust control framework. Two numerical examples illustrate the applicability of theoretical results provided.},
year = {2017}
}