Zubov's equation is a Hamilton-Jacobi type partial differential equation whose solution yields a Lyapunov function for an asymptotically stable equilibrium on its domain of attraction, i.e., on the maximal possible domain. The equation can be set up for general nonlinear ordinary differential equations subject to (deterministic or random) perturbations. In low space dimensions it offers a possibility for the numerical approximation of Lyapunov functions and domains of attraction.
In most practical applications, state constraints are imposed in order to prevent the system state from entering unwanted or physically meaningless regions of the state space. It is then of interest to know the set of initial states for which the solution converges to an asymptotically stable equilibrium without violating the state constraints. To this end, state constraints can be incorporated into Zubov's equation. Unfortunately, unless rather strong controllability properties hold, introducing state constraints into Hamilton-Jacobi equations renders the solutions discontinuous which causes severe problems for both analysis and numerics.
In this talk, we present a method for incorporating state constraints into Zubov's equation which does not lead to discontinuities while at the same time it maintains the Lyapunov function property of its solution and the characterization of the domain of attraction. The method is inspired from a similar approach for the minimum time optimal control problem and uses that the solution of Zubov's equation can be characterized as an optimal value function for a suitable optimal control problem. The talk is based on a joint paper with Hasnaa Zidani (ENSTA-ParisTech).