Improving validated computation of Viability Kernels

The study of viability kernels can be of critical importance for the verification of control systems. A viability kernel over a set of safe states is the set of initial states for which the trajectory can be controlled so as to stay within the safe set for an indefinite amount of time. This paper investigates improvements of the rigorous method from Monnet et al. [19, 20]. This method computes an inner-approximation of the viability kernel of a continuous time control system using methods based on interval analysis. It consists of two phases: first an initial inner-approximation of the viability kernel is computed via Lyapunov-like functions; second the initial inner-approximation is improved by finding other states that can reach the inner-approximation, without exiting the safe set, using validated numerical integration. Among the improvements, we discuss an approach inspired by an interval method using barrier functions for computing a good initial inner-approximation of the viability kernel, easing the improvement phase.