Revisiting the Viability Algorithm for Hybrid Systems Using Optimal Control*

In this paper, we revisit the problem of approximating viability sets for hybrid systems with nonlinear continuous dynamics and competing inputs. As usual in the literature, an iterative algorithm, based on the alternating application of a continuous and a discrete operator, is employed. Three different cases, based on whether the continuous evolution and the number of discrete transitions are finite or infinite, are considered. A complete characterization of the reach-avoid computation (involved in the continuous time calculation) is provided based entirely on optimal control. Moreover, we show convergence of the iterative process by using a constructive version of Tarski's fixed point theorem, to determine the maximal fixed point of a monotone operator on a complete lattice of closed sets. To illustrate its performance, the viability algorithm is applied to investigate voltage stability for a single machine-load system in case of a line fault.