Abstraction-based solution of optimal stopping problems under uncertainty

In this paper we present novel results on the solution of optimal control problems with the help of finite-state approximations (“symbolic models”) of infinite-state plants. We investigate optimal stopping problems in the minimax sense, with undiscounted running and terminal costs, for nonlinear discrete-time plants subject to perturbations and constraints. This problem class includes finite-horizon and exit-(entry-)time problems as well as pursuit-evasion and reach-avoid games as special cases. We utilize symbolic models of the plant to upper bound the value function, i.e., the achievable closed-loop performance, and to compute controllers realizing the bounds. The symbolic models are obtained from suitable discretizations of the state and input spaces, and we prove that the computed bounds converge to the value function as the discretization errors approach zero. The value function is in general discontinuous, and the convergence (in the hypographical sense) is uniform on every compact subset of the state space. We apply the proposed method to design an approximately optimal feedback controller that starts up a DC-DC converter and is robust against supply voltage as well as load fluctuations.