Symbolic Optimal Control

TL;DR

This paper introduces a method for solving complex nonlinear optimal control problems using symbolic abstractions, providing convergent bounds and controllers without requiring continuity assumptions.

Contribution

It develops a novel abstraction-based approach for nonlinear optimal control, applicable to a wide class of problems including pursuit-evasion and reach-avoid games.

Findings

Bounds and controllers converge as discretization refines

Method applies to problems with hard constraints

No continuity assumptions needed

Abstract

We present novel results on the solution of a class of leavable, undiscounted optimal control problems in the minimax sense for nonlinear, continuous-state, discrete-time plants. The problem class includes entry-(exit-)time problems as well as minimum time, pursuit-evasion and reach-avoid games as special cases. We utilize auxiliary optimal control problems (`abstractions') to compute both upper bounds of the value function, i.e., of the achievable closed-loop performance, and symbolic feedback controllers realizing those bounds. The abstractions are obtained from discretizing the problem data, and we prove that the computed bounds and the performance of the symbolic controllers converge to the value function as the discretization parameters approach zero. In particular, if the optimal control problem is solvable on some compact subset of the state space, and if the discretization parameters are sufficiently small, then we obtain a symbolic feedback controller solving the problem on that subset. These results do not assume the continuity of the value function or any problem data, and they fully apply in the presence of hard state and control constraints.