Efficient Value Function Upper Bounds for a Class of Constrained Linear Time-Varying Optimal Control Problems

This paper develops an algorithm for upper-bounding the value function for a class of continuous-time optimal control problems. The upper bound can be used as a conservative estimate for the minimum cost that can be attained by any constraint admissible control from some initial state. Linear time-varying systems subject to convex input constraints and a state-independent running cost are considered. A collection of solutions of an augmented dynamical system is used to characterise viscosity supersolutions of a Hamilton-Jacobi-Bellman equation, which in turn yields an upper bound for the value function. The proposed algorithm has a computational complexity that scales in the number of these solutions as opposed to the dimension of the system, making the algorithm tractable for high dimensional systems.