A Combinatorial, Graph-Based Solution Method for a Class of Continuous-Time Optimal Control Problems

The paper addresses a class of continuous-time, optimal control problems whose solutions are typically characterized by both bang-bang and "singular" control regimes. Analytical study and numerical computation of such solutions are very difficult and far from complete when only techniques from control theory are used. This paper solves optimal control problems by reducing them to the combinatorial search for the shortest path in a specially constructed graph. Since the nodes of the graph are weighted in a sequence-dependent manner, we extend the classical, shortest-path algorithm to our case. The proposed solution method is currently limited to single-state problems with multiple control functions. A production planning problem and a train operation problem are optimally solved to illustrate the method.