Method of evolving junctions: A new approach to optimal control with constraints

We propose a new strategy, called method of evolving junctions (MEJ), to compute the solutions for a class of optimal control problems with constraints on both state and control variables. Our main idea is that by leveraging the geometric structures of the optimal solutions, we recast the infinite dimensional optimal control problem into an optimization problem depending on a finite number of points, called junctions. Then, using a modified gradient flow method, whose dimension can change dynamically, we find local solutions for the optimal control problem. We also employ intermittent diffusion, a global optimization method based on stochastic differential equations, to obtain the global optimal solution. We demonstrate, via a numerical example, that MEJ can effectively solve path planning problems in dynamical environments.