A Homotopy Method Based on Theory of Functional Connections

Abstract

Discrete continuation method (DCM) is a widely used homotopy method to solve optimal control problems. Starting from auxiliary problem, DCM monotonously varies homotopy parameter until the objective problem is solved. However, DCM often fails when the initially and manually defined homotopy path exists limit points or goes off to infinity. In this paper, a novel homotopy method, called TFC homotopy method, is presented to enhance DCM performance in aspect of homotopy function construction. Firstly, Theory of Functional Connections (TFC) is introduced and TFC homotopy function is established, which implicitly defines infinite homotopy paths that connect the auxiliary problem with the objective problem for the first time in literature. Secondly, a novel two-layer path tracking method is designed. DCM is used in the first layer and the second layer is triggered when unfavorable situations are met, then another feasible homotopy path is found by solving an optimization problem and tracked by DCM. Compared with pseudo-arclength method (PAM), TFC homotopy method remains the simplicity of DCM method while owing its capability of flexible path switching. Multiple simulations are conducted against PAM, illustrating the effectiveness of the proposed method.