Heat-Equation-Based Smoothing Homotopy Method for Nonlinear Optimal Control Problems

This paper presents a new heat-equation-based smoothing homotopy method for solving nonlinear optimal control problems with the indirect method. The surrogates, derived from the heat equation solution, are first incorporated into the necessary conditions as replacement of the terminal state and costate variables. The homotopy process is then applied to the heat conduction time: a longer time results in a more uniform temperature distribution and greater stability against initial temperature variations, thereby making the corresponding homotopy problem much easier to solve; zero time implies no heat conduction and reverts to the original problem. Furthermore, capitalizing on the heat equation’s boundedness characteristic, an efficient approach for determining the hypersensitive parameters is proposed, thus obviating the necessity of manual tuning. Challenging numerical examples are provided to demonstrate the superior performance of the proposed method, indicating that its convergence and efficiency are significantly enhanced compared to the original smoothing homotopy methods.