A numerical algorithm for singular optimal control synthesis using continuation methods

This paper presents an efficient computational method for the synthesis of singular optimal control problems. The proposed numerical procedure consists of two phases. In the first phase the original singular optimal control problem is converted into a non-singular one by adding to the performance index a perturbed (or weighted) energy term. The resultant boundary value problem can easily be solved for an appropriately large value of the perturbation parameter. In the second phase the solution obtained from the first phase is refined in a systematic manner based on continuation methods (imbedding methods or homotopy methods) until the optimal (or suboptimal) solution to the original problem is achieved. One of the major advantages of the proposed algorithm is that the resultant two-point boundary value problem need be solved just once for a properly large perturbation parameter and the refinement of the solution is accomplished by solving a set of initial value problems sequentially and/or in parallel as the perturbation parameter goes to zero. The proposed algorithm is therefore computationally efficient and applicable to a large class of optimal control problems with various boundary conditions (e.g. fixed and free terminal time). The practicability of the method is demonstrated by computer simulations on an example problem.