Gradient Method for Solving Singular Optimal Control Problems

Solving an optimal control problem consists in finding a control structure and corresponding switching times. Unlike in a bang-bang case, switching to a singular control perturbs the control structure. The perturbation of one of the switching times affects any subsequent singular intervals in the control, as the trajectories move along different singular arcs with different values of singular controls. It makes the problem of finding optimal solutions extremely difficult. In this paper, we discuss a gradient method for solving optimal control problems, when singular intervals are present in the optimal structure. The method is based on applying the necessary conditions of optimality given by the Pontryagin Maximum Principle, where the control variable enters the Hamiltonian linearly. To demonstrate the method, we formulate a nonlinear optimal control problem and then, using the proposed algorithm, we solve the problem and find the optimal control structure and corresponding switching times. Lastly, we compare the results with results obtained using three popular optimisation modelling languages: Pyomo, AMPL and JuMP. These languages serve as interfaces for solving the optimal control problem with the non-linear optimisation algorithm Ipopt. Our case study shows that the presented method not only computes the switching times accurately, but also moves precisely along the singular arc.