A composite pseudospectral method for optimal control problems with piecewise smooth solutions

In this paper, we develop a composite collocation approximation scheme for solving optimal control problems governed by ordinary differential equations with piecewise smooth solutions. For this purpose, we divide the time interval of the problem into some nonequal subintervals and define a piecewise interpolating polynomial on the base of transformed Legendre-Gauss nodes in subintervals. According to the weak representations approach, we derive the corresponding operational matrix of derivative. Using the Legendre-Gauss quadrature formula and the obtained operational matrix, the optimal control problem is discretized as a nonlinear programming problem. In this approach, the time locations in which corners happen in the state and control functions, are considered as unknown parameters. Therefore, the problem can be solved as a nonlinear programming problem with respect to these parameters. Four examples are investigated to demonstrate the validity and applicability of the proposed technique.