To the optimal control(OC) problems whose optimal solutions are weakly discontinuous, an adaptive algorithm
is introduced. The time interval is divided into several subintervals. The piecewise polynomials are used to approximate the
solutions of OC problems. In each subinterval, the considered OC problems are discretized by using the pseudospectral
method and the Chebyshev-Gauss-Lobatto points are the collocation points. According to the numerical solutions, the
adaptive algorithm divides the subintervals into new subintervals, and increases the degrees of the approximating polynomials
in some subintervals. Finally, several examples are given to demonstrate the high accuracy and effectiveness of the proposed
algorithm.