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Solutions to optimal control problems can be discontinuous, even if all thefunctionals defining the problem are smooth. This can cause difficulties whennumerically computing solutions to these problems. While conventional numericalmethods assume state and input trajectories are continuous and differentiableor smooth, our method is able to capture discontinuities in the solution byintroducing time-mesh nodes as decision variables. This allows one to obtain ahigher accuracy solution for the same number of mesh nodes compared to a fixedtime-mesh approach. Furthermore, we propose to first solve a sequence ofsuitably-defined least-squares problems to ensure that the error in the dynamicequation is below a given tolerance. The cost functional is then minimizedsubject to an inequality constraint on the dynamic equation residual. Wedemonstrate our implementation on an optimal control problem that has achattering solution. Solving such a problem is difficult, since the solutioninvolves infinitely many switches of decreasing duration. This simulation showshow the flexible mesh is able to capture discontinuities present in thesolution and achieve superlinear convergence as the number of mesh intervals isincreased.

Solving optimal control problems with non-smooth solutions using an integrated residual method and flexible mesh