Numerical Methods for Optimal Control Problems with Application to Autonomous Vehicles

In the present PhD thesis an optimal problem suite is proposed as benchmark for the test of numerical solvers. The problems are divided in four categories, classic, singular, constrained and hard problems. Apart from the hard problems, where it is not possible to give the analytical solution but only some details, all other problems are supplied with the derivation of the solution. The exact solution allows a precise comparison of the performance of the considered software. All of the proposed problems were taken from published papers or books, but it turned out that an analytic exact solution was only rarely provided, thus a true and reliable comparison among numerical solvers could not be done before. A typical wrong conclusion when a solver obtains a lower value of the target functional with respect to other solvers is to claim it better than the others, but it is not recognized that it has only underestimated the true value. In this thesis, a cutting edge application of optimal control to vehicles is showed: the optimization of the lap time in a race circuit track considering a number of realistic constraints. A new algorithm for path planning is completely described for the construction of a quasi G2 fitting of the GPS data with a clothoid spline in terms of the G1 Hermite interpolation problem. In particular the present algorithm is proved to work better than state of the art algorithms in terms of both efficiency and precision.