In this talk we investigate some direct methods to solve a class of optimization problems in a Hilbert space framework. In particular, we discuss some methods based on proximal splitting techinques and propose some applications to PDE-constrained optimization. We conclude the talk with a discussion about stochastic methods for problems with random data, numerical simulations, and some future research directions. The talk is based on a joint work with Caroline Geiersbach.
joint work with Maria do Rosário de Pinho
We formulate the path planning of AUVs under currents as a free end time multiprocess optimal control problem with state constraints. Reformulating it as a fixed time problem, we derive first order necessary conditions in the form of a maximum principle. The notable feature of this result is that it covers the case when the time spent in an intermediate system is reduced to zero. It also provides us with the tools to design codes to solve the problem numerically and extract information on the optimal solution and its multipliers to partially validate the numerical findings.
We propose a numerical scheme for the approximation of high-dimensional, nonlinear Isaacs PDEs arising in robust optimal feedback control of nonlinear dynamics. The numerical method consists of a global polynomial ansatz together separability assumptions for the calculation of high-dimensional integrals. The resulting Galerkin residual equation is solved by means of an alternating Newton-type/policy iteration method for differential games. We present numerical experiments illustrating the applicability of our approach in robust optimal control of nonlinear parabolic PDEs.