Approximation of Optimally Controlled Ordinary and Partial Differential Equations

In this thesis, which consists of four papers, approximation of optimal control problems is studied. In Paper I the Symplectic Pontryagin method for approximation of optimally controlled ordinary differential equations is presented. The method consists of a Symplectic Euler time stepping scheme for a Hamiltonian system with a regularized Hamiltonian. Under some assumptions it is shown that the approximate value function associated with this scheme converges to the original value function with a linear rate. In Paper II the ideas from Paper I are extended to approximation of an optimally controlled partial differential equation, a one-dimensional Ginzburg-Landau equation. The approximation is performed in two steps. In the first step a value function associated with a finite element spatial discretization is shown to converge quadratically in the mesh size to the original value function. In the second step a Symplectic Euler discretization in time is shown to converge with a linear rate. The behavior of optimal solutions is shown by numerical examples. In Paper III the same approximation method as in Paper II is applied to three other problems; the optimal design of an electric conductor, the design of an elastic domain, and the problem of reconstructing the interior of an object from measured electrical surface currents. Since these problems are time-independent the Hamilton-Jacobi theory can not be used directly. In order to be able to obtain error bounds the problems are therefore transferred to a setting where time plays a role. Computer experiments with the Symplectic Pontryagin method is performed for all three problems. Common to the three first papers is that the convergence proofs use that the approximate value functions solve Hamilton-Jacobi equations consistent with the original Hamilton-Jacobi equations. Paper IV concerns convergence of attainable sets for non-convex differential inclusions. When the right hand side in the differential inclusion is a bounded, Lipschitz set-valued function it is shown that the convergence in Hausdorff-distance of attainable sets for a Forward Euler discretization is linear in the time step. This implies that dynamic programming using Forward Euler discretizations of optimal control problems converge with a linear rate when all the functions involved are bounded and Lipschitz continuous.