Optimal Control of Initial-Boundary Value Problems for Hyperbolic Balance Laws with Switching Controls and State Constraints

The subject of this thesis is the treatment of pointwise state constraints in the context of the optimal control of nonlinear hyperbolic balance laws with switching controls. This kind of problem appears in several examples, e. g., gas transport in networks or the optimal control of traffic flow. The main goal of this thesis is to derive necessary optimality conditions for the problem with state constraints and, as a further step, to approximate the corresponding optimality system by Moreau-Yosida type regularizations. The main difficulty is the fact that solutions of hyperbolic balance laws may develop moving discontinuities even in the case of smooth data. Therefore, the control-to-state mapping is in general not differentiable to L¹ such that the Fréchet-differentiability of the reduced cost functional does not follow by standard arguments. Moreover, due to the moving discontinuities, the control-to-state mapping is in general not continuous to L-infinity. In the context of optimal control problems with state constraints, this in particular poses a difficulty since the solution of the state equation has to be considered at least in L-infinity in order to assure that the Robinson's CQ is possible to hold. To overcome these problems, we use the recently developed sensitivity- and adjoint calculus by Pfaff and Ulbrich, which is based on a more general notion of differentiability of the control-to-state mapping, the so-called shift-differentiability. Building upon these techniques, which in particular use Dafermos' theory of generalized characteristics, we first derive necessary optimality condition for the optimal control problem with state constraints. As a second step, we use the Moreau-Yosida regularization method for the algorithmic treatment of the pointwise state constraints, which was originally introduced by Ito and Kunisch. Here, we prove strong convergence of the optimal controls of the regularized problems to an optimal control to the problem with state constraints and weak convergence of the corresponding Lagrange multiplier estimates.