Optimal control of hyperbolic conservation laws on bounded domains with switching controls (Q2827276)

The aim of the dissertation is a study on optimal control problems for systems governed by scalar conservation laws on bounded domains, in one dimension, including switching controls. The main issue is the proof of a generalized differentiability for the control -- to -- state mapping. This generalized differentiability is understood as the so-called shift-differentiability which has been introduced by \textit{S. Ulbrich} in [Optimal control of nonlinear hyperbolic conservation laws with source terms. München: Technische Universität München (Habilitation) (2001)]. Two main optimal control problems are addressed. One considers the case where the controls are the initial and boundary data and the case of controlling the traffic light where the control variables are the switching times between red and green phases. Control can act in the source term too. In both cases the shift-differentiability of the control-to state mapping is proved. As usual, the first chapter of the work is devoted to Introduction, Motivation, Literature. The second chapter deals with notations, definitions and basic results used in the next chapters. In the third chapter optimal control problems for hyperbolic conservation law are introduced. Mathematical models for hyperbolic conservation laws and existence and uniqueness of solutions are discussed first. As an application of hyperbolic conservation laws to the real-world problems the LWR-model, the traffic light problem, their solutions and structural properties of solutions to Cauchy problems, initial boundary value problems and BV solutions to the traffic light problem are considered. In the second part of this chapter general optimal control problems, results on the existence of optimal controls and the standard adjoint calculus are presented. The optimal control for an IBVP for a hyperbolic conservation law and optimal control of an on/off switching, the way they are addressed in this work, are introduced. The issue of linear transport equations with discontinuous coefficients is discussed in the forth chapter. The notion of reversible solution and results on its existence, solvability and regularity are presented. The fifth chapter is devoted to the shift-differentiability. The definition of this notion and some properties connected to it are introduces first. Finally, the main results of this thesis, the shift-differentiability of the control-to-state mapping for the initial-boundary value problem and the traffic light problem including a formula for the shift derivative as well as an adjoint-based formula for the gradient for the reduced objective function are presented. Chapter 6 and 7 are concerned with the proof of the main results for the initial-boundary value problem respectively for the traffic light problem, presented in the previous chapter. The strategy in both cases is similar: investigation of differentiability in the neighborhood of continuity points and investigation of differentiability around shock points. Possible extensions of the research are discussed in the eight chapter and conclusions in the last chapter. References extents to 92 titles.