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978-3-8439-2409-2, Reihe Mathematik

Sebastian Pfaff Optimal Control of Hyperbolic Conservation Laws on Bounded Domains with Switching Controls

209 Seiten, Dissertation Technische Universität Darmstadt (2015), Softcover, B5

Hyperbolic conservation laws are frequently used to model physical problems. In the past decades optimal control problems governed by this type of partial differential equations have become an active field of research. The crucial issue of nonlinear hyperbolic conservation laws is the development of shocks in the entropy solutions in finite time. The discontinuity of the solutions leads to the problem that the solution operators are not differentiable in a sufficiently strong sense, which has to be overcome by using generalized notions of differentiability.

This thesis is concerned with the analysis of optimal control problems governed by scalar conservation laws on bounded one-dimensional spatial domains, including switching controls. It focuses on the proof of generalized differentiability for the control-to-state mapping. Our investigations are based on the notion of Shift-Differentiability, introduced by Stefan Ulbrich in 2001. This differential concept is strong enough to directly imply Fréchet-differentiability of the composition of a usual tracking-type functional with a shift-differentiable function.

We consider two problems. The first problem is the initial-boundary value problem for scalar conservation laws with source terms in one space dimension. We show that at some fixed observation time the solution depends shift-differentiably on the control, which acts on the initial and boundary data, as well as on the source term. Initial and boundary data are assumed to be piecewise continuously differentiable functions, where the shock-creating discontinuities are considered as additional control variables. The second problem under consideration is the optimal control of an on/off-node condition for a simple linear network of scalar conservation laws, which we illustrate by a traffic light in the context of the LWR-traffic flow model. Here, the switching times between red and green phases serve as control variables. We show shift-differentiability to hold for the control-to-state mapping of this so called traffic light problem, too.

Many arguments in the proofs of the shift-differentiability are based on Dafermos' theory of generalized characteristics. For the analysis of the shock sensitivities we additionally utilize an appropriate adjoint calculus, which we also employ to derive an adjoint-based representation of the gradient of the reduced objective function.