Optimal Control of Nonlinear Hyperbolic Conservation Laws with Switching
We consider optimal control problems governed by nonlinear hyperbolic conservation laws at junctions and analyze in particular the Fréchet-differentiability of the reduced objective functional. This is done by showing that the control-to-state mapping of the considered problems satisfies a generalized notion of differentiability. We consider both, the case where the controls are the initial and the boundary data as well as the case where the system is controlled by the switching times of the node condition. We present differentiability results for the considered problems in a quite general setting including an adjoint-based gradient representation of the reduced objective function.
KeywordsOptimal control Scalar conservation law Network
The authors gratefully acknowledge the support of the German Research Foundation (DFG) within the Priority Program 1253 “Optimization with Partial Differential Equations” under grant UL158/8-1. Moreover, we gratefully acknowledge discussions with T. I. Seidman.
- 1.F. Ancona, G.M. Coclite, On the attainable set for Temple class systems with boundary controls. SIAM J. Control Optim. 43(6), 2166–2190 (2005). (electronic)Google Scholar
- 2.F. Ancona, A. Marson, On the attainable set for scalar nonlinear conservation laws with boundary control. SIAM J. Control Optim. 36(1), 290–312 (1998). (electronic)Google Scholar
- 7.A. Bressan, Hyperbolic Systems of Conservation Laws. Volume 20 of Oxford Lecture Series in Mathematics and Its Applications (Oxford University Press, Oxford, 2000). The one-dimensional Cauchy problem.Google Scholar
- 19.M. Giles, S. Ulbrich, Convergence of linearized and adjoint approximations for discontinuous solutions of conservation laws. Part 1: linearized approximations and linearized output functionals. SIAM J. Numer. Anal. 48(3), 882–904 (2010)Google Scholar
- 20.M. Giles, S. Ulbrich, Convergence of linearized and adjoint approximations for discontinuous solutions of conservation laws. Part 2: adjoint approximations and extensions. SIAM J. Numer. Anal. 48(3), 905–921 (2010)Google Scholar
- 26.S.N. Kružkov, Quasilinear parabolic equations and systems with two independent variables. Trudy Sem. Petrovsk. 5, 217–272 (1979)Google Scholar
- 30.J. Málek, J. Nečas, M. Rokyta, M. R uužička, Weak and Measure-Valued Solutions to Evolutionary PDEs. Volume 13 of Applied Mathematics and Mathematical Computation (Chapman & Hall, London, 1996)Google Scholar
- 33.E. Polak, Optimization. Volume 124 of Applied Mathematical Sciences (Springer, New York, 1997). Algorithms and consistent approximationsGoogle Scholar
- 35.S. Ulbrich, On the existence and approximation of solutions for the optimal control of nonlinear hyperbolic conservation laws, in Optimal Control of Partial Differential Equations (Chemnitz, 1998). Volume 133 of International Series of Numerical Mathematics (Birkhäuser, Basel, 1999), pp. 287–299Google Scholar
- 36.S. Ulbrich, Optimal control of nonlinear hyperbolic conservation laws with source terms. Habilitation, Zentrum Mathematik, Technische Universität München, 2001Google Scholar
- 37.S. Ulbrich, A sensitivity and adjoint calculus for discontinuous solutions of hyperbolic conservation laws with source terms. SIAM J. Control Optim. 41(3), 740–797 (2002). (electronic)Google Scholar
- 38.S. Ulbrich, Adjoint-based derivative computations for the optimal control of discontinuous solutions of hyperbolic conservation laws. Syst. Control Lett. 48(3–4), 313–328 (2003). Optimization and control of distributed systemsGoogle Scholar