Adjoint-based derivative computations for the optimal control of discontinuous solutions of hyperbolic conservation laws

Systems and Control Letters

Volumn 48, Issue 3-4, 2003, Pages 313-328

  Ulbrich, Stefan ^a  

^a TECHNICAL UNIVERSITY OF MUNICH   (Germany)

Author keywords

Adjoint state; Conservation laws; Differentiability; Discontinuous coefficients; Linear transport equations; Optimal control

Indexed keywords

COMPUTATIONAL METHODS; INTERPOLATION; LINEAR EQUATIONS; STABILITY;

CONSERVATION LAWS;

OPTIMAL CONTROL SYSTEMS;

EID: 0037445051     PISSN: 01676911     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0167-6911(02)00275-X     Document Type: Article

References (21)

1. Bouchut F., James F. One-dimensional transport equations with discontinuous coefficients. Nonlinear Anal. 32:1998;891-933.
2. F. Bouchut, F. James, Differentiability with respect to initial data for a scalar conservation law, in: M. Fey, R. Jeltsch (Eds.), Hyperbolic Problems: Theory, Numerics, Applications, Vol. I (Zürich, 1998), Birkhäuser, Basel, 1999, pp. 113-118.
3. Cliff E.M., Heinkenschloss M., Shenoy A.R. An optimal control problem for flows with discontinuities. J. Optim. Theory Appl. 94:1997;273-309.
4. Conway E.D. Generalized solutions of linear differential equations with discontinuous coefficients and the uniqueness question for multidimensional quasilinear conservation laws. J. Math. Anal. Appl. 18:1967;238-251.
5. Dafermos C.M. Generalized characteristics and the structure of solutions of hyperbolic conservation laws. Indiana Univ. Math. J. 26:1977;1097-1119.
6. Filippov A.F. Differential equations with discontinuous right-hand side. Amer. Math. Soc. Transl. Ser. 2. 42:1964;199-231.
7. Frank P.D., Shubin G.R. A comparison of optimization-based approaches for a model computational aerodynamics design problem. J. Comput. Phys. 98:1992;74-89.
8. Giusti E. Minimal Surfaces and Functions of Bounded Variation. 1984;Birkhäuser, Basel.
9. Godlewski E., Raviart P.-A. The linearized stability of solutions of nonlinear hyperbolic systems of conservation laws. A general numerical approach. Math. Comput. Simulation. 50:1999;77-95. (Modelling '98, Prague).
10. Hoff D. The sharp form of Olenik's entropy condition in several space variables. Trans. Amer. Math. Soc. 276:1983;707-714.
11. Kružkov S.N. First order quasilinear equations in several independent variables. Math. USSR-Sb. 10:1970;217-243.
12. Liu T.-P. Nonlinear resonance for quasilinear hyperbolic equation. J. Math. Phys. 28:1987;2593-2602.
13. Málek J., Nečas J., Rokyta M., Růžička M. Weak and Measure-Valued Solutions to Evolutionary PDEs. 1996;Chapman & Hall, London.
14. Olenik O.A. Discontinuous solutions of non-linear differential equations. Amer. Math. Soc. Transl. Ser. 2. 26:1963;95-172. (Translation from Usp. Mat. Nauk 12 (1957) 3-73).
15. Olenik O.A. Discontinuous solutions of non-linear differential equations. Amer. Math. Soc. Transl. Ser. 2. 26:1963;95-172. (Translation from Usp. Mat. Nauk 12 (1957) 3-73).
16. Slemrod M. Existence of optimal controls for control systems governed by nonlinear partial differential equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 1(4):1974;229-246.
17. Smoller J. Shock Waves and Reaction-Diffusion Equations. 2nd Edition:1994;Springer, New York.
18. Tadmor E. Local error estimates for discontinuous solutions of nonlinear hyperbolic equations. SIAM J. Numer. Anal. 28:1991;891-906.
19. S. Ulbrich, On the existence and approximation of solutions for the optimal control of nonlinear hyperbolic conservation laws, in: K.-H. Hoffmann, G. Leugering, F. Tröltzch (Eds.), Optimal Control of Partial Differential Equations (Chemnitz, 1998), International Series on Numerical Mathematics, Vol. 133, Birkhäuser, Basel, 1999, pp. 287-299.
20. S. Ulbrich, Optimal control of nonlinear hyperbolic conservation laws with source terms, Habilitation Thesis, Zentrum Mathematik, Technische Universität München, Germany, 2001.
21. S. Ulbrich, A sensitivity and adjoint calculus for discontinuous solutions of hyperbolic conservation laws with source terms, SIAM J. Control Optim. 41 (2002) 740-797.