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On time optimal control of the wave equation,its regularization and optimality system∗

Published online by Cambridge University Press:  21 June 2012

Karl Kunisch  and
Daniel Wachsmuth
Karl Kunisch
Affiliation:
Institute for Mathematics and Scientific Computing, Heinrichstraße 36, 8010 Graz, Austria. karl.kunisch@uni-graz.at; http://www.kfunigraz.ac.at/imawww/kunisch
Daniel Wachsmuth
Affiliation:
Institute for Mathematics and Scientific Computing, Heinrichstraße 36, 8010 Graz, Austria. karl.kunisch@uni-graz.at; http://www.kfunigraz.ac.at/imawww/kunisch
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Abstract

An approximation procedure for time optimal control problems for the linear wave equation is analyzed. Its asymptotic behavior is investigated and an optimality system including the maximum principle and the transversality conditions for the regularized and unregularized problems are derived.

Keywords

Time optimal controlwave equationoptimality conditiontransversality condition
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 19 , Issue 2 , April 2013 , pp. 317 - 336
Copyright
© EDP Sciences, SMAI, 2012

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References

Ahmed, N.U., Finite-time null controllability for a class of linear evolution equations on a Banach space with control constraints. J. Optim. Theory Appl. 47 (1985) 129–158. Google Scholar
V. Barbu. Optimal control of variational inequalities. Res. Notes Math. 100 (1984).
Barcenas, D., Leiva, H. and Maya, T., The transversality condition for infinite dimensional control systems. Revista Notas de Matematica 4 (2008) 25–36. Google Scholar
Bardos, C., Lebeau, G. and Rauch, J., Un exemple d’utilisation des notions de propagation pour le contrôle et la stabilisation de problèmes hyperboliques, Nonlinear hyperbolic equations in applied sciences. Rend. Sem. Mat. Univ. Politec. Torino 1988 (1989) 11–31. Google Scholar
H.O. Fattorini, The time optimal problem for distributed control of systems described by the wave equation, in Control theory of systems governed by partial differential equations (Conf. Naval Surface Weapons Center, Silver Spring, Md., 1976). Academic Press, New York (1977) 151–175.
H.O. Fattorini, Infinite-dimensional optimization and control theory. Cambridge University Press, Cambridge. Encyclopedia of Mathematics and its Applications. 62 (1999).
H.O. Fattorini, Infinite dimensional linear control systems, The time optimal and norm optimal problems. Elsevier Science B.V., Amsterdam. North-Holland Mathematics Studies. 201 (2005).
Gugat, M., Penalty techniques for state constrained optimal control problems with the wave equation. SIAM J. Control Optim. 48 (2009/2010) 3026–3051. Google Scholar
Gugat, M. and Leugering, G., L ∞-norm minimal control of the wave equation : on the weakness of the bang-bang principle. ESAIM : COCV 14 (2008) 254–283. Google Scholar
H. Hermes and J.P. LaSalle, Functional analysis and time optimal control. Academic Press, New York. Math. Sci. Eng. 56 (1969).
K. Ito and K. Kunisch, Lagrange multiplier approach to variational problems and applications. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA. Advances in Design and Control. 15 (2008).
Krabs, W., On time-minimal distributed control of vibrating systems governed by an abstract wave equation. Appl. Math. Optim. 13 (1985) 137–149. Google Scholar
Krabs, W., On time-minimal distributed control of vibrations. Appl. Math. Optim. 19 (1989) 65–73. Google Scholar
Lasiecka, I., Lions, J.-L. and Triggiani, R., Nonhomogeneous boundary value problems for second order hyperbolic operators. J. Math. Pures Appl. 65 (1986) 149–192. Google Scholar
E.B. Lee and L. Markus, Foundations of optimal control theory. John Wiley & Sons Inc., New York (1967).
Lions, J.-L., Exact controllability, stabilization and perturbations for distributed systems. SIAM Rev. 30 (1988) 1–68. Google Scholar
Peichl, G. and Schappacher, W., Constrained controllability in Banach spaces. SIAM J. Control Optim. 24 (1986) 1261–1275. Google Scholar
Phung, K.D., Wang, G. and Zhang, X., On the existence of time optimal controls for linear evolution equations. Discrete Contin. Dyn. Syst. Ser. B 8 (2007) 925–941. Google Scholar
Zuazua, E., Propagation, observation, and control of waves approximated by finite difference methods. SIAM Rev. 47 (2005) 197–243. Google Scholar