Optimal boundary control of the wave equation with pointwise control constraints

Article

Abstract

In optimal control problems frequently pointwise control constraints appear. We consider a finite string that is fixed at one end and controlled via Dirichlet conditions at the other end with a given upper bound M for the L -norm of the control. The problem is to control the string to the zero state in a given finite time. If M is too small, no feasible control exists. If M is large enough, the optimal control problem to find an admissible control with minimal L 2-norm has a solution that we present in this paper.

A finite difference discretization of the optimal control problem is considered and we prove that for Lipschitz continuous data the discretization error is of the order of the stepsize.

Keywords

Optimal control Boundary control Dirichlet control Wave equation Control constraints Discretization Finite differences Discretization error 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Avdonin, S.A., Ivanov, S.A.: Families of Exponentials. Cambridge University Press, Cambridge (1995) MATHGoogle Scholar
  2. 2.
    Castro, C., Micu, S.: Boundary controllability of a linear semi-discrete 1-D wave equation derived from a mixed finite element method. Numer. Math. 102, 413–462 (2006) MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Glowinski, R., Lions, J.-L.: Exact and approximate controllability for distributed parameter systems. Acta Numer., 159–333 (1996) Google Scholar
  4. 4.
    Greif, G.: Numerische Simulation und optimale Steuerung der Wellengleichung, Diplomarbeit Universität Bayreuth, Bayreuth (2005) Google Scholar
  5. 5.
    Gugat, M.: Analytic solutions of L -optimal control problems for the wave equation. J. Optim. Theory Appl. 114, 397–421 (2002) MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Gugat, M.: L 1-optimal boundary control of a string to rest in finite time. In: Seeger, A. (ed.) Recent Advances in Optimization. Lecture Notes in Economics and Mathematical Systems, vol. 563, pp. 149–162. Springer, Berlin (2006) CrossRefGoogle Scholar
  7. 7.
    Gugat, M., Leugering, G.: Solutions of L p-norm-minimal control problems for the wave equation. Comput. Appl. Math. 21, 227–244 (2002) MathSciNetMATHGoogle Scholar
  8. 8.
    Gugat, M., Leugering, G., Sklyar, G.: L p-optimal boundary control for the wave equation. SIAM J. Control Optim. 44, 49–74 (2005) MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Gugat, M., Leugering, G.: L norm minimal control of the wave equation: on the weakness of the bang-bang principle. ESAIM COCV 14, 254–283 (2008) MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Gugat, M.: Optimal switching boundary control of a string to rest in finite time. ZAMM 88, 283–305 (2008) MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Infante, J.A., Zuazua, E.: Boundary Observability for the space discretization of the one-dimensional wave equation. Math. Model. Numer. Anal. 33, 407–438 (1999) MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Krabs, W.: Optimal control of processes governed by partial differential equations part II: Vibrations. Z. Oper. Res. 26, 63–86 (1982) MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Krabs, W.: On Moment Theory and Controllability of One-Dimensional Vibrating Systems and Heating Processes, Lecture Notes in Control and Information Science, vol. 173. Springer, Heidelberg (1992) MATHCrossRefGoogle Scholar
  14. 14.
    Lions, J.L.: Exact controllability, stabilization and perturbations of distributed systems. SIAM Rev. 30, 1–68 (1988) MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Micu, S.: Uniform boundary controllability of a semi-discrete 1-D wave equation. Numer. Math. 91, 723–768 (2002) MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Russell, D.L.: Nonharmonic Fourier series in the control theory of distributed parameter systems. J. Math. Anal. Appl. 18, 542–560 (1967) MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Zuazua, E.: Propagation, observation and control of waves approximated by finite difference methods. SIAM Rev. 47, 197–243 (2005) MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Negreanu, M., Zuazua, E.: Uniform boundary controllability of a discrete 1-D wave equation. Syst. Control Lett. 48, 261–280 (2003) MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Department MathematikFriedrich-Alexander Universität Erlangen-NürnbergNürnbergGermany

Personalised recommendations