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Optimal Final Value Boundary Control of Conservative Wave Equations

  • John E. Lagnese
  • Günter Leugering
Chapter
Part of the ISNM International Series of Numerical Mathematics book series (ISNM, volume 148)

Abstract

This chapter is concerned with the problem with of optimal final value control of the second-order hyperbolic system
$$ \begin{gathered} \frac{{\partial ^2 w}} {{\partial t^2 }} - \nabla \cdot (A\nabla w) + cw = F in \Omega \times (0,T) \hfill \\ w = 0 on \Gamma ^D \times (0,T) \hfill \\ \frac{{\partial w}} {{\partial \nu _A }} = f on \Gamma ^N \times (0,T) \hfill \\ w(0) = w_0 , \frac{{\partial w}} {{\partial t}}(0) = v_0 in \Omega . \hfill \\ \end{gathered} $$
(8.1.1.1)
where fL2 N × (0,T)) is the control input and F is a given distributed system input. We retain the notation of Section 6.2.1, as well as the assumptions on the coefficients A and c, the region Ω and its boundary components Γ D , Γ N delineated there. When f =0 andF =0, this system conserves energy: E(t) ≡ E(0), where the energy functional E(t) is given by
$$ E(t) = \int_\Omega {\left[ {\left| {\frac{{\partial w}} {{\partial t}}(x,t)} \right|^2 + A\nabla w(x,t) \cdot \nabla w(x,t) + c\left| {w(x,t)} \right|^2 } \right]} dx. $$

Keywords

Optimal Control Problem Optimality System Domain Decomposition Posteriori Error Estimate Observability Estimate 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Basel AG 2004

Authors and Affiliations

  • John E. Lagnese
    • 1
  • Günter Leugering
    • 2
  1. 1.Department of MathematicsGeorgetown UniversityUSA
  2. 2.Angewandte Mathematik IIUniversität Erlangen NürnbergErlangenGermany

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