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Applied Mathematics and Optimization

, Volume 16, Issue 1, pp 147–168 | Cite as

The regulator problem for parabolic equations with dirichlet boundary control

Part I: Riccati's Feedback Synthesis and Regularity of Optimal Solution
  • I. Lasiecka
  • R. Triggiani
Article

Abstract

This paper considers the regulator problem for a parabolic equation (generally unstable), defined on an open, bounded domain Ω, with control functionu acting in the Dirichlet boundary condition: minimize the quadratic functional which penalizes theL2(0, ∞; L2(Ω))-norm of the solutiony and theL2(0, ∞; L2(Γ))-norm of the Dirichlet controlu. The paper is divided in two parts. Part I derives, in a constructive way, the algebraic Riccati equation satisfied by the candidate Riccati operator solution (unique in our case) and, moreover, studies the regularity properties of the optimal pairu0, y0. Part II studies a Galerkin approximation of the regulator problem. It shows first the uniform analyticity and the uniform exponential stability of the underlying discrete (approximating) semigroups. Then it establishes the desired convergence properties, in particular, pointwise Riccati operators convergence and, as a final goal, convergence of the original dynamics acted upon by the discrete feedbacks.

Keywords

Parabolic Equation Dirichlet Boundary Dirichlet Boundary Condition Convergence Property Exponential Stability 
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-Verlag New York Inc. 1987

Authors and Affiliations

  • I. Lasiecka
    • 1
  • R. Triggiani
    • 1
  1. 1.Department of Applied MathematicsUniversity of VirginiaCharlottesvilleUSA

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