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Boundary feedback stabilization of a parabolic equation

  • Thomas I. Seidman
Session 7 Distributed Parameter Systems
  • 136 Downloads
Part of the Lecture Notes in Control and Information Sciences book series (LNCIS, volume 62)

Abstract

The equation (*) ut=uxx+qu+f will, in general, be unstable for positive q. We consider control through the boundary conditions u(·,0)=0, ux(·,1)=ф with observation available of φ :=3 u(·,1) and no knowledge of the initial state or of the input f. It is shown that one can construct a linear feedback law of the form (**) ф(t)=〈λ,φt〉+〈μ,фt〉 (фt, φt are intervals of past history) which stabilizes (*).

Keywords

Parabolic Equation Boundary Control Unstable Mode Finite Order Quadrature Point 
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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References

  1. [1]
    R. CURTAIN, Finite dimensional compensators for parabolic distributed systems with unbounded control and observation, TW-234, Rijks-universitat Groningen, 1982.Google Scholar
  2. [2]
    S. DOLECKI and D. RUSSELL, A general theory of observation and control, SIAM J. Contr. Opt. 15(1977), pp. 185–220.Google Scholar
  3. [3]
    A. ICHIKAWA, Quadratic control of evolution equations with delays in control, SIAM J. Contr. Opt. 20 (1982), pp. 645–668.Google Scholar
  4. [4]
    I. LASIECKA and R. TRIGGIANI, Feedback semigroups and cosine operators for boundary feedback parabolic and hyperbolic equations, JDE 47(1983), pp. 246–272.Google Scholar
  5. [5]
    J.-L. LIONS and E. MAGENES, Non-Homogeneous Boundary Value Problems, v. 2, Springer-Verlag, Berlin, 1972.Google Scholar
  6. [6]
    J.M. SCHUMACHER, A direct approach to compensator design for distributed parameter systems, SIAM J. Contr. Opt.Google Scholar
  7. [7]
    T.I. SEIDMAN, Problems of boundary control and observation for diffusion processes, MRR 73-10, UMBC, 1973.Google Scholar
  8. [8]
    __, Observation and prediction for one-dimensional diffusion equations, JMAA 51(1975), pp. 165–175.Google Scholar
  9. [9]
    __. Regularity of optimal boundary controls for parabolic equations, in Analysis and Optimization of Systems (edit. A. Bensoussan and J.-L. Lions; Lecture Notes Cont. and Inf. Sci. #28), Springer Verlag, Berlin, 1980, pp. 536–550.Google Scholar
  10. [10]
    __, Regularity of optimal boundary controls for parabolic equations, I: analyticity, SIAM J. Cont. Opt. 20(1982), pp. 428–453.Google Scholar
  11. [11]
    T. I. SEIDMAN, Construction of stabilizing control laws, in preparation.Google Scholar

Copyright information

© Springer-Verlag 1984

Authors and Affiliations

  • Thomas I. Seidman
    • 1
  1. 1.Department of Mathematics and Computer ScienceUniversity of Maryland Baltimore CountyBaltimoreU.S.A.

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