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 theL 2(0, ∞; L2(Ω))-norm of the solutiony and theL 2(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 pairu 0, 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.
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References
Anselone PM (1971) Collectively Compact Operator Approximation Theory. Prentice-Hall, Englewood Cliffs, NJ
Balakrishnan AV (1981) Applied Functional Analysis, 2nd ed. Springer-Verlag, New York
Banks TH, Kunish K (1984) The linear regulator problem for parabolic systems. SIAM J Control Optim 22:684–699
Bramble J, Schatz A (1970) Rayleigh-Ritz-Galerkin methods for Dirichlet's problem using subspaces without boundary conditions. Comm Pure Appl Math 23:653–657
Datko R (1970) Extending a theorem of Liapunov to Hilbert spaces, J Math Anal Appl 32:610–616
Dunford N, Schwartz J (1963) Linear Operators, II. Wiley-Interscience, New York
Fattorini HO (1983) The Cauchy problem. In Encylcopedia of Mathematics and Its Applications, vol. 18. Addison-Wesley, Reading, MA
Flandoli F (1984) Riccati equation arising in a boundary control problem with distributed parameters. SIAM J Control Optim 22:76–86
Flandoli F (1985) Algebraic Riccati equation arising in boundary control problems. Preprint
Friedman A (1976) Partial Differential Equations (reprint). R. Knegar Huntington, New York
Fujiwara D (1967) Concrete characterization of the domains of fractional powers of some elliptic differential operators of the second order. Proc. Japan Acad 43:82–86
Kato T (1966) Perturbation Theory of Linear Operators. Springer-Verlag, New York
Kondratiev VA (1967) Boundary problems for elliptic equations in domains with conical or angular points. Trans Moscow Math Soc 16
Lasiecka I (1980) Unified theory for abstract parabolic boundary problems; a semigroup approach. Appl Math Optimiz 6:287–333
Lasiecka I (1984) Convergence estimates for semidiscrete approximations of non-selfadjoint parabolic equations. SIAM J Numer Anal 21:894–909
Lasiecka I (1986) Galerkin approximations of abstract parabolic boundary value problems with rough boundary data,L p-theory. Math Comput 47:55–75
Lasiecka I, Lions JL, Triggiani R (1986) Non-homogeneous boundary value problems for second-order hyperbolic operators. J Math Pures Appl 65:149–192
Lasiecka I, Manitius A (1985) Differentiability and convergence rates of approximating semigroups for retarded functional differential equations. SIAM J Numer Anal (to appear)
Lasiecka I, Triggiani R (1983) Dirichlet boundary control problem for parabolic equations with quadratic cost: analyticity and Riccati's feedback synthesis. SIAM J Control Optim 21:41–68
Lasiecka I, Triggiani R (1983) Stabilization and structural assignement of Dirichlet boundary feedback parabolic equations. SIAM J Control Optim 21:766–803
Lasiecka I, Triggiani R (1983) Feedback semigroups and cosine operators for boundary feedback parabolic and hyperbolic equations. J Differential Equations 47:245–272
Lasiecka I, Triggiani R (1987) Riccati equations for hyperbolic partial differential equations withL 2(0, T; L2(Γ))—Dirichlet boundary terms. SIAM J Control Optim (to appear)
Lions JL, Magenes E (1972) Non-homogeneous Boundary Value Problems, I, II. Springer-Verlag, New York
Necas J (1967) Les Methodes Direct en Theorie des Equations Elliptiques. Masson et Cie (Édit.) Paris-Vie
Nitsche J (1971) Über ein Variationsprincip der Lösung von Dirichlet Problem bei Verwendung von Teilräumen, die keinen Randbegingungen unterworfen sind. Abh Math Sem Univ Hamburg 36:9–15
Pazy A (1974) Semigroups of operators and applications to partial differential equations. Lecture Notes. Department of Mathematics, University of Maryland
Sadosky C (1979) Interpolation of operators and singular integrals. In Pure and Applied Mathematics. Marcel Dekker, New York
Sorine M (1981) Une resultat d'existence et unicité pour l'equation de Riccati stationnaire. Rapport INRIA No 55.
Triebel H (1978) Interpolation Theory, Function Spaces, Differential Operators. VEB Deutscher Verlag der Wissenshaften, Berlin (licenced edition of North-Holland Publishing Company, 1978)
Triggiani R (1980) Boundary feedback stabilizability of parabolic equations. Appl Math Optim 6:201–220
Zabczyk J (1978) On decomposition of generators. SIAM J Control Optim 16:523–534
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Research partially supported by the National Science Foundation under Grant DMS-8301668.
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Lasiecka, I., Triggiani, R. The regulator problem for parabolic equations with dirichlet boundary control. Appl Math Optim 16, 147–168 (1987). https://doi.org/10.1007/BF01442189
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DOI: https://doi.org/10.1007/BF01442189