Abstract
Galerkin (finite elements) approximations of compensators/estimators for partially observed infinite-dimensional systems with unbounded control operators are considered. It is shown that these approximations enjoy two features: (i) they provide a near-optimal performance, and (ii) they retain uniform asymptotic stability properties (uniform with respect to the parameter of discretization) of the entire closed loop system. Examples of hyperbolic equations with boundary controls and boundary observations are provided.
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References
Balakrishnan, A. V.: Applied Functional Analysis, Springer-Verlag, New York, 1975.
Curtain, R.: Finite dimensional compensators for parabolic distributed systems with unbounded control and observation, SIAM J. Control 22 (1984), 255–277.
Da Prato, G., Lasiecka, I. and Triggiani, R.: A direct study of Riccati equations arising in hyperbolic boundary control problems, J. Differential Equations, 64(1) (1986), 26–47.
Datko, R.: Extending a theorem of Liapunov to Hilbert spaces, J. Math. Anal. Appl. 32 (1970), 610–613.
Desch, W., Lasiecka, I., and Schapacher, W.: Feedback boundary control problems for linear semigroups, Israel J. Math. 51(3) (1985).
Flandoli, F., Lasiecka, I., and Triggiani, R.: Algebraic Riccati equations with nonsmoothing observation arising in hyperbolic and Euler-Bernoulli boundary control problems, Ann. Mat. Pura Appl. 153 (1981), 307–382.
Gibson, J. S.: An analysis of optimal model regulation: Convergence and stability, SIAM J. Control Optim. 19 (1981), 686–707.
Gibson, J. S.: Approximation theory for linear quadratic Gaussian control of flexible structures. SIAM J. Control Optim. 29 (1990), 1–38.
Gibson, J. S.: The Riccati integral equations for optimal control problems in Hilbert spaces, SIAM J. Control Optim. 17 (1979), 537–565.
Grenier, G. and Nagel, R.: On the stability of strongly continuous semigroups of positive operators on L 2(μ), Ann. Schuola Norm. Sup. Pisa X.2 (1983), 257–262.
Itô, K.: Finite dimensional compensators for infinite dimensional systems via Galerkin-type approximations, J. Control Optim. 28 (1990), 1251–1269.
Kato, T.: Perturbations Theory for Linear Operators, Springer-Verlag, New York, Berlin, 1976.
Lasiecka, I.: Approximations of solutions to infinite-dimensional Algebraic Riccati equations with unbounded input operators, Numer. Fund. Anal. Optim. 11(304), (1990), 303–378.
Lasiecka, I.: Stabilization of hyperbolic and parabolic systems with nonlinearly perturbed boundary conditions. J. Differential Equation 75 (1988), 53–87.
Lasiecka, I and Triggiani, R.: Differential and Algebraic Riccoh Equations with Applications to Boundary Point Control Problems: Continuous and Approximation Theory. LNCIS vol 164, Springer-Verlag, New York, 1991.
Monauni, L.: On the abstract Cauchy problem and the generator problem for semigroups of bounded operators. Technical Report No. 90 (1980), Controfl Theory Centre, University of Warwick, England.
Pazy, A.: Semigroups of Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1986.
Schumacher, J. M.: A direct approach to compensator design for distributed parameter systems, SIAM J. Control 21 (1983), 823–837.
Weiss, G.: Two conjectures on the admissibility of control operators. Proc. Conf. Distributed Parameter Systems, Vorau (Austria), July 1990.
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Lasiecka, I. Galerkin approximations of infinite-dimensional compensators for flexible structures with unbounded control action. Acta Appl Math 28, 101–133 (1992). https://doi.org/10.1007/BF00047552
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DOI: https://doi.org/10.1007/BF00047552