Uniform controllability of semidiscrete approximations of parabolic control systems

Systems and Control Letters

Volumn 55, Issue 7, 2006, Pages 597-609

  Labbé, Stéphane ^a   Trélat, Emmanuel ^a  

^a UNIVERSITÉ PARIS SACLAY   (France)

Author keywords

Controllability; Discretization; Hilbert uniqueness method; Observability inequality; Partial differential equation

Indexed keywords

APPROXIMATION THEORY; CONTROLLABILITY; DIFFERENTIAL EQUATIONS; LINEAR CONTROL SYSTEMS; MATHEMATICAL MODELS;

DISCRETIZATION; HILBERT UNIQUENESS METHOD; OBSERVABILITY INEQUALITY; PARTIAL DIFFERENTIAL EQUATION;

CONTROL SYSTEMS;

EID: 33646164179     PISSN: 01676911     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.sysconle.2006.01.004     Document Type: Article

References (35)

1. H.T. Banks, K. Ito, Approximation in LQR problems for infinite dimensional systems with unbounded input operators, J. Math. Systems Estim. Control 7 (1) (1997), 119-122 (short version), 34 p, (Retrieval code 62459).
2. Banks H.T., and Kunisch K. The linear regulator problem for parabolic systems. SIAM J. Control Optim. 22 5 (1984) 684-698
3. Bramble J., Shatz A., Thomee V., and Wahlbin L. Some convergence estimates for semidiscrete Galerkin type approximations for parabolic equations. SIAM J. Numer. Anal. 14 (1977) 218-241
4. C. Castro, S. Micu, Boundary controllability of a linear semi-discrete 1-D wave equation derived from a mixed finite element method, Numer. Math., to appear.
5. A. Ern, J.L. Guermond, Theory and Practice of Finite Elements, Applied Mathematical Series vol. 159, Springer, New York, NY, 2004.
6. Gibson J.S. The Riccati integral equations for optimal control problems on Hilbert spaces. SIAM J. Control Optim. 17 (1979) 537-565
7. R. Glowinski, J.-L. Lions, Exact and approximate controllability of distributed parameter systems, Acta Numerica (1996) 159-333.
8. Infante J.A., and Zuazua E. Boundary observability for the space semi-discretizations of the 1-D wave equation. M2AN Math. Model. Numer. Anal. 33 2 (1999) 407-438
9. Kappel F., and Salamon D. An approximation theorem for the algebraic Riccati equation. SIAM J. Control Optim. 28 5 (1990) 1136-1147
10. J. Lagnese, Boundary Stabilization of Thin Plates, SIAM, Philadelphia, 1989.
11. Lasiecka I. Convergence estimates for semidiscrete approximations of nonselfadjoint parabolic equations. SIAM J. Numer. Anal. 21 5 (1977) 894-908
12. I. Lasiecka, R. Triggiani, Control theory for partial differential equations: continuous and approximation theories. I. Abstract parabolic systems, Encyclopedia of Mathematics and its Applications, vol. 74, Cambridge University Press, Cambridge, 2000.
13. Leon L., and Zuazua E. Boundary controllability of the finite-difference space semi-discretizations of the beam equation. ESAIM Control Optim. Calc. Var. 8 (2002) 827-862
14. Lions J.-L. Exact controllability, stabilization and perturbations for distributed systems. SIAM Rev. 30 (1988) 1-68
15. Liu Z., and Zheng S. Semigroups associated with dissipative systems. Res. Notes Math. 398 (1999)
16. A. Lopez, E. Zuazua, Some new results related to the null controllability of the 1-d heat equation, Sém. EDP, Ecole Polytechnique, VIII, 1998, 1-22.
17. Mizel V.J., and Seidman T.I. Observation and prediction for the heat equation. J. Math. Anal. Appl. 28 (1969) 303-312
18. Negreanu M., and Zuazua E. Uniform boundary controllability of a discrete 1-D wave equation. Systems Control Lett. 48 3-4 (2003) 261-279
19. Pazy A. Semigroups of linear operators and applications to partial differential equations. Appl. Math. Sci. 44 (1983)
20. K. Ramdani, T. Takahashi, M. Tucsnak, Uniformly exponentially stable approximations for a class of second order evolution equations, Preprint University Nancy, 2004.
21. Rebarber R., and Weiss G. Necessary conditions for exact controllability with a finite-dimensional input space. Systems Control Lett. 40 3 (2000) 217-227
22. Russell D.L. A unified boundary controllability theory for hyperbolic and parabolic partial differential equations. Stud. Appl. Math. 52 (1973) 189-211
23. Russell D.L. Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions. SIAM Rev. 20 4 (1978) 639-739
24. Seidman T.I. Observation and prediction for the heat equation, III. J. Differential Equation 20 1 (1976) 18-27
25. Seidman T.I. Observation and prediction for the heat equation, IV. Patch observability and controllability, SIAM J. Control. Optim. 15 3 (1977) 412-427
26. Tcheugoué Tébou L.R., and Zuazua E. Uniform exponential long time decay for the space semi-discretization of a locally damped wave equation via an artificial numerical viscosity. Numer. Math. 95 3 (2003) 563-598
27. V. Thomée, Galerkin Finite Element Methods for Parabolic Problems, Lecture Notes in Mathematics, Springer, Berlin, 1984.
28. Weiss G. Admissible observation operators for linear semigroups. Israel J. Math. 65 1 (1989) 17-43
29. Weiss G. Admissibility of unbounded control operators. SIAM J. Control. Optim. 27 3 (1989) 527-545
30. Zuazua E. Boundary observability for the finite-difference space semi-discretizations of the 2-D wave equation in the square. J. Math. Pures Appl. (9) 78 5 (1999) 523-563
31. Zuazua E. Controllability of partial differential equations and its semi-discrete approximations. Discrete Contin. Dyn. Syst. 8 2 (2002) 469-513
32. Zuazua E. Optimal and approximate control of finite-difference approximation schemes for the 1-D wave equation. Rendiconti di Matematica VIII 24 II (2004) 201-237
33. Zuazua E. Propagation, observation, control and numerical approximation of waves approximated by finite difference method. SIAM Rev. 47 2 (2005) 197-243
34. E. Zuazua, Controllability and observability of partial differential equations: some results and open problems, in: C. Dafermos, E. Feireisl (Eds.), Handbook of Differential Equations: Evolutionary Differential Equations, Elsevier, Amsterdam, to appear.
35. E. Zuazua, Control and numerical approximation of the wave and the heat equations, Proc. Internat. Congress Math. (2006) to appear.