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Exact controllability of semilinear abstract systems with application to waves and plates boundary control problems

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Abstract

This paper studies (global) exact controllability of abstract semilinear equations. Applications include boundary control problems for wave and plate equations on the explicitly identified spaces of exact controllability of the corresponding linear systems.

Contents. 1. Motivating examples, corresponding results, literature. 1.1. Motivating examples and corresponding results. 1.2. Literature. 2. Abstract formulation. Statement of main result. Proof. 2.1. Abstract formulation. Exact controllability problem. 2.2. Assumptions and statement of main result. 2.3. Proof of Theorem 2.1. 3. Application: a semilinear wave equation with Dirichlet boundary control. Problem (1.1). 3.1. The caseγ = 1 in Theorem 1.1 for problem (1.1). 3.2. The caseγ = 0 in Theorem 1.1 for problem (1.1). 4. Application: a semilinear Euler—Bernoulli equation with boundary controls. Problem (1.14). 4.1. Verification of assumption (C.1): exact controllability of the linear system. 4.2. Abstract setting for problem (1.14). 4.3. Verification of assumptions (A.1)–(A.5). 4.4. Verification of assumption (C.2). 5. Proof of Theorem 1.2 and of Remark 1.2. Appendix A: Proof of Theorem 3.1. Appendix B: Proof of (4.9) and of (4.10b). References.

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References

  1. P. M. Anselone, Collectively Compact Operator Approximation Theory, Prentice-Hall, Englewood Cliffs, New Jersey, 1971.

    Google Scholar 

  2. J. P. Aubin, Un theoreme de compacite, C. R. Acad. Sci. Paris 256, 1963, 5042–5044.

    Google Scholar 

  3. C. Bardos, G. Lebeau, and R. Rauch, Controle et stabilization dans des problems hyperboliques, Appendix II in [L5].

  4. M. Berger, Nonlinearity and Functional Analysis, Academic Press, New York, 1977.

    Google Scholar 

  5. N. Carmichael and M. D. Quinn, Fixed-point methods in nonlinear control, Proceedings of the 2nd International Conference, Vorau, Austria, 1984, Lecture Notes in Control and Information Sciences, Vol. 75, Springer-Verlag, Berlin, 1985, pp. 24–51.

    Google Scholar 

  6. G. Chen, Energy decay estimates and exact boundary controllability of the wave equations in a bounded domain, J. Math. Pures Appl. (9), 58, 1979, 249–274.

    Google Scholar 

  7. G. Da Prato, I. Lasiecka, and R. Triggiani, A direct study of Riccati equations arising in boundary control problems for hyperbolic equations, J. Differential Equations 64(1), 1986, 26–47.

    Google Scholar 

  8. K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, 1980, Section 15.2, p. 152.

    Google Scholar 

  9. F. Flandoli, I. Lasiecka, and R. Triggiani, Algebraic Riccati equations with non-smoothing observation arising in hyperbolic and Euler—Bernoulli equations, Ann. Mat. Pura Appl. (IV) CLIII, 1988, 307–382.

    Google Scholar 

  10. H. Frankowska, Some inverse mapping theorems, SISSA 113 M, September 1988, Trieste.

  11. P. Grisvard, A caracterization de quelques espaces d'interpolation, Arch. Rational Mech. Anal. 25, 1967, 40–63.

    Google Scholar 

  12. H. Hermes, Controllability and the singular problem, SIAM J. Control Optim. 2, 1965, 241–260.

    Google Scholar 

  13. L. F. Ho, Observabilite frontiere de l'equation des ondes, C. R. Acad. Sci. Paris 302, 1986, 443–446.

    Google Scholar 

  14. M. A. Horn, Exact controllability of the Euler—Bernoulli plate via bending moments only, Preprint, University of Virginia, August 1990.

  15. L. Hormander, Linear Partial Differential Operators, Springer-Verlag, Berlin, 1969.

    Google Scholar 

  16. L. Hormander, The Analysis of Linear Partial Differential Operators, Vol. III, Springer-Verlag, Berlin, 1985.

    Google Scholar 

  17. V. M. Isakov, On the uniqueness of the solution of the Cauchy problem, Dokl. Akad. Nauka SSSR 225, 1980. English translation: Soviet Math, Dokl. 22 1980, 639–642.

    Google Scholar 

  18. T. Kato, Perturbation Theory of Linear Operators, Springer-Verlag, Berlin, 1966.

    Google Scholar 

  19. E. E. Kenig, A. Ruiz, and L. D. Sogge, Uniform Sobolev inequalities and unique continuation for second-order constant coefficient differential operators, Duke Math. J. 1987, 329–397.

  20. V. Komornik, Controlabilite exacte en un temps minimal, C. R. Acad. Sci. Paris Ser. I 304, 1987, 223–225.

    Google Scholar 

  21. J. Lagnese, Decay of solutions of wave equations in a bounded region with boundary dissispation, J. Differential Equations 50, 1983, 163–182.

    Google Scholar 

  22. I. Lasiecka, Exact controllability of a plate equation with one control acting as a bending moment, in Differential Equations, Stability and Control, Lectures Notes in Pure and Applied Mathematics, vol. 127, September 1990, Proceedings of International Conference, June 1989, Colorado Springs.

  23. I. Lasiecka, Stabilization of the semilinear wave equation with viscous damping, J. Differential Equations 85, 1990, 73–87.

    Google Scholar 

  24. I. Lasiecka, J. L. Lions, and R. Triggiani, Nonhomogeneous boundary value problems for second-order hyperbolic operators, J. Math. Pures Appl. 69, 1986, 149–192.

    Google Scholar 

  25. I. Lasiecka and R. Triggiani, A cosine operator approach to modelingL 2(0,T; L 2(Γ))-boundary input hyperbolic equations, Appl. Math. Optim. 7, 1981, 35–83.

    Google Scholar 

  26. I. Lasiecka and R. Triggiani, Regularity of hyperbolic equations underL 2(0,T; L 2(Γ))-boundary terms, Appl. Math. Optim. 10, 1983, 275–286.

    Google Scholar 

  27. I. Lasiecka and R. Triggiani, Riccati equations for hyperbolic partial differential equations withL 2(0,T; L 2(Γ))-Dirichlet boundary terms, SIAM J. Control Optim. 24, 1986, 884–926.

    Google Scholar 

  28. I. Lasiecka and R. Triggiani, Uniform exponential energy decay of the wave equation in a bounded region withL 2(0, ∞:L 2(Γ))-feedback control, J. Differential Equations 66, 1987, 340–390.

    Google Scholar 

  29. I. Lasiecka and R. Triggiani, Exact controllability for the wave equation with Neumann boundary control, Appl. Math. Optim. 19, 1989, 243–290.

    Google Scholar 

  30. I. Lasiecka and R. Triggiani, Exact controllability of the Euler—Bernoulli equation with controls in the Dirichlet and Neumann boundary conditions: a non-conservative case, SIAM J. Control Optim. 27, 1989, 330–373.

    Google Scholar 

  31. I. Lasiecka and R. Triggiani, Regularity theory for a class of Euler—Bernoulli equations: a cosine operator approach, Boll. Un. Mat. Ital. B (7) 3, 1989, 199–228.

    Google Scholar 

  32. I. Lasiecka and R. Triggiani, Exact controllability of Euler—Bernoulli equations with boundary controls for displacement and moments, J. Math. Anal. Appl. 145, 1990, 1–33.

    Google Scholar 

  33. I. Lasiecka and R. Triggiani, Sharp regularity results for mixed second-order hyperbolic equations of Neumann type: theL 2-boundary case, Ann. Mat. Pura Appl., to appear Nov. 1990.

  34. I. Lasiecka and R. Triggiani, Exact controllability and uniform stabilization of Kirchoff plates with only one active control in Δw∣Σ. J. Differential Equations, to appear. Announcement in CDC Proceedings, Tampa, Florida, Dec. 1989, pp. 2280–2281.

  35. I. Lasiecka and R. Triggiani, Exact controllability and uniform stabilization of Euler—Bernoulli equations with only one active control in Δw∣Σ. Boll. Un. Mat. Ital., to appear. Short announcement in Proceedings of Vorau Conference, July 1988, International Series of Numerical Mathematics, Vol. 91, Birkhäuser, Basch, pp. 391–400; and Rendiconti Accad. Nazionale Lincei, Roma.

  36. J. L. Lions, Control of Singular Distributed Systems, Gauthier-Villars, Paris, 1983.

    Google Scholar 

  37. J. L. Lions, Exact controllability, stabilization and perturbations, SIAM Rev. 30, 1988, 1–68, and Masson, 1989.

    Google Scholar 

  38. J. L. Lions, and E. Magenes, Nonhomogeneous boundary value problems and applications, Vols I and II, Springer-Verlag, Berlin, 1972.

    Google Scholar 

  39. W. Littman, Boundary control theory for beams and plates. Proceedings CDC Conference, Fort Lauderdale, 1985, pp. 2007–2009.

  40. W. Littman, Near Optimal Time Boundary Controllability for a Class of Hyperbolic Equations, Lecture Notes in Control and Information Sciences, Vol. 97, Springer-Verlag, Berlin, pp. 307–312.

  41. W. Littman, Private communication.

  42. C. Parenti and F. Segala, Propagation and reflection of singularities for a class of evolution equations, Comm. Partial Differential Equations 1981, 741–782.

  43. A. Ruiz, Unique continuation for weak solutions of the wave equation plus a potential, Preprint.

  44. J. T. Schwartz, Nonlinear Functional Analysis, Gordon and Breach, New York, 1969.

    Google Scholar 

  45. T. Seidman, Invariance of the reachable set under nonlinear perturbations, SIAM J. Control Optim. 23, 1987, 1173–1191.

    Google Scholar 

  46. J. Simon, Compact sets in the spaceL P(0,T; B), Ann. Mat. Pura Appl. (4) CXLVI, 1987, 65–96.

    Google Scholar 

  47. R. Triggiani, Controllability and observability in Banach space with bounded operators, SIAM J. Control Optim. 13, 1975, 462–491.

    Google Scholar 

  48. R. Triggiani, A cosine operator approach to modeling boundary inputs problems for hyperbolic systems, Proceedings of the 8th IFIP Conference on Optimization Techniques, University of Wuzburg, West Germany, September 1977, Lecture Notes in Control and Information Sciences, Vol. 6, Springer-Verlag, Berlin, 1978, pp. 380–390.

    Google Scholar 

  49. R. Triggiani, Exact controllability onL 2(Ω) ×H −1(Ω) of the wave equation with Dirichlet boundary control acting on a portion of the boundary, and related problems, Appl. Math Optim. 18, 1988, 241–277.

    Google Scholar 

  50. R. Triggiani, Wave equation on a bounded domain with boundary dissipation: an operator approach, J. Math. Anal. Appl. 137, 1989, 438–451.

    Google Scholar 

  51. E. Zuazua, Exact boundary controllability for the semilinear wave equation, Preprint 1988; presented at CDC Conference, Austin, December 1988; also in Non-Linear PDE and Their Applications, Research Notes in Mathematics, Pitman, Boston, pp. 1265–1268.

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  1. Department of Applied Mathematics, University of Virginia, Thornton Hall, 22903, Charlottesville, VA, USA

    I. Lasiecka & R. Triggiani

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Research partially supported by the National Science Foundation under Grant DMS-8902811 and by the Air Force Office of Scientific Research under Grant AFOSR-87-0321. The main results of this paper are announced in: Proceedings of the 28th Conference on Decision and Control, Tampa, Florida, December 1989, pp 2291–2294.

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Lasiecka, I., Triggiani, R. Exact controllability of semilinear abstract systems with application to waves and plates boundary control problems. Appl Math Optim 23, 109–154 (1991). https://doi.org/10.1007/BF01442394

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