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

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.

[1]  H. Frankowska.  Some inverse mapping theorems , 1990 .

[2]  K. Deimling.  Nonlinear functional analysis , 1985 .

[3]  R. Triggiani.  Controllability and Observability in Banach Space with Bounded Operators , 1975 .

[4]  Irena Lasiecka,et al.  A cosine operator approach to modelingL2(0,T; L2 (Γ))—Boundary input hyperbolic equations , 1981 .

[5]  Roberto Triggiani,et al.  Exact boundary controllability onL2(Ω) ×H−1(Ω) of the wave equation with dirichlet boundary control acting on a portion of the boundary∂Ω, and related problems , 1988 .

[6]  N. Carmichael,et al.  Fixed point methods in nonlinear control , 1988 .

[7]  Propagation and reflection of singularities for a class of evolution equations , 1981 .

[8]  R. Triggiani,et al.  Exact controllability of the wave equation with Neumann boundary control , 1989 .

[9]  P. Grisvard,et al.  Caractérisation de quelques espaces d'interpolation , 1967 .

[10]  H. Hermes.  Controllability and the Singular Problem , 1964 .

[11]  J. Lions,et al.  Non-homogeneous boundary value problems and applications , 1972 .

[12]  J. Lions.  Exact controllability, stabilization and perturbations for distributed systems , 1988 .

[13]  Irena Lasiecka,et al.  Uniform exponential energy decay of wave equations in a bounded region with L2(0, ∞; L2 (Γ))-feedback control in the Dirichlet boundary conditions , 1987 .

[14]  R. Triggiani,et al.  Riccati equations for hyperbolic partial differential equations with L2(O,T; L2(T)) - Dirichlet boundary terms , 1985, 1985 24th IEEE Conference on Decision and Control.

[15]  Irena Lasiecka,et al.  Algebraic Riccati equations with non-smoothing observation arising in hyperbolic and Euler-Bernoulli boundary control problems , 1988 .

[16]  R. Triggiani,et al.  Exact controllability of the Euler-Bernoulli equation with controls in the Dirichlet and Neumann boundary conditions: a nonconservative case , 1989 .

[17]  J. Simon.  Compact sets in the spaceLp(O,T; B) , 1986 .

[18]  REGULARITY THEORY FOR A CLASS OF NONHOMOGENEOUS EULER-BERNOULLI EQUATIONS: A COSINE OPERATOR APPROACH , 1989 .

[19]  J. Schwartz.  Nonlinear Functional Analysis , 1969 .

[20]  R. Triggiani.  A cosine operator approach to modelling boundary input hyperbolic systems , 1978 .

[21]  J. Lagnese.  Decay of solutions of wave equations in a bounded region with boundary dissipation , 1983 .

[22]  Christopher D. Sogge,et al.  Uniform Sobolev inequalities and unique continuation for second order constant coefficient differential operators , 1987 .

[23]  Wan-Se Kim.  Double-periodic boundary value problem for non-linear dissipative hyperbolic equations , 1990 .

[24]  R. Triggiani,et al.  Regularity of hyperbolic equations underL2(0,T; L2(Γ))-Dirichlet boundary terms , 1983 .

[25]  Jacques-Louis Lions,et al.  Modelling Analysis and Control of Thin Plates , 1988 .

[26]  Lop Fat Ho.  Observabilité frontière de l'équation des ondes , 1986 .

[27]  W. Littman.  Boundary control theory for beams and plates , 1985, 1985 24th IEEE Conference on Decision and Control.

[28]  W. Littman.  Near optimal time boundary controllability for a class of hyperbolic equations , 1987 .

[29]  Irena Lasiecka,et al.  Exact controllability of the Euler-Bernoulli equation with boundary controls for displacement and moment , 1990 .

[30]  Irena Lasiecka,et al.  Sharp regularity theory for second order hyperbolic equations of Neumann type , 1990 .

[31]  R. Triggiani.  Wave equation on a bounded domain with boundary dissipation: An operator approach☆ , 1989 .

[32]  R. Triggiani,et al.  Exact controllability for wave equation with Neumann boundary control , 1988 .

[33]  Stabilization of the semilinear wave equation with viscous damping , 1990 .

[34]  P. H. Müller,et al.  L. Hörmander, Linear Partial Differential Operators. VIII + 284 S. m. 1 Fig. Berlin/Göttingen/Heidelberg 1963. Springer-Verlag. Preis geb. DM 42,- . , 1964 .

[35]  L. Hörmander.  Linear Partial Differential Operators , 1963 .

[36]  Thomas I. Seidman,et al.  Invariance of the reachable set under nonlinear perturbations , 1987 .

[37]  J. Lions,et al.  Non homogeneous boundary value problems for second order hyperbolic operators , 1986 .

[38]  L. Hörmander.  The analysis of linear partial differential operators , 1990 .

[39]  Irena Lasiecka,et al.  A direct study of the Riccati equation arising in hyperbolic boundary control problems , 1986 .