Uniform energy decay rates of hyperbolic equations with nonlinear boundary and interior dissipation

• Autorzy: Lasiecka I. , Triggiani R.
• Języki publikacji: EN

Abstrakty

EN

We consider the problem of uniform stabilization of nonlinear hyperbolic equations, epitomized by the following three canonical dynamics: (1) the wave equation in the natural state space L2(Ω) x H^-1(Ω), under nonlinear (and non-local) boundary dissipation in the Dirichlet B.C., as well as nonlinear internal damping; (2) a corresponding Kirchhoff equation in the natural state space [wzór), under nonlinear boundary dissipation in the 'moment' B.C. as well as nonlinear internal damping; (3) the system of dynamic elasticity corresponding to (1). All three dynamics possess a strong, hard-to-show 'boundary → boundary' regularity property, which was proved, also by invoking a micro-local argument, in Lasiecka and Triggiani (2004, 2008). This is by no means a general property of hyperbolic or hyperbolic-like dynamics (Lasiecka and Triggiani, 2003, 2008). The present paper, as a continuation of Lasiecka and Triggiani (2008), seeks to take advantage of this strong regularity property in the case of those PDE dynamics where it holds true. Thus, under the above boundary → boundary regularity, as well as exact controllability of the corresponding linear model, uniform stabilization of nonlinear models is obtained under minimal nonlinear assumptions, provided that a corresponding unique continuation property holds true. The treatment of the present paper is cast in the abstract setting (Lasiecka, 1989, 2001; Lasiecka and Triggiani, 2000, Ch. 7, 2003, 2008), which is proper for these hyperbolic dynamics and recovers the results of Lasiecka and Triggiani (2003, 2008) in the absence of the nonlinear interior damping, in particular in the linear case.

Słowa kluczowe

EN

non-linear hyperbolic equations; uniform energy decay rates

• Wydawca: Systems Research Institute, Polish Academy of Sciences
• Czasopismo: Control and Cybernetics
• Rocznik: 2008
• Tom: Vol. 37, no 4
• Strony: 935--969
• Opis fizyczny: Bibliogr. 49 poz.

Twórcy

autor

Lasiecka I.

autor

Triggiani R.

• Mathematics Department, University of Virginia Charlottesville, VA 22903, USA

Bibliografia

• AMMARI, K. (2002) Dirichlet boundary stabilization of the wave equation. Asymptotic Analysis. 30, 117-130.
• AUBIN, J.P. (1963) Un theoreme de compacite. C. R. Acad. Sci. Paris 56, 5042-5044.
• ALABAU, F. and KOMORNIK, V. (1998) Boundary observability, controllability, and stabilization of linear elastodynamic systems. SIAM J. Contr. & Optim. 37, 521-542.
• BARBU, V. (1976) Nonlinear Semigroups and Differential Equations in Banach Spaces. Noordhoff International Publishing, Leyden, The Netherlands.
• BELISHEV, M. and LASIECKA, I. (2002) The dynamical Lame system: Regularity of solutions, boundary control and boundary data continuation. ESAIM 8, 143-167.
• BARDOS, C., LEBEAU, G. and RAUCH, J. (1992) Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary. SIAM J. Control & Optimiz. 30, 1024-1065.
• CHUESHOV, I., ELLER, M. and LASIECKA, I. (2002) On the attractor fora semilinear wave equation with critical exponent and nonlinear boundary dissipation. Comm. Partial Differential Equations 27, 1901-1951.
• ELLER, M., ISAKOV, V., NAKAMURA, G. and TATARU, D. (2002) Uniqueness and stability in the Cauchy problem for Maxwell and elasticity systems. In: Nonlinear PDE, College de France Seminar, J.L. Lions Series in Applied Math. 7.
• FLANDOLI, F., LASIECKA, I. and TRIGGIANI, R. (1988) Algebraic Riccati equations with non-smoothing observation arising in hyperbolic and Euler-Bernoulli boundary control problems. Ann. Matem. Pura Appl. 307-382.
• GULLIVER, R., LASIECKA, I., LITTMAN, W. and TRIGGIANI, R. (2003) The case for differential geometry for the control of single and coupled PDEs: The structural acoustic chamber. The IMA Volumes in Mathematics and its Applications, 137: Geometric Methods in Inverse Problems and PDE Control. Springer-Verlag, 73-181.
• GUO, G.Z. and LUO, Y.H. (2002) Controllability and stability of a second-order hyperbolic system with collocated sensor/actuator. Systems & Control Letters 46, 45-65.
• HO, F.L. (1986) Observabilité frontiére de L’équation des ondes. C.R.Acad. Sci. Paris Ser I Math. 306, 443-446.
• HORN, M.A. (1998a) Sharp trace regularity for the solutions of equations of dynamic elasticity. J. Math. Sys., Estimation & Control 8 (12), 217-219.
• HORN, M.A. (1998b) Implications of sharp trace regularity results on boundary stabilization of the system of linear elasticity. J. Math. Anal. & Appl. 223,126-150.
• HORN, M.A. and LASIECKA, I. (1994) Asymptotic behavior with respect to thickness of boundary stabilizing feedback for the Kirchhoff plate. J. Diff. Eqns. 114, 396-433.
• LASIECKA, I. (1989) Stabilization of wave and plate-like equations with nonlinear dissipation. J. Diff. Eqns. 79, 340-381.
• LASIECKA, I. (1999) Uniform decay rates for the full von Karman system of dynamic thermoelasticity, with free boundary conditions and partial boundary dissipation. Comm. in PDEs 24, 1801-1849.
• LASIECKA, I. (2001) Mathematical Control Theory of Coupled Systems. SIAM Publications, CMBS-NSF Lecture Notes.
• LASIECKA, I., LIONS, J.L. and TRIGGIANI, R. (1986) Non-homogeneous boundary value problems for second order hyperbolic operators. J. Math. Pures et Appl. 65, 149-192.
• LASIECKA, I. and TATARU, D. (1993) Uniform boundary stabilization of semi-linear wave equations with nonlinear boundary damping. Diff. Int. Eqns. 6, 507-533.
• LASIECKA, I. and TOUNDYKOV, D. (2006) Energy decay rate for semilinear wave equations with nonlinear localized damping and source terms. Non-linear Analysis 64, 1757-1797.
• LASIECKA, I. and TRIGGIANI, R. (1981) A cosine operator approach to modeling L2(0,T;L2(Γ))-boundary input hyperbolic equations. AMO 7, 35-93.
• LASIECKA, I. and TRIGGIANI, R. (1983) Regularity of hyperbolic equations under L2(0,T;L2(Γ))-Dirichlet boundary terms. Appl. Math. & Optimiz. 10, 275-286.
• LASIECKA, I. and TRIGGIANI, R. (1987) Uniform exponential energy decay of wave equations in a bounded region with L2(0,∞;L2(Γ))-feedback control in the Dirichlet boundary conditions. J. Dig. Eqns. 66, 340-390.
• LASIECKA, I. and TRIGGIANI, R. (1991a) Differential and Algebraic Riccati Equations with Applications to Boundary/Point Control Problems: Continuous Theory and Approximation Theory. Lecture Notes in Control and Information Sciences 164, Springer Verlag.
• LASIECKA, I. and TRIGGIANI, R. (1991b) Exact controllability and uniform stabilization of Kirchhoff plates with boundary control only in ∆ω|∑ and homogeneous boundary displacement. J. Diff. Eqns. 93, 62-101.
• LASIECKA, I. and TRIGGIANI, R. (1992) Uniform stabilization of the wave equation with Dirichlet or Neumann feedback control without geometrical conditions. Applied Math, and Optimiz. 25, 189-224.
• LASIECKA, I. and TRIGGIANI, R. (1994) Recent advances in regularity theory of second order hyperbolic mixed problems and applications. Dynamics Reported, Expositions in Dynamical Systems 3, Springer-Verlag.
• LASIECKA. I. and TRIGGIANI, R. (2000) Control Theory for Partial Differential Equations, Vol. I and II, Encyclopedia of Mathematics and its Applications. Cambridge University Press.
• LASIECKA, I. and TRIGGIANI, R. (2002) Uniform stabilization of a shallow shell model with nonlinear boundary feedbacks. J. Math. Anal. Appl. 269, 642-688.
• LASIECKA, I. and TRIGGIANI, R. (2003) L2(Σ)-regularity of the boundary to boundary operator for hyperbolic and Petrowski-type PDEs. Abstr. Appl. Anal. 19, 1061-1139.
• LASIECKA, I. and TRIGGIANI, R. (2004) The operator B*L for the wave equation with Dirichlet control. Abstr. Appl. Anal. 7, 625-634.
• LASIECKA, I. and TRIGGIANI, R. (2006) Well-posedness and uniform decay rates at the L2(Ω)-level of Schrödinger equations with nonlinear boundary dissipation. J. Evol. Eqns. 6, 485-537.
• LASIECKA, I. and TRIGGIANI, R. (2008) Linear hyperbolic and Petrowski-type PDEs with continuous boundary control → boundary observation open loop map: Implication on nonlinear boundary stabilization with optimal decay rates. In: V. Isakov, ed., Sobolev Spaces in Mathematics III: Applications of Functional Analysis in Mathematical Physics. Springer-Tamara Rozhkovskaya Publishers, 187-276.
• LASIECKA, I., TRIGGIANI, R. and YAO, P.P. (1999) Inverse/observability estimates for second order hyperbolic equations with variable coefficients. J. Math. Anal. Appl. 235, 13-57.
• LASIECKA, I., TRIGGIANI, R. and ZHANG, X. (2000) Nonconservative wave equations with unobserved Neumann B.C.: Global uniqueness and observability in one shot. Contemporary Mathematics AMS 268, 227-326
• LIEB, E.H. and Loss, M. (1996) Analysis. Graduate Studies in Mathematics 14, AMS.
• LIONS, J.L. (1988) Controlabilité exacte des systèmes distribués. Masson, Paris.
• RUSSELL, D. (1978) Controllability and stabilizability theory for linear PDEs: Recent progress and open questions. SIAM Review 20, 639-739.
• SIMON, J. (1987) Compact sets on the space LP(0,T;B). Ann. Mat. Pura et Appl. CXLVI (4), 65-96.
• TATARU, D. (1995) Boundary controllability for conservative PDEs. Appl. Math. & Optim. 31, 257-295.
• TRIGGIANI, R. (1978) A cosine operator approach to modeling L2(0,T;L2(Γ))-boundary input problems for hyperbolic systems. Lecture Notes LNCIS, Springer-Verlag, 380-390. Proceedings of 8th IFIP Conference, University of Würzburg, Germany, July 1977.
• TRIGGIANI, R. (1989/1990) Lack of uniform stabilization for non-contractive semigroups under compact perturbation. Proc. Amer. Math. Soc. 105, (1989), 375-383; (1990) 503-522. Preliminary version: Proceedings INRIA Conference, Paris, France, June 1988, Springer-Verlag Lecture Notes.
• TRIGGIANI, R. (1988) Exact boundary controllability on L2(Ω) x H-1(Ω) of the wave equation with Dirichlet boundary control. Appl. Math. & Optimiz. 18, 241-277. Preliminary version: Lecture Notes in Control and Information Sciences 102, Springer-Verlag 1987, 291-332, Proceedings Workshop on Control for Distributed Parameter Systems, University of Graz, Austria, July 1986.
• TRIGGIANI, R. (1989) Wave equation on a bounded domain with boundary dissipation: An operator approach. J. Math. Anal. Appl. 137, 438-461. (Also, preliminary versions in Sung J. Lee, ed., Operator Methods for Optimal Control Problems, Marcel Dekker (1988), Lecture Notes Pure Appl. Math. 108, 283-310; Proceedings of Special Session of the Annual Meeting of the American Mathematical Society, New Orleans, LA (1986).
• TRIGGIANI, R. (1990) Finite rank, relatively bounded perturbations of semigroup generators, Part III: A sharp result on the lack of uniform stabilization. Diff. & Int. Eqns. 3, 503-522.
• TRIGGIANI, R. (1991) Lack of exact controllability for wave and plate equations with finitely many boundary controls. Diff. & Int. Eqns. 4, 683-705.
• TRIGGIANI, R. (2007) The role of an L2(Ω)-energy estimate in the theories of uniform stabilization and exact controllability for Schrödinger equations with Neumann boundary control. Boletin de Sociedade Patanaense de Matematica 25 (1-2), Series 3, 111-160.
• TRIGGIANI, R. and YAO, P.P. (2002) Carleman estimates with no lower-order terms for general Riemann wave equations. Global uniqueness and observability in one shot. Appl. Math. & Optim. 46, 2002 (special issue dedicated to the memory of J. L. Lions).