Optimal regularity and optimal control of a thermoelastic structural acoustic model with point control and clamped boundary conditions

• Autorzy: Lebiedzik C. , Triggiani R.
• Języki publikacji: EN

Abstrakty

EN

In this paper we consider point control of a structural acoustic model with thermoelastic effects. The key feature of this paper is that the two-dimensional plate modeling the active wall of the acoustic chamber has clamped boundary conditions. For this case a new optimal regularity result has recently become available (Triggiani, 2008). Using this new result for the plate alone, we derive a sharp (optimal) regularity result for the overall coupled system of wave and thermoelastic plate equations, after overcoming a series of additional technical difficulties. This allows for the study of an optimal control problem of the coupled system.

Słowa kluczowe

EN

point control; parabolic/hyperbolic thermoelastic system; hyperbolic/hyperbolic chamber/wall coupling

• Wydawca: Systems Research Institute, Polish Academy of Sciences
• Czasopismo: Control and Cybernetics
• Rocznik: 2009
• Tom: Vol. 38, no 4B
• Strony: 1461--1499
• Opis fizyczny: Bibliogr. 35 poz.

Twórcy

autor

Lebiedzik C.

autor

Triggiani R.

• Department of Mathematics, Wayne State University, Detroit, MI 48201, USA

Bibliografia

• AVALOS, G. and LASIECKA, I. (1996) Differential Riccati equation for the active control of a problem in structural acoustics. J. Optim. Theory Appl. 91 (3), 695-728.
• AVALOS, G. and LASIECKA, I. (1997) Exponential stability of a thermoelastic system without mechanical dissipation. Dedicated to the memory of Pierre Grisvard. Rend. Istit. Mat. Univ. Trieste 28 (suppl.) 1-27.
• AVALOS, G. and LASIECKA, I. (2003) Exact controllability of structural acoustic interactions. J. M. Pures. Appl 82, 1047-1073.
• BUCCI, F. (2007) Control theoretic properties of structural acoustic models with thermal effects. I: Singular estimates. J. Evol. Eqns. 7, 387-414.
• CAMURDAN, M. (1998) Uniform stabilization of a coupled structural acoustic system with boundary dissipation. Abstract & Appl. Anal. 3 (3-4), 377-400.
• CAMURDAN, M. and JI, G. (2000) A noise reduction problem arising in structural acoustic: A 3-d solution. In: R. Gulliver, W. Littman and R. Triggiani, eds., Differential Geometric Methods in the Control of PDE. AMS, Contemporary Mathematics, 268.
• CAMURDAN, M. and TRIGGIANI, R. (1999) Sharp regularity of a coupled system of a wave equation and a Kirchhoff equation with point control arising in noise reduction. Diff. Int. Eqns. 12, 101-118.
• LAGNESE, J. (1989) Boundary Stabilization of Thin Plates. SIAM, Philadelphia.
• LASIECKA, I. (2002) Mathematical Control Theory of Coupled PDE. CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia.
• LASIECKA, I. and LEBIEDZIK, C. (1999) Uniform stability in structural acoustic systems with thermal effects and nonlinear boundary damping. Control and Cybernetics 28 (3), 557-581.
• LASIECKA, I., LIONS, J.L. and TRIGGIANI, R. (1986) Nonhomogeneous boundary value problems for second-order hyperbolic operators. J. Math. Pures Appl. 65, 149-192.
• LASIECKA, I. and TRIGGIANI, R. (1990) Sharp regularity for mixed second order hyperbolic equations of Neumann type. Part I: The L2-boundary case. Annali Matem. Pura Appl. (IV) CLVII, 285-367.
• LASIECKA, I. and TRIGGIANI, R. (1991a) Sharp regularity theory for second-order hyperbolic equations of Neumann type. Part II: General boundary data. Diff. Int. Eqns. 94 (1), 112-164.
• LASIECKA, I. and TRIGGIANI, R. (1991b) Exact controllability and uniform stabilization of Euler-Bernoulli equations with only one active control in ∆w|∑- Bollettino Unione Matem. Ital. 5-B, (7), 665-702.
• LASIECKA, I. and TRIGGIANI, R. (1994) Recent advances in regularity of second-order hyperbolic mixed problems, and applications. Dynamics Reported, Springer-Verlag, 3, 104-158.
• LASIECKA, I. and TRIGGIANI, R. (1998a) Two direct proofs on the analyticity of the s.c. semigroup arising in abstract thermo-elastic equations. Advances Diff. Eqns. 3 (3), 387 416.
• LASIECKA, I. and TRIGGIANI, R. (1998b) Analyticity of thermo-elastic semigroups with coupled hinged/ Neumann B.C. Abstract Appl. Anal. 3 (1-2), 153-169.
• LASIECKA, I. and TRIGGIANI, R. (1998c) Analyticity of thermo-elastic semigroups with free B.C. Annali Scuola Normale Superiors, Pisa, Cl. Sci. (4), XXVII, 457-482.
• LASIECKA, I. and TRIGGIANI, R. (2000a) A sharp trace regularity result of Kirchhoff and thermoelastic plate equations with free boundary conditions. Rocky Mount. J. Math. 30 (3), 981-1023.
• LASIECKA, I. and TRIGGIANI, R. (2000b) Structural decomposition of thermo-elastic semigroups with rotational forces. Emigroup Forum 60, 16-66.
• LASIECKA, I. and TRIGGIANI, R. (2000c) Control Theory for Partial Differential Equations. Vol. I: Abstract Parabolic-Like Systems. Encyclopedia of Mathematics and its Applications, Cambridge University Press.
• LASIECKA, I. and TRIGGIANI, R. (2000d) Control Theory for Partial Differential Equations. Vol. II: Abstract Hyperbolic-Like Systems over a Finite Time Horizon. Encyclopedia of Mathematics and its Applications, Cambridge University Press.
• LASIECKA, I. and TRIGGIANI, R. (2001) Factor spaces and implications on Kirchhoff elastic and thermoelastic systems with clamped boundary conditions. Abstract and Applied Analysis 8 (1), 1-48.
• LEBIEDZIK, C. (2000) Uniform stability of a coupled structural acoustic system with thermoelastic effects. Dynam. Cont. Discr. and Impulsive Sys. 7 (3), 369-385.
• LEBIEDZIK, C. (2001) Boundary stabilizability of a nonlinear structural acoustic model including thermoelastic effects. In: Control of nonlinear distributed parameter systems: Lecture Notes in Pure and Applied Mathematics 218, Marcel Dekker, 177-197.
• LEBIEDZIK, C. and TRIGGIANI, R. (2010) The optimal interior regularity for the critical case of a clamped thermoelastic system with point control revisited. Submitted to Proc. of ISAAC Conference, Imperial College, London, July 2009.
• LIONS, J.L. and MAGENES, E. (1972) Non-Homogeneous Boundary Value Problems and Applications. Vol. I. Springer-Verlag, New York.
• OURADA, N. and TRIGGIANI, R. (1991) Uniform stabilization of the Euler-Bernoulli equation with feedback only in the Neumann boundary conditions. Diff. Int. Eqns. 4, 277-292.
• TATARU, D. (1998) On the regularity of boundary traces for the wave equation. Annali della Scuola Normale Superiore.
• TRIGGIANI, R. (1993) Interior and boundary regularity of the wave equations with point control. Diff. Int. Eqns. (6), 111-129.
• TRIGGIANI, R. (1993) Regularity with point control. Part II: Kirchhoff equations. J. Diff. Eqns. 103, 394-420.
• TRIGGIANI, R. (1997) Control problems in noise reduction: the case of two hyperbolic equations. Math. Contr. Smart Structures, SPIE 3039, 382-392.
• TRIGGIANI, R. (2007a) Sharp regularity results for hyperbolic-dominated ther-moelastic systems with point control: The hinged case. J. Math. Anal. & Appl. 333, 530-542.
• TRIGGIANI, R. (2007b) Sharp regularity of hyperbolic-dominated thermoelastic systems with point control: The clamped case. Discr. & Cont. Dynam. Systems (September), 993-1004.
• TRIGGIANI, R. (2008) The critical case of clamped thermoelastic systems with interior point control: Optimal interior and boundary regularity results. J. Diff. Eqns. 245, 3764-3805.