Differential
Inclusions, Control and Optimization 20 (2000) 113-140
doi: 10.7151/dmdico.1008
Irena Lasiecka
Department of Mathematics, University of Virginia
Charlottesville, Virginia 22903, USA
e-mail: il2v@virginia.edu
Optimization problem for a structural acoustic model with controls governed by unbounded operators on the state space is considered. This type of controls arises naturally in the context of ßmart material technology". The main result of the paper provides an optimal synthesis and solvability of associated nonstandard Riccati equations. It is shown that in spite of the unboundedness of control operators, the resulting gain operators (feedbacks) are bounded on the state space. This allows to provide full solvability of the associated Riccati equations. The proof of the main result is based on exploiting propagation of analyticity from the structural component of the model into an acoustic medium.
Keywords: structural acoustic model with thermal effects, optimal control problem, smart controls, nonstandard Riccati equations, analyticity of semigroups.
1991 Mathematics Subject Classification: 35L70, 93D15, 35B40.
[1] | G. Avalos and I. Lasiecka, Differential Riccati equation for the active control of a problem in structural acoustics, JOTA 91 (1996), 695-728. |
[2] | G. Avalos and I. Lasiecka, The strong stability of a semigroup arising from a coupled hyperbolic/parabolic system, Semigroup Forum 57 (1997), 278-292. |
[3] | A.V. Balakrishnan, Applied Functional Analysis, Springer Verlag 1975. |
[4] | H.T. Banks, R.J. Silcox and R.C. Smith, The modeling and control of acoustic/structure interaction problems via piezoceramic actuat ors: 2-D numerical examples, ASME Journal of Vibration and Acoustics 2 (1993), 343-390. |
[5] | H.T. Banks and R. Smith, Active control of acoustic pressure fields using smart material technology in flow control, IMA, (M. Gunzburger, ed.) 68 Springer Verlag 1995. |
[6] | H.T. Banks and R.C. Smith, Well-Posedness of a model for structural acoustic coupling in a cavity enclosed by a thin cylindrical shell, Journal of Mathematical Analysis and Applications 191 (1995), 1-25. |
[7] | H.T. Banks, R.C. Smith and Y. Wang, The modeling of piezoceramic patch interactions with shells, plates and beams, Quaterly of Applied Mathematics 53, 2 (1995), 353-381. |
[8] | V. Barbu, I. Lasiecka and R. Triggiani, Extended Algebraic Riccati Equations arising in hyperbolic dynamics with unbounded controls, Nonlinear Analysis, to appear. |
[9] | J. Beale, Spectral properties of an acoustic boundary condition, Indiana Univ. Math. J. 9 (1976), 895-917. |
[10] | A. Bensoussan, G. Da Prato, M.C. Delfour and S.K. Mitter, Representation and Control of Infinite Dimensional Systems, vol. II, Birkhauser, Boston 1993. |
[11] | M. Camurdan and R. Triggiani, Control Problems in Noise Reduction: The case of two coupled hyperbolic equations, in: Proceedings of SPIE's 4-th Annual Symposium on Smart Structures and Materials, Mathematic and Control in Smart Structures 1997. |
[12] | M. Camurdan and R. Triggiani, Sharp Regularity of a coupled system of a wave and Kirchhoff equation with point control, arising in noise reduction, Diff and Integral Equations 1998. |
[13] | E.F. Crawley and E. H. Anderson, Detailed models of piezoceramic actuation of beams, in: Proc. of AIAA Conference 1989. |
[14] | E.F. Crawley and J. de Luis, Use of piezoelectric actuators as elements of intelligent structures, AIAA Journal 25 (1987), 1373-1385. |
[15] | E.K. Dimitriadis, C.R. Fuller and C.A. Rogers, Piezoelectric Actuators for Distributed Noise and Vibration Excitation of Thin Plates, Journal of Vibration and Acoustics 13 (1991), 100-107. |
[16] | F. Flandoli, Riccati equations arising in boundary control problems with distributed parameters, SIAM J. Control 22 (1984), 76-86. |
[17] | F. Flandoli, I. Lasiecka and R. Triggiani, Algebraic Riccati Equations with non-smoothing observations arising in hyperbolic and Euler-Bernoulli boundary control problems, Annali di Matematica Pura et. Applicata 153 (1988), 307-382. |
[18] | P. Grisvard, Characterization de quelques espaces d'interpolation, Archive rational Mechanics and Analysis 26 (1967), 40-63. |
[19] | I. Lasiecka, NSF-CBMS Lecture Notes on Mathematical Control Theory of Coupled PDE's, SIAM, Philadelphia 2000. |
[20] | J. Lagnese, Boundary Stabilization of Thin Plates, SIAM, Philadelphia 1989. |
[21] | I. Lasiecka, Mathematical control theory in structural acoustic problems, Mathematical Models and Methods in Applied Sciences 8 (1998), 1119-1153. |
[22] | I. Lasiecka, Boundary stabilization of a 3-dimensional structural acoustic model, Journal de Math. Pure et Appli. 78 (1999), 203-232. |
[23] | I. Lasiecka, Uniform decay rates for full von Karman system of dynamic elasticity with free boundary conditions and partial boundary dissipation, Communications in PDE 24 (1999), 1801-1849. |
[24] | I. Lasiecka and C. Lebiedzik, Decay rates in nonlinear structural acoustic models with thermal effects and without a damping on the interface, Preprint, 1999. |
[25] | I. Lasiecka and C. Lebiedzik, Boundary stabilizability of nonlinear structural acoustic models with thermal effects on the interface, C.R. Acad. Sci. Paris, serie II b, (2000), 1-6. |
[26] | I. Lasiecka and R. Triggiani, Differential and Algebraic Riccati Equations with Applications to Boundary/Point Control Problems, Continuous Theory and Approximation Theory, Springer Verlag, LNCIS 164, 1991. |
[27] | I. Lasiecka and R. Triggiani, Riccati differential equations with unbounded coefficients and nonsmooth terminal condition-the case of analytic semigroups, SIAM J. Math. Analysis 23 (2) (1992), 448-481. |
[28] | I. Lasiecka and R. Triggiani, Analyticity and lack thereof, of thermoelastic semigroups, ESAIM 4 (1998), 199-222. |
[29] | I. Lasiecka and R. Triggiani, Analyticity of thermo-elastic semigroups with free B.C., Annali di Scuola Normale Superiore XXVII (1998), 457-482. |
[30] | I. Lasiecka and R. Triggiani, Two direct proofs on the analyticity of the S.C. semigroup arising in abstract thermo-elastic equations, Advances in Differential Equations 3 (1998), 387-416. |
[31] | I. Lasiecka and R. Triggiani, Control Therory for Partial Differential Equations, Cambridge University Press, Cambridge 1999. |
[32] | C. Lebiedzik, Exponential stability in structural acoustic models with thermoelasticity, Dynamics of Continous, Discrete and Impulsive Systems, 1999. |
[33] | H.C. Lester and C.R. Fuller, Active control of propeller induced noise fields inside a flexible cylinder, in: Proc. of AIAA Tenth Aeroacoustics Conference, Seattle, WA 1986. |
[34] | J.L. Lions and E. Magenes, Non-homogenous Boundary Value Problems and Applications, Springer Verlag, New York 1972. |
[35] | Z. Liu and M. Renardy, A note on the equation of thermoelastic plate, Applied Math. Letters 8 (1995), 1-6. |
[36] | P.M. Morse and K.U. Ingard, Theoretical Acoustics, McGraw-Hill, New York 1968. |
[37] | A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer Verlag, New York 1986. |
[38] | G. Da Prato, I. Lasiecka and R. Triggiani, A Direct Study of Riccati Equations arising in Hyperbolic Boundary Control Problems, Journal of Differential Equations 64 (1986), 26-47. |
[39] | D.L. Russell, Controllability and stabilizability theory for linear partial differential equations. recent progress and open questions, SIAM Review 20 (1978), 639-739. |
[40] | D.L. Russell, Mathematical models for the elastic beam and their control-theoretic properties, Semigroups Theory and Applications, Pitman Research Notes 152 (1986), 177-217. |
[41] | R. Triggiani, Lack of exact controllability for wave and plate equations with finitely many boundary controls, Differential and Integral Equations 4 (1991), 683-705. |
[42] | R. Triggiani, Interior and boundary regularity of the wave equation with interior point control, Differential and Integral Equations 6 (1993), 111-129. |
[43] | R. Triggiani, Regularity with interior point control, part II: Kirchhoff equations, Journal of Differential Equations 103 (1993), 394-420. |
[44] | R. Triggiani, The Algebraic Riccati Equations with Unbounded Coefficients; Hyperbolic Case Revisited, Contemporary Mathematics: Optimization Methods in PDE's, AMS, Providence 209 (1997), 315-339. |
[45] | R. Triggiani, Sharp regularity of thermoelastic plates with point controls, Abstract and Applied Analysis, to appear. |
[46] | M. Tucsnak, Control of plates vibrations by means of piezoelectric actuators, Discrete and Continuous Dynamical Systems 2 (1996), 281-293. |
[47] | G. Weiss and H. Zwart, An example in LQ optimal control, Systems and Control Letters 33 (1998), 339-349. |
Received 29 November 1999
Revised 22 February 2000