Exact boundary controllability of coupled hyperbolic equations

• Autorzy: Sergei Avdonin , Abdon Choque Rivero , Luz de Teresa
• Języki publikacji: EN

Abstrakty

EN

We study the exact boundary controllability of two coupled one dimensional wave equations with a control acting only in one equation. The problem is transformed into a moment problem. This framework has been used in control theory of distributed parameter systems since the classical works of A.G. Butkovsky, H.O. Fattorini and D.L. Russell in the late 1960s to the early 1970s. We use recent results on the Riesz basis property of exponential divided differences.

Słowa kluczowe

EN

coupled wave equations; controllability; Riesz bases; moment problem

• Wydawca: University of Zielona Gora Press
• Czasopismo: International Journal of Applied Mathematics and Computer Science
• Rocznik: 2013
• Tom: 23
• Numer: 4
• Strony: 701-710

Daty

wydano

2013

otrzymano

2012-09-18

poprawiono

2013-03-25

Twórcy

autor

Sergei Avdonin

• Department of Mathematics and Statistics, University of Alaska Fairbanks, Fairbanks, AK 99775-6660, USA

autor

Abdon Choque Rivero

• Institute of Physics and Mathematics, Universidad Michoacana de San Nicolás de Hidalgo Ciudad Universitaria, Ed. C-3, C.P. 58040 Morelia, Mich., México

autor

Luz de Teresa

• Institute of Mathematics, Universidad Nacional Autonoma de México, Circuito Exterior, C.U. 04510 D.F., México

Bibliografia

• Alabau-Boussouira, F. (2003). A two-level energy method for indirect boundary observability and controllability of weakly coupled hyperbolic systems, SIAM Journal on Control and Optimization 42(3): 871-906.
• Alabau-Boussouira, F. and Leautaud, M. (2011). Indirect controllability of locally coupled systems under geometric conditions, Comptes Rendus Mathematique 349(7-8): 395-400.
• Ammar-Kohdja, A., Benabdallah, M., González-Burgos, L. and de Teresa, L. (2011). Recent results on the controllability of coupled parabolic problems: A survey, Mathematical Control and Related Fields 1(3): 267-306
• Avdonin, S.A. and Ivanov, S.A., (1995). Families of Exponentials. The Method of Moments in Controllability Problems for Distributed Parameter Systems, Cambridge University Press, New York, NY.
• Avdonin, S.A. and Ivanov, S.A. (2001). Exponential Riesz bases of subspaces and divided differences, St Petersburg Mathematical Journal 13(3): 339-351.
• Avdonin, S. and Moran, W. (2001a). Ingham-type inequalities and Riesz bases of divided differences, International Journal of Applied Mathematics and Computer Science 11(4): 803-820.
• Avdonin, S. and Moran, W. (2001b). Simultaneous control problems for systems of elastic strings and beams, Systems and Control Letters 44(2): 147-155.
• Avdonin, S. and Pandolfi, L. (2011). Temperature and heat flux dependence independence for heat equations with memory, in R. Sipahi, T. Vyhlidal, S.-I. Niculescu and P. Pepe (Eds.), Time Delay Systems-Methods, Applications and New Trends, Lecture Notes in Control and Information Sciences, Vol. 423, Springer-Verlag, Berlin/Heidelberg, pp. 87-101.
• Biot, M. (1962). Generalized theory of acoustic propagation in porous dissipative media, The Journal of the Acoustical Society of America 34(9): 1254-1264.
• Bodart, O. and Fabre, C. (1995). Controls insensitizing the norm of the solution of a semilinear heat equation, Journal of Mathematical Analysis and Applications 195(3): 658-683.
• Dáger, R. (2006). Insensitizing controls for the 1-D wave equation, SIAM Journal on Control and Optimization 45(5): 1758-1768.
• El Jai, A. and Hamzaoui, H. (2009). Regional observation and sensors, International Journal of Applied Mathematics and Computer Science 19(1): 5-14, DOI: 10.2478/v10006-009-0001-y.
• Hansen, S. and Zuazua E. (1995). Exact controllability and stabilization of a vibrating string with an interior point mass, SIAM Journal on Control and Optimization 33(5): 1357-1391.
• Holger, S., Frehner, M. and Schmalholz S. (2010). Waves in residual-saturated porous media, in G.A. Maugin and A.V. Metrikine (Eds.), Mechanics of Generalized Continua, Advances in Mechanics and Mathematics, Vol. 21, Springer, New York, NY, pp. 179-187.
• Kavian, O. and de Teresa, L. (2010). Unique continuation principle for systems of parabolic equations, ESAIM: Control, Optimisation and Calculus of Variations 16(2): 247-274.
• Khapalov, A. (2010). Source localization and sensor placement in environmental monitoring, International Journal of Applied Mathematics and Computer Science 20(3): 445-458, DOI: 10.2478/v10006-010-0033-3.
• Leonard, E. (1996). The matrix exponential, SIAM Review 38(3): 507-512.
• Lions, J.L. (1989). Remarques préliminaires sur le contrˆole des systèmes à données incomplètes, XI Congreso de Ecuaciones Diferenciales y Aplicaciones (CEDYA), Málaga, Spain pp. 43-54.
• Najafi, M. (2001). Study of exponential stability of coupled wave systems via distributed stabilizer, International Journal of Mathematics and Mathematical Sciences 28(8): 479-491.
• Najafi, M., Sarhangi G.R. and Wang H. (1997). Stabilizability of coupled wave equations in parallel under various boundary conditions, IEEE Transactions on Automatic Control 42(9): 1308-1312.
• Pandolfi, L. (2009). Riesz system and the controllability of heat equations with memory Integral Equations and Operator Theory 64(3): 429-453.
• Rosier, L. and de Teresa, L. (2011). Exact controllability of a cascade system of conservative equations, Comptes Rendus Mathematique 349(5): 291-296.
• Russell, D. (1978). Controllability and stabilizability theory for linear partial differential equations, SIAM Review 20(4): 639-739.
• Tebou, L. (2008). Locally distributed desensitizing controls for the wave equation, Comptes Rendus Mathematique 346(7): 407-412.
• de Teresa, L. (2000). Insensitizing controls for a semilinear heat equation, Communications in Partial Differential Equations 25(1-2): 39-72.
• Uciński, D. and Patan, M. (2010). Sensor network design for the estimation of spatially distributed processes, International Journal of Mathematics and Mathematical Sciences 20(3): 459-481, DOI: 10.2478/v10006-010-0034-2.

Identyfikatory

DOI

10.2478/amcs-2013-0052