Interior controllability of the Benjamin-Bona-Mahony equation

• Autorzy: Leiva H. , Merentes N. , Sanchez J.
• Języki publikacji: EN

Abstrakty

EN

In this paper we prove the interior approximate controllability of the following Generalized Benjamin-Bona-Mahony type equation (BBM) with homogeneous Dirichlet boundary conditions [formula/wzór] where a(mniejszy-równy) and b > 0 are constants, Ω is a domain in IR(N), ω is an open nonempty subset of Ω denotes the characteristic function of the set ω and the distributed control [formula/wzór]. We prove that for all r>0 and any nonempty open subset ω of Ω the system is approximately controllable on [0, r]. Moreover, we exhibit a sequence of controls steering the system from an initial state to a final state in a prefixed time. As a consequence of this result we obtain the interior approximate controllability of the heat equation by putting a = 0 and b = 1.

Słowa kluczowe

EN

interior controllability; reaction-diffusion equations; strongly continuous semigroups

PL

sterowalność; równanie dyfuzji; równanie Benjamin-Bona-Mahony; równanie BBM

• Wydawca: Oficyna Wydawnicza Politechniki Rzeszowskiej
• Czasopismo: Journal of Mathematics and Applications
• Rocznik: 2010
• Tom: Vol. 33
• Strony: 51--59
• Opis fizyczny: Bibliogr. 19 poz.

Twórcy

autor

Leiva H.

autor

Merentes N.

autor

Sanchez J.

• Universidadde los Andes, Facultad de Ciencias, Departamento de Matemática, Mérida 5101-Venezuela,

Bibliografia

• [1] Adames N., Leiva H. and J. Sanchez J., Ccontrollability of the Benjamin-Bona-Mohany Equation. Divulgaciones Matemáticas Vol. 13 N. 2(2005), pp. 1-9.
• [2] Axler S., Bourdon P. and Ramey W., Harmonic Fucntion Theory. Graduate Texts in Math., 137. Springer Verlag, New york (1992).
• [3] Avrin J. and Goldtaein J. A., Global Existence for the Benjamin- Bona-Mahony Equation in Arbitrary Dimensions. Nonlinear Anal. 9(1995), 861-865.
• [4] Benjamin T. B. , Bona J. L. and Mahony J. J., Model Equations for Long Waves in Nonlinear Dispersive Systems. Philos. Trans. Roy. Soc. London Ser. A 272(1972), 47-78.
• [5] P. Biler. Long Time Behavior of Solutions of the Generalized Benjamin- Bona-Mahony Equation in Two Space Dimensions. Differential Integral Equations 5(1992), 891-901.
• [6] Caldas C. S. Q., Limaco J. and Barreto R. K., About the Benjamin-Bona-Mahony Equation in Domians with Moving Boundary. TEMA Tend. Mat. Apl. Comput., 8. N.3(2007), 329-339.
• [7] A.O. Celebei A. O., Kalantarov V. K. and Polat M., Attractors for the Generalized Benjamin-Bona-Mahony Equation. J. Differential Equations 157(1999), 439-451.
• [8] Chueshov I., Polat M. and Siegmund S. Gevrey Regularity of Global Attractor for Generalized Benjamin-Bona-Mahony Equation. Submitted to J.D.E (2002).
• [9] Curtain R.F. and Pritchard A. J., Infinite Dimensional Linear Systems. Lecture Notes in Control and Information Sciences, 8 Springer Verlag, Berlin, 1978.
• [10] Curtain R. F. and Zwart H. J., An Introduction to Infinite Dimensional Linear Systems Theory. 21 Springer Verlag, Berlin, 1995.
• [11] Hormander L., Linear Partial Differential Equations. Springer Verlag,(1969).
• [12] Larkin N. A. and Vishnevskii M. P., Disspative Initial Boundary Value Problem for the BBM-Equation. Electronic J. of Diff. Eqs., Vol. 2008(2008), N. 149, pp.1-10.
• [13] Micu S., On the Controllability of the Linearized Benjamin-Bona- Mahony Equetion. SIAM J. Control Optim. 39(6)(2001), 1677-1696,
• [14] Russell D. L., Controllability and Stabilizability Theory for Linear Partial Differential Equations: Recent Progress and Open Questions. SIAM Rev. 20 No. 4 (1978), 636-739.
• [15] Stanislavova M., On the Global Attractor for the Damped Benjamin-Bona-Mahony Equation. Proceedings of the Fifth Int. Conf. on Dynamical Systems and Diff. Eqs, June 16-19, 2004, Pomona, CA, USA.
• [16] Triggiani R., Extensions of Rank Conditions for Controllability and Observability to Banach Spaces and Unbounded Operators. SIAM J. Control Optimization 14 No. 2 (1976), 313-338.
• [17] Xu Zhang, A Remark on Null Exact Controllability of the Heat Equation. IAM J. CONTROL OPTIM. Vol. 40, No. 1(2001), pp. 39-53.
• [18] Zuazua E., Controllability of a System of Linear Thermoelasticity, J. Math. Pures Appl., 74, (1995), 291-315.
• [19] Zuazua E., Control of Partial Di_erential Equations and its Semi-Discrete Approximation. Discrete and Continuous Dynamical Systems, vol. 8, No. 2. April (2002), 469-513.