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Controllability of the semilinear Benjamin-Bona-Mahony equation

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In this paper we prove the interior approximate controllability of the following Generalized Semilinear Benjamin-Bona-Mahony type equation (BBM) with homogeneous Dirichlet boundary conditions [formula/wzor] where a ≥ 0 and b > 0 are constans, Ω is a domain in IRN, ω is an open nonempty subset of Ω, 1ω denotes the characteristic function of the set ω, the distributed control u belongs to L2(0, τ;L2(Ω)) and the nonlinear function ƒ:[0, τ] x IR x IR → IR is smooth enough and there are c, d, e ∈ IR, with c ≠ -1, ea + b > 0 such that [formula/wzor] where Qr = [0, τ] x IR x IR. We prove that for all τ > 0 and any nonempty open subset ω of Ω the system the system is approximately controllable on [0, τ]. Moreover, we exhibit a sequence of controls steering the system from an initial state z0 to an ε-neighborhood of the final state z1 on time > 0. As a consequence of this result we obtain the interior approximate controllability of the semilinear heat equation by putting a = 0 and b = 1.
Rocznik
Tom
Strony
39--51
Opis fizyczny
Bibliogr. 17 poz.
Twórcy
autor
  • Departamento de Matemáticas, Universidad de Los Andes, Mérida 5101 – VENEZUELA
autor
  • Departamento de Matemáticas, Facultad de Ciencias, Universidad Central de Venezuela, Caracas 1051 VENEZUELA
  • Departamento de Matemáticas, Facultad de Ciencias, Universidad Central de Venezuela, Caracas 1051 VENEZUELA
Bibliografia
  • [1] J. Appel, H.Leiva, N. Merentes, A. Vignoli, Un espectro de compresión no lineal con aplicaciones a la controlabilidad aproximada de sistemas semilineales, preprint
  • [2] S. Axler, P. Bourdon and W. Ramey, Harmonic Function Theory. Graduate Texts in Math., 137. Springer Verlag, New York (1992).
  • [3] D. Barcenas, H. Leiva AND Z. Sivoli, A Broad Class of Evolution Equations are Approximately Controllable, but Never Exactly Controllable. IMA J. Math. Control Inform. 22, no. 3 (2005), 310-320.
  • [4] R.F. Curtain, A.J. Pritchard, In nite Dimensional Linear Systems. Lecture Notes in Control and Information Sciences, 8. Springer Verlag, Berlin (1978).
  • [5] R.F. Curtain, H.J. Zwart, An Introduction to Infinite Dimensional Linear Systems Theory. Text in Applied Mathematics, 21. Springer Verlag, New York (1995).
  • [6] J.I. DIAZ, J.Henry and A.M. Ramos, On the Approximate Controllability of Some Semilinear Parabolic Boundary-Value Problemas, Appl. Math. Optim 37-71 (1998).
  • [7] E. Fernandez-Cara, Remark on Approximate and Null Controllability of Semilinear Parabolic Equations ESAIM:Proceeding OF CONTROLE ET EQUATIONS AUX DE-RIVEES PARTIELLES, Vol. 4, 1998, 73-81.
  • [8] E. Fernandez-Cara and E. Zuazua, Controllability for Blowing up Semilinear Parabolic Equations, C.R. Acad. Sci. Paris, t. 330, serie I, p. 199-204, 2000.
  • [9] Luiz A. F. de Oliveira, On Reaction-Diffusion Systems E. Journal of Differential Equations, Vol. 1998(1998), N0. 24, pp. 1-10.
  • [10] H. Leiva, A Lemma on C0-Semigroups and Applications PDEs Systems, Quaestions Mathematicae, Vol. 26, pp. 247-265 (2003).
  • [11] H. Leiva, N. Merentes and J.L. Sanchez, Interior Controllability of the Benjamin-Bona-Mahony Equation, Journal of Mathematis and Applications, No 33,pp. 51-59 (2010).
  • [12] H. H. Leiva, N. Merentes and J.L. Sanchez, Interior Controllability of the nD Semilinear Heat Equation, African Diaspora of Mathematics. Special Volume in Honor to Profs. C. Carduneanu, A. Fink, and S. Zaideman. Vol. 12, Number 2, pp. 1-12(2011).
  • [13] H. Leiva, Controllability of a System of Parabolic equation with non-diagonal di usion matrix. IMA Journal of Mathematical Control and Information; Vol. 32, 2005, pp. 187-199.
  • [14] H. Leiva and Y. Quintana, Interior Controllability of a Broad Class of Reaction Diffusion Equations, Mathematical Problems in Engineering, Vol. 2009, Article ID 708516, 8 pages, doi:10.1155/2009/708516.
  • [15] Xu Zhang, A Remark on Null Exact Controllability of the Heat Equation. IAM J. CONTROL OPTIM. Vol. 40, No. 1(2001), pp. 39-53.
  • [16] E. Zuazua, Controllability of a System of Linear Thermoelasticity, J. Math. Pures Appl., 74, (1995), 291-315.
  • [17] E. Zuazua, Control of Partial Differential Equations and its Semi-Discrete Approximation. Discrete and Continuous Dynamical Systems, vol. 8, No. 2. April (2002), 469-513.
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Bibliografia
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