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Some control results for simplified one-dimensional models of fluid-solid interaction. (English) Zbl 1122.93008

Summary: We analyze the null controllability of a one-dimensional nonlinear system which models the interaction of a fluid and a particle. This can be viewed as a first step in the control analysis of fluid-solid systems. The fluid is governed by the Burgers equation and the control is exerted at the boundary points. We present two main results: the global null controllability of a linearized system and the local null controllability of the nonlinear original model. The proofs rely on appropriate global Carleman inequalities, observability estimates and fixed point arguments.

MSC:

93B05 Controllability
35Q35 PDEs in connection with fluid mechanics
35Q53 KdV equations (Korteweg-de Vries equations)
76T99 Multiphase and multicomponent flows
93C20 Control/observation systems governed by partial differential equations
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[1] Anita S., Appl. Math. Optim. 46 pp 97–
[2] Aubin J. P., L’Analyse Non Linéaire et ses Motivations Économiques (1984)
[3] Conca C., Comm. Partial Diff. Eqns. 25 pp 1019–
[4] DOI: 10.1007/s002050050136 · Zbl 0943.35063
[5] Desjardins B., Comm. Partial. Diff. Eqns. 25 pp 1399–
[6] Desjardins B., Rev. Mat. Comput. 14 pp 523–
[7] DOI: 10.1137/S0363012901386465 · Zbl 1038.93041
[8] Fabre C., Proc. R. Soc. Edinburgh 125 pp 31– · Zbl 0818.93032
[9] DOI: 10.1016/j.matpur.2004.02.010 · Zbl 1267.93020
[10] Fernández-Cara E., Comp. Appl. Math. 21 pp 167–
[11] Fursikov A. V., Lecture Notes 34, in: Controllability of Evolution Equations (1996) · Zbl 0862.49004
[12] DOI: 10.1007/s00021-002-8536-9 · Zbl 1009.76016
[13] DOI: 10.1137/S0363012993248347 · Zbl 0853.93018
[14] Yu. Imanuvilov O., Mat. Sb. 186 pp 102–
[15] DOI: 10.1051/cocv:2001103 · Zbl 0961.35104
[16] Ladyzenskaya O. A., Trans. Math. Monographs 23, in: Linear and Quasilinear Equations of Parabolic Type (1968)
[17] DOI: 10.1137/1030001 · Zbl 0644.49028
[18] Lions J.-L., Contrôlabilité Exacte, Perturbation et Stabilisation de Systèmes Distribués (1988)
[19] DOI: 10.1016/j.crma.2004.07.005 · Zbl 1060.93015
[20] DOI: 10.1002/sapm1973523189 · Zbl 0274.35041
[21] DOI: 10.1137/1020095 · Zbl 0397.93001
[22] Takahashi T., Adv. Diff. Eqns. 8 pp 1499–
[23] DOI: 10.1007/s00021-003-0083-4 · Zbl 1054.35061
[24] DOI: 10.1081/PDE-120024530 · Zbl 1071.74017
[25] E. Zuazua, Nonlinear Partial Differential Equations and their Applications X, eds. H. Brezis and J.L. Lions (Pitman, 1991) pp. 357–391.
[26] Zuazua E., J. Math. Pures Appl. 69 pp 1–
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