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Control of blow-up singularities for nonlinear wave equations

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  • While the global boundary control of nonlinear wave equations that exhibit blow-up is generally impossible, we show on a typical example, motivated by laser breakdown, that it is possible to control solutions with small data so that they blow up on a prescribed compact set bounded away from the boundary of the domain. This is achieved using the representation of singular solutions with prescribed blow-up surface given by Fuchsian reduction. We outline on this example simple methods that may be of wider applicability.
    Mathematics Subject Classification: Primary: 93B05, 93C20; Secondary: 35L71.


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  • [1]

    C. Bardos, Distributed control and observation, in Control of fluid flow, (eds. Koumoutsakos, Petros et al. (ed.),), Lecture Notes in Control and Information Sciences 330, Springer, (2006), 139-156.doi: 10.1007/978-3-540-36085-8_6.


    G. CabartSingularités en Optique Non Linéaire: Etude Mathématique, Thèse de Doctorat, Université de Reims (defended 2/9/2005). Available from http://theses.univ-reims.fr/exl-doc/GED00000105.pdf


    G. Cabart and S. Kichenassamy, Explosion et normes $L^p$ pour l'équation des ondes non linéaire cubique, C. R. Acad. Sci. Paris, Séer. I, 335 (2002), 903-908.doi: 10.1016/S1631-073X(02)02606-7.


    W. C. Chewning, Controllability of the nonlinear wave equation in several space variables, SIAM J. Control, 14 (1976), 19-25.doi: 10.1137/0314002.


    M. Cirinà, Boundary controllability of nonlinear hyperbolic systems, SIAM J. Control, 7 (1969), 198-212.doi: 10.1137/0307014.


    B. Dehman, G. Lebeau and E. Zuazua, Stabilization and control for the subcritical semilinear wave equation, Ann. Sci. ENS, (4), 36 (2003), 525-551.doi: 10.1016/S0012-9593(03)00021-1.


    D. Ebin and J. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid, Ann. Math., Second Series, 92 (1970), 102-163.doi: 10.2307/1970699.


    H. O. Fattorini, Local controllability of a nonlinear wave equation, Mathem. Systems Theory, 9 (1975), 30-45.doi: 10.1007/BF01698123.


    S. Kichenassamy, Fuchsian Reduction: Applications to Geometry, Cosmology and Mathematical Physics, Progress in Nonlinear Differential Equations and their Applications, 71. Birkhäuser Boston, Inc., Boston, MA, 2007.


    S. Kichenassamy and W. Littman, Blow-up surfaces for nonlinear wave equations, Part I, Commun. in P. D. E., 18 (1993), 431-452.doi: 10.1080/03605309308820936.


    S. Kichenassamy and W. Littman, Blow-up surfaces for nonlinear wave equations, Part II, Commun. in P. D. E., 18 (1993), 1869-1899.doi: 10.1080/03605309308820997.


    I. Lasiecka and R. Triggiani, Exact controllability of semilinear abstract systems with applications to waves and plates, Appl. Math. Optim., 23 (1991), 109-154.doi: 10.1007/BF01442394.


    I. Lasiecka and R. Triggiani, Global exact controllability of semilinear wave equations by a double compactness/uniqueness argument, Discr. Cont. Dyn. Syst., Suppl., (2005), 556-565.


    J. L. Lions, Contrôlabilité exacte, perturbations et systèmes distribués, Tome 1, Rech. Math. Appl. 8, Masson, Paris, 1988.


    W. Littman, Aspects of boundary control theory, in Differential Equations and Mathematical Physics, (ed. C. Bennewitz) Math. in Sci. and Engineering, 186 (1992), 201-215.doi: 10.1016/S0076-5392(08)63381-0.


    W. Littman, Boundary control theory for hyperbolic and parabolic linear partial differential equations with constant coefficients, Ann. Sc. Norm. Sup. Pisa, ser IV, 5 (1978), 567-580.


    D. L. Russell, Controllability and stabilizability theory for linear partial differential equations: Recent progress and open questions, SIAM Review, 20 (1978), 679-739.doi: 10.1137/1020095.


    D. L. Russell, A unified boundary controllability theory for hyperbolic and parabolic equations, Studies in Appl. Math., 52 (1973), 189-211.


    M. E. Taylor, Pseudodifferential Operators and Nonlinear PDE, Birkhäuser, Boston, 1991.doi: 10.1007/978-1-4612-0431-2.


    Y. Zhou and Z. Lei, Local exact boundary controllability for nonlinear wave equations, SIAM J. Control Optim., 46 (2007), 1022-1051.doi: 10.1137/060650222.


    E. Zuazua, Exact controllability for the semilinear wave equation, J. Math. Pures Appl., 69 (1990), 1-31.


    E. Zuazua, Exact boundary controllability for the semilinear wave equation, in Nonlinear partial differential equations and their applications, Collège de France Seminar, Vol. X (Paris, 1987-1988), (eds. H. Brezis and J.-L. Lions), Pitman, (1991), 357-391.


    E. Zuazua, Controllability and Observability of Partial Differential Equations: Some results and open problems, in Handbook of Differential Equations: Evolutionary Differential Equations, vol. 3(eds. C. M. Dafermos and E. Feireisl eds.), Elsevier Science, (2006), 527-621.doi: 10.1016/S1874-5717(07)80010-7.


    X. Zhang and E. Zuazua, Exact Controllability of the Semi-Linear Wave Equation, (2010), available from http://institucional.us.es/doc-course-imus/PDF/XZhang-EZ_Open_Problems.pdf

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