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December  2013, 2(4): 669-677. doi: 10.3934/eect.2013.2.669

## Control of blow-up singularities for nonlinear wave equations

 1 Laboratoire de Mathématiques, Université de Reims Champagne-Ardenne, Moulin de la Housse, B.P. 1039, F-51687 Reims Cedex 2, France

Received  October 2012 Revised  August 2013 Published  November 2013

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.
Citation: Satyanad Kichenassamy. Control of blow-up singularities for nonlinear wave equations. Evolution Equations & Control Theory, 2013, 2 (4) : 669-677. doi: 10.3934/eect.2013.2.669
##### References:
 [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.  Google Scholar [2] G. Cabart, Singularités en Optique Non Linéaire: Etude Mathématique,, Thèse de Doctorat, ().   Google Scholar [3] 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.  Google Scholar [4] W. C. Chewning, Controllability of the nonlinear wave equation in several space variables, SIAM J. Control, 14 (1976), 19-25. doi: 10.1137/0314002.  Google Scholar [5] M. Cirinà, Boundary controllability of nonlinear hyperbolic systems, SIAM J. Control, 7 (1969), 198-212. doi: 10.1137/0307014.  Google Scholar [6] 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.  Google Scholar [7] 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.  Google Scholar [8] H. O. Fattorini, Local controllability of a nonlinear wave equation, Mathem. Systems Theory, 9 (1975), 30-45. doi: 10.1007/BF01698123.  Google Scholar [9] 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.  Google Scholar [10] 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.  Google Scholar [11] 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.  Google Scholar [12] 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.  Google Scholar [13] 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.  Google Scholar [14] J. L. Lions, Contrôlabilité exacte, perturbations et systèmes distribués, Tome 1, Rech. Math. Appl. 8, Masson, Paris, 1988. Google Scholar [15] 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.  Google Scholar [16] 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.  Google Scholar [17] 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.  Google Scholar [18] D. L. Russell, A unified boundary controllability theory for hyperbolic and parabolic equations, Studies in Appl. Math., 52 (1973), 189-211.  Google Scholar [19] M. E. Taylor, Pseudodifferential Operators and Nonlinear PDE, Birkhäuser, Boston, 1991. doi: 10.1007/978-1-4612-0431-2.  Google Scholar [20] 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.  Google Scholar [21] E. Zuazua, Exact controllability for the semilinear wave equation, J. Math. Pures Appl., 69 (1990), 1-31.  Google Scholar [22] 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.  Google Scholar [23] 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.  Google Scholar [24] 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 Google Scholar

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##### References:
 [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.  Google Scholar [2] G. Cabart, Singularités en Optique Non Linéaire: Etude Mathématique,, Thèse de Doctorat, ().   Google Scholar [3] 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.  Google Scholar [4] W. C. Chewning, Controllability of the nonlinear wave equation in several space variables, SIAM J. Control, 14 (1976), 19-25. doi: 10.1137/0314002.  Google Scholar [5] M. Cirinà, Boundary controllability of nonlinear hyperbolic systems, SIAM J. Control, 7 (1969), 198-212. doi: 10.1137/0307014.  Google Scholar [6] 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.  Google Scholar [7] 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.  Google Scholar [8] H. O. Fattorini, Local controllability of a nonlinear wave equation, Mathem. Systems Theory, 9 (1975), 30-45. doi: 10.1007/BF01698123.  Google Scholar [9] 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.  Google Scholar [10] 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.  Google Scholar [11] 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.  Google Scholar [12] 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.  Google Scholar [13] 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.  Google Scholar [14] J. L. Lions, Contrôlabilité exacte, perturbations et systèmes distribués, Tome 1, Rech. Math. Appl. 8, Masson, Paris, 1988. Google Scholar [15] 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.  Google Scholar [16] 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.  Google Scholar [17] 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.  Google Scholar [18] D. L. Russell, A unified boundary controllability theory for hyperbolic and parabolic equations, Studies in Appl. Math., 52 (1973), 189-211.  Google Scholar [19] M. E. Taylor, Pseudodifferential Operators and Nonlinear PDE, Birkhäuser, Boston, 1991. doi: 10.1007/978-1-4612-0431-2.  Google Scholar [20] 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.  Google Scholar [21] E. Zuazua, Exact controllability for the semilinear wave equation, J. Math. Pures Appl., 69 (1990), 1-31.  Google Scholar [22] 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.  Google Scholar [23] 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.  Google Scholar [24] 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 Google Scholar
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