By Piermarco Cannarsa, Jean-Michel Coron, Fatiha Alabau-Boussouira, Roger Brockett, Olivier Glass, Jérôme Le Rousseau, Enrique Zuazua
The time period “control concept” refers back to the physique of effects - theoretical, numerical and algorithmic - that have been constructed to persuade the evolution of the country of a given approach which will meet a prescribed functionality criterion. platforms of curiosity to manage conception can be of very diverse natures. This monograph is anxious with versions that may be defined by means of partial differential equations of evolution. It includes 5 significant contributions and is hooked up to the CIME path on keep watch over of Partial Differential Equations that happened in Cetraro (CS, Italy), July 19 - 23, 2010. in particular, it covers the stabilization of evolution equations, keep watch over of the Liouville equation, keep watch over in fluid mechanics, keep watch over and numerics for the wave equation, and Carleman estimates for elliptic and parabolic equations with software to regulate. we're convinced this paintings will offer an authoritative reference paintings for all scientists who're drawn to this box, representing whilst a pleasant creation to, and an up-to-date account of, probably the most lively tendencies in present research.
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This publication has been provided the Ferran Sunyer i Balaguer 2005 prize. the purpose of this monograph is to debate numerous elliptic difficulties on Rn with major features: they are variational and perturbative in nature, and traditional instruments of nonlinear research in response to compactness arguments can't be utilized in normal.
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Extra info for Control of Partial Differential Equations: Cetraro, Italy 2010, Editors: Piermarco Cannarsa, Jean-Michel Coron
For the sake of clearness, the proof of this theorem is divided in three lemmas. The first Lemma below establishes an energy comparison principle and allows us to give a lower estimate of the energy of the solutions. 13 (An energy comparison principle). HS1/. 0/ D 0. 113) and E be its energy. 117). 14. The above lower estimate does not require the hypotheses on the behavior of H in a neighbourhood of 0. Proof. 119) holds. 15. HS1/. 118) is strictly convex on Œ0; r02 . 83). 113) corresponding to this initial data, and E be its energy.
28:265–268, 1989), Komornik (Exact Controllability and Stabilization. The Multiplier Method, Wiley, Masson, Paris, 1994), Nakao  (Math. Ann. 305:403–417, 1996), Conrad and Rao (Asymptot. Anal. 7:159–177, 1993) and the references therein. These results concern localized and boundary damped wave-like equations and use either the method of perturbed energy combined with nonlinear differential inequalities or polynomial Gronwall inequalities for the natural energy together with the multiplier method.
0/. t/ Ä 2ˇL r Á 1 ; 1. 91) 32 F. 92). Proof. 71). 3. Then, L is a strictly increasing onto function from Œ0; C1/ on Œ0; Á/. 0/ b < Á: s E. E. r02 //. 94), r 2 ŒB; Á/. t/ r Á 1 ; t 1. 69). L. 91) in the general case. x/ < 1. v/ Á dv : for x sufficiently large ; where D is a positive constant which depends on r0 and ı. Since increasing, we deduce that 1 1 t r . 90). 5. Note also that for general types of dampings, the required weight function L 1 is not defined on all RC . This is not surprising in view of this degree of generality.