By Piermarco Cannarsa, Jean-Michel Coron, Fatiha Alabau-Boussouira, Roger Brockett, Olivier Glass, Jérôme Le Rousseau, Enrique Zuazua

The time period 'control thought' refers back to the physique of effects - theoretical, numerical and algorithmic - which were built to steer the evolution of the kingdom of a given approach with a view to meet a prescribed functionality criterion. structures of curiosity to manage idea will be of very varied natures. This monograph is anxious with versions that may be defined via partial differential equations of evolution. It comprises 5 significant contributions and is hooked up to the CIME path on keep an eye on of Partial Differential Equations that happened in Cetraro (CS, Italy), July 19 - 23, 2010. in particular, it covers the stabilization of evolution equations, regulate of the Liouville equation, keep an eye on in fluid mechanics, keep an eye on and numerics for the wave equation, and Carleman estimates for elliptic and parabolic equations with program 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 advent to, and an up to date account of, one of the most energetic tendencies in present examine.

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**Additional resources for Control of Partial Differential Equations: Cetraro, Italy 2010, Editors: Piermarco Cannarsa, Jean-Michel Coron (Lecture Notes in Mathematics)**

**Example text**

U02 C jruj2 / ; s S ˝\Q1 where C is a positive constant which depends only on j and mj . 25) S ˝ u. 26) as follows. 26). We just follow here the usual technique for the wave equation as developed in [84]. We give the steps for the sake of the completeness. 31) S ˝\Q2 Since RN n! RN / such that 0 Ä ˇ Ä 1 ; ˇ D 1 on Q2 ; ˇ D 0 on RN n! 36) ˝ 18 F. x; u0 /j2 C Á s E ÁS ! 26). x; u0 /j2 ı S ! x; u0 /j2 : ! 17). Once this estimate is proved, one can use the dissipation relation to prove that the energy satisfies integral inequalities of Gronwall type.

This result is proved in Zuazua [105]. 56) does not hold. s/ D 1 f. 51). vn /t j2 ! 55) with a constant C0 which still only depends on ı. 0; T / ˝/. vn /n such that 8 1 ˆ ˆ

0; r0 . u0 ; u1 / in R2 . ln. 0; r0 . p 1/ uniformly for large t. 0; r0 . t// 1 ; uniformly for large t. ln. x // ; 1 < p < 2 ; x 2 Œ0; r0 . t // uniformly for large t. ln. 0/ e pt 0 2. 0; r0 , where p > 0. 141) for t sufficiently large. Optimality cannot be asserted for this latter example. Proof. 11. 11 is satisfied. Applying this theorem we obtain the announced characterization of the asymptotic behavior of the energy at infinity for these two examples. For Examples 3 and 4, H tends to 0 at 0.