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

The time period “control idea” refers back to the physique of effects - theoretical, numerical and algorithmic - that have been built to persuade the evolution of the nation of a given procedure to be able to meet a prescribed functionality criterion. structures of curiosity to manage conception should 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 attached to the CIME direction 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 an eye on 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 manage. we're convinced this paintings will supply an authoritative reference paintings for all scientists who're drawn to this box, representing while a pleasant creation to, and an up to date account of, the most energetic developments in present research.

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**Additional info for Control of Partial Differential Equations: Cetraro, Italy 2010, Editors: Piermarco Cannarsa, Jean-Michel Coron**

**Example text**

T /. 51). The last term on the right hand side involves the L2 norm of the solution. 56) 1 On Some Recent Advances on Stabilization for Hyperbolic Equations 23 for all solutions. 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

143). 146) S Proof. t D fx 2 ! ; jut j Ä 1g. t : 48 F. t S 1/=2 . pC1/ dt S 1/=2 . 146). 4. The idea of splitting the set ˝ in two subsets, one with velocities close to zero and its complementary goes back to an original idea of Zuazua. 5. 2. 143). ˝/. Proof. 0/ D 0 then the result holds easily. 0/ ¤ 0. p 1/=2, we conclude. 1 Introduction and Scope Abstract The main purpose of this section is to present the optimal-weight convexity method based on the construction of an optimal-weight function for general Gronwall inequalities, determined as an unknown of an explicit equation, thanks to convexity properties of a suitable feedback-dependent function H .

We recall some definitions and introduce some notation for convex functions. We recall that if is a proper convex function from R on R [ fC1g, then its convex conjugate ? is defined as: ? x/g : x2R Most of the properties given below, are established in [17] but in a somehow different context. We prefer therefore to reformulate the requested results for clarity in the next proposition, which proof is left to the reader. 3. 0/ D 0, where r0 > 0 is sufficiently small. 81) and 8 ? 82) b ? stands for the convex conjugate function of H b .