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

On a few contemporary Advances on Stabilization for Hyperbolic Equations / Fatiha Alabau-Boussouira -- Notes at the keep an eye on of the Liouville Equation / Roger Brockett -- a few Questions of regulate in Fluid Mechanics / Olivier Glass -- Carleman Estimates and a few functions to manage concept / Jérôme Le Rousseau -- The Wave Equation: regulate and Numerics / Sylvain Ervedoza and Enrique Zuazua

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**Extra info for Control of partial differential equations : Cetraro, Italy 2010**

**Example text**

Hence . n /n is bounded, so that up to a subsequence we can assume that either that n ! > 0 or n ! 0. Let us first assume that n ! > 0. 0; 1/. vn //n converges strongly to q D f . ˝// for every r 1. 0; T / ˝; We set w D vt . Then w satisfies w C f 0 . e. 0; T / !. Let us now assume that n ! 0. 0/. e. 0; T / !. e. e. 0; T / !. 0; T / ˝. t; x/v D 0 in ˝ ; 8 t 0 where q 0. 62). 56) holds for all solutions. 55). C /. 7. In his paper Zuazua’s result is presented in a larger manner, so that if a unique continuation result is available for the linear wave equation with nonnegative and time and space dependent potential essentially bounded, then his compactness–uniqueness method can be used to derive an exponential decay of the semilinear wave equation.

Equat. 248: 1473–1517, 2010) for its version in the finite dimensional case, already presented in Sect. 3, and its simplification. The extension of this method to nonfrictional dampings such as linear memory-damping with general decaying kernels, is presented in [10]. This approach is build through several new results: optimal decay of solutions of general Gronwall inequalities, feedback classification through a new function H (introduced in (Alabau-Boussouira, J. Differ. Equat. 248: 1473– 1517, 2010)), constructive and simplified decay rates, comparison arguments.

Optim. 51(1):61–105, 2005)] and on a new criteria to classify the feedbacks’ behavior based on the behavior at 0 of a function H introduced for the first time, as far as we know, in (Alabau-Boussouira, J. Differ. Equat. 248:1473–1517, 2010) . We combine these new mathematical tools to establish optimal upper energy decay rates and energy comparison principles. These tools will also be used in the infinite dimensional case combined with the multiplier method to handle geometrical aspects (see Sect.