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 keep an eye on in Fluid Mechanics / Olivier Glass -- Carleman Estimates and a few purposes to manage idea / Jérôme Le Rousseau -- The Wave Equation: keep an eye on and Numerics / Sylvain Ervedoza and Enrique Zuazua

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

**Example text**

Optim. 51(1):61– 105, 2005), adapted to the finite dimensional case in (Alabau-Boussouira, J. Differ. Equat. 248:1473–1517, 2010) with optimality results in this latter case. Hence, we consider in this section the case of nonlinear stabilization for ordinary differential equations. The aim is to give a complete characterization (optimal) of the energy decay rates for general damping functions with applications to the semidiscretization of PDE’s. We will give general tools based on nonlinear Gronwall inequalities, convexity properties and a key comparison lemma (Alabau-Boussouira, J.

98) 34 F. Alabau-Boussouira where z0 > 0 and Ä > 0 are given. t/ is defined for every t to 0 at infinity. 0 > 0 such that Á H. 0; z0 (arbitrary). 7. This lemma allows us to compare time-pointwise estimates to energy types estimates. The upper estimates obtained by the optimal-weight convexity method are time-pointwise estimates, whereas the lower estimates derived by an energy comparison principle are in an energy formulation. To get optimality results, it is essential to be able to compare these two kind of estimates.

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. 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 Example 5, now H tends to 1 at 0. 91). 23. Let us comment Example 5 for which optimality cannot be asserted. 113), then we can give easily examples of parameters such that there exist two branches of solutions with a different asymptotic behavior (see [8] for more details).