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

The time period “control conception” refers back to the physique of effects - theoretical, numerical and algorithmic - that have been built to persuade the evolution of the kingdom of a given method with the intention to meet a prescribed functionality criterion. structures of curiosity to regulate concept will be of very various natures. This monograph is worried with types that may be defined by way of partial differential equations of evolution. It comprises 5 significant contributions and is attached to the CIME direction on keep an eye on of Partial Differential Equations that came about 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 an eye on in fluid mechanics, regulate 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 attracted to this box, representing while a pleasant creation to, and an up-to-date account of, the most lively tendencies in present research.

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(1) Pu == f in (9, an open set in R N ,

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lor on a subset of the boundary 8(9 1 < i < v,
where P is a linear differential operator in (9 and the place the Q/s are linear
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**Extra resources for Control of Partial Differential Equations: Cetraro, Italy 2010, Editors: Piermarco Cannarsa, Jean-Michel Coron**

**Example text**

Convexity properties for general feedbacks have been first introduced in the context of damped hyperbolic PDE’s by Lasiecka and Tataru [70], who used an ODE approach rather than integral inequalities. Liu and Zuazua [80] and Martinez [85], and Eller Lagnese and Nicaise for Maxwell equations used convexity properties in different ways. A more general nonlinear Gronwall inequality than stated above is proved in [17] (see also [18]). 2 A Comparison Lemma The following result will be determinant for optimality results and energy comparison principles as introduced in [7, 8].

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.

Explicit computable decay estimates were given for the case of linear or polynomial feedbacks as applications. A similar approach was given by Liu and Zuazua in [80] including in addition feedbacks with nonlinear growth at infinity. The authors used sharper properties of convex functions and in particular their convex conjugates. Simple explicit and easily computable formula for upper energy decay estimates for general feedbacks is given in Martinez in [84, 85]. His method also involves convexity properties but in a different way than in [17, 70, 80].