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Funzioni semiconcave, singolarità e pile di sabbia

Piermarco Cannarsa — 2005

Bollettino dell'Unione Matematica Italiana

La semiconcavità è una nozione che generalizza quella di concavità conservandone la maggior parte delle proprietà ma permettendo di ampliarne le applicazioni. Questa è una rassegna dei punti più salienti della teoria delle funzioni semiconcave, con particolare riguardo allo studio dei loro insiemi singolari. Come applicazione, si discuterà una formula di rappresentazione per la soluzione di un modello dinamico per la materia granulare.

Generalized gradient flow and singularities of the Riemannian distance function

Piermarco Cannarsa

Séminaire Laurent Schwartz — EDP et applications

Significant information about the topology of a bounded domain Ω of a Riemannian manifold M is encoded into the properties of the distance, d Ω , from the boundary of Ω . We discuss recent results showing the invariance of the singular set of the distance function with respect to the generalized gradient flow of d Ω , as well as applications to homotopy equivalence.

Representation of equilibrium solutions to the table problem of growing sandpiles

Piermarco CannarsaPierre Cardaliaguet — 2004

Journal of the European Mathematical Society

In the dynamical theory of granular matter the so-called table problem consists in studying the evolution of a heap of matter poured continuously onto a bounded domain Ω 2 . The mathematical description of the table problem, at an equilibrium configuration, can be reduced to a boundary value problem for a system of partial differential equations. The analysis of such a system, also connected with other mathematical models such as the Monge–Kantorovich problem, is the object of this paper. Our main...

Interior sphere property of attainable sets and time optimal control problems

Piermarco CannarsaHélène Frankowska — 2006

ESAIM: Control, Optimisation and Calculus of Variations

This paper studies the attainable set at time for the control system y ˙ ( t ) = f ( y ( t ) , u ( t ) ) u ( t ) U showing that, under suitable assumptions on , such a set satisfies a uniform interior sphere condition. The interior sphere property is then applied to recover a semiconcavity result for the value function of time optimal control problems with a general target, and to deduce C-regularity for boundaries of attainable sets.

Null controllability of the heat equation in unbounded domains by a finite measure control region

Piermarco CannarsaPatrick MartinezJudith Vancostenoble — 2004

ESAIM: Control, Optimisation and Calculus of Variations

Motivated by two recent works of Micu and Zuazua and Cabanillas, De Menezes and Zuazua, we study the null controllability of the heat equation in unbounded domains, typically + or N . Considering an unbounded and disconnected control region of the form ω : = n ω n , we prove two null controllability results: under some technical assumption on the control parts ω n , we prove that every initial datum in some weighted L 2 space can be controlled to zero by usual control functions, and every initial datum in L 2 ( Ω ) can...

The vanishing viscosity method in infinite dimensions

Piermarco CannarsaGiuseppe Da Prato — 1989

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

The vanishing viscosity method is adapted to the infinite dimensional case, by showing that the value function of a deterministic optimal control problem can be approximated by the solutions of suitable parabolic equations in Hilbert spaces.

The vanishing viscosity method in infinite dimensions

Piermarco CannarsaGiuseppe Da Prato — 1989

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

The vanishing viscosity method is adapted to the infinite dimensional case, by showing that the value function of a deterministic optimal control problem can be approximated by the solutions of suitable parabolic equations in Hilbert spaces.

Null controllability of Grushin-type operators in dimension two

Karine BeauchardPiermarco CannarsaRoberto Guglielmi — 2014

Journal of the European Mathematical Society

We study the null controllability of the parabolic equation associated with the Grushin-type operator A = x 2 + x 2 γ γ 2 , ( γ > 0 ) , in the rectangle Ω = ( - 1 , 1 ) × ( 0 , 1 ) , under an additive control supported in an open subset ω of Ω . We prove that the equation is null controllable in any positive time for γ < 1 and that there is no time for which it is null controllable for γ > 1 . In the transition regime γ = 1 and when ω is a strip ω = ( a , b ) × ( 0 , 1 ) ( 0 < a , b 1 ) ), a positive minimal time is required for null controllability. Our approach is based on the fact that, thanks to the particular...

Null controllability of the heat equation in unbounded domains by a finite measure control region

Piermarco CannarsaPatrick MartinezJudith Vancostenoble — 2010

ESAIM: Control, Optimisation and Calculus of Variations

Motivated by two recent works of Micu and Zuazua and Cabanillas, De Menezes and Zuazua, we study the null controllability of the heat equation in unbounded domains, typically + or  N . Considering an unbounded and disconnected control region of the form ω : = n ω n , we prove two null controllability results: under some technical assumption on the control parts ω n , we prove that every initial datum in some weighted space can be controlled to zero by usual control functions, and every initial...

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