On Evolution Equations Having Hypo- monotonicities of Opposite Sign On Evolution Equations Having Tahar Haddad and Chems Hypomonotonicities of Opposite Sign Eddine Arroud Introduction Preliminaries Tahar Haddad and Chems Eddine Arroud Main result Comments and open Departement of Mathematics, University of Jijel, Algeria problem Padova University, Italy Bibliographie september 25-29,2017
Outline On Evolution Equations Having Hypo- monotonicities of Opposite Introduction 1 Sign Tahar Haddad and Chems Eddine Arroud Introduction Preliminaries Main result Comments and open problem Bibliographie
Outline On Evolution Equations Having Hypo- monotonicities of Opposite Introduction 1 Sign Tahar Haddad and Chems Eddine Arroud Preliminaries 2 Introduction Preliminaries Main result Comments and open problem Bibliographie
Outline On Evolution Equations Having Hypo- monotonicities of Opposite Introduction 1 Sign Tahar Haddad and Chems Eddine Arroud Preliminaries 2 Introduction Preliminaries Main result 3 Main result Comments and open problem Bibliographie
Outline On Evolution Equations Having Hypo- monotonicities of Opposite Introduction 1 Sign Tahar Haddad and Chems Eddine Arroud Preliminaries 2 Introduction Preliminaries Main result 3 Main result Comments and open Comments and open problem 4 problem Bibliographie
Outline On Evolution Equations Having Hypo- monotonicities of Opposite Introduction 1 Sign Tahar Haddad and Chems Eddine Arroud Preliminaries 2 Introduction Preliminaries Main result 3 Main result Comments and open Comments and open problem 4 problem Bibliographie Bibliographie 5
Introduction On Evolution Sweeping processes were proposed and thoroughly studied by Equations Having Hypo- J.J. Moreau in the seventies, monotonicities of Opposite Sign Tahar Haddad and Chems Eddine Arroud Introduction Preliminaries Main result Comments and open problem Bibliographie
Introduction On Evolution Sweeping processes were proposed and thoroughly studied by Equations Having Hypo- J.J. Moreau in the seventies, As a partial viewpoint, consider a monotonicities of Opposite time-moving closed convex set C ( t ) which drags a point u ( t ) , Sign so this point must stay in C ( t ) at every time t , and the Tahar Haddad and Chems opposite of its velocity, say − du dt ( t ) , has to be normal to the set Eddine Arroud C ( t ) . Introduction Preliminaries Main result Comments and open problem Bibliographie
Introduction On Evolution Sweeping processes were proposed and thoroughly studied by Equations Having Hypo- J.J. Moreau in the seventies, As a partial viewpoint, consider a monotonicities of Opposite time-moving closed convex set C ( t ) which drags a point u ( t ) , Sign so this point must stay in C ( t ) at every time t , and the Tahar Haddad and Chems opposite of its velocity, say − du dt ( t ) , has to be normal to the set Eddine Arroud C ( t ) . Introduction To take into account the nonsmoothness of the boundary Preliminaries of the convex set C ( t ) , the law of motion is formulated as Main result Comments and open du dt ( t ) ∈ − N C ( t ) ( u ( t )) , a . e . t ∈ [ 0 , T ] problem u ( 0 ) = u 0 ∈ C ( 0 ) (1) Bibliographie u ( t ) ∈ C ( t ) ∀ t ∈ [ 0 , T ] , where N C ( t ) ( · ) denotes the normal cone operator of the convex set C ( t ) in the sense of convex analysis in a Hilbert space H.
Existence and uniqueness of solutions of such systems and their On Evolution Equations classical variants : Having Hypo- monotonicities ⊲ subjected to perturbation forces, of Opposite Sign Tahar Haddad and Chems Eddine Arroud ⊲ non convex prox-regular sets C , Introduction Preliminaries Main result ⊲ state dependent set C ( t , x ) Comments and open problem Bibliographie ⊲ second-order sweeping processes ... have been considered by many authors in the literature.
Introduction On Evolution Equations Having Hypo- On the other hand, monotonicities of Opposite in [A. Bressan, A. Cellina, and G. Colombo ;’89] Sign Tahar Haddad and Chems x ( t ) ∈ F ( x ( t )) , ˙ F ( x ) ⊂ ∂ g ( x ) , x ( 0 ) = x 0 , (2) Eddine Arroud Introduction � F is a monotonic upper semicontinuous ( not necessarily Preliminaries convex valued, hence not maximal) map contained in the Main result subdifferential of a locally bounded convex function. Comments and open problem Bibliographie
Introduction On Evolution Equations Having Hypo- On the other hand, monotonicities of Opposite in [A. Bressan, A. Cellina, and G. Colombo ;’89] Sign Tahar Haddad and Chems x ( t ) ∈ F ( x ( t )) , ˙ F ( x ) ⊂ ∂ g ( x ) , x ( 0 ) = x 0 , (2) Eddine Arroud Introduction � F is a monotonic upper semicontinuous ( not necessarily Preliminaries convex valued, hence not maximal) map contained in the Main result subdifferential of a locally bounded convex function. Comments • The lack of the minus sign in (2) yields that the distance and open problem among solutions increases and typically there is no uniqueness. Bibliographie
Introduction On Evolution Equations Having Hypo- On the other hand, monotonicities of Opposite in [A. Bressan, A. Cellina, and G. Colombo ;’89] Sign Tahar Haddad and Chems x ( t ) ∈ F ( x ( t )) , ˙ F ( x ) ⊂ ∂ g ( x ) , x ( 0 ) = x 0 , (2) Eddine Arroud Introduction � F is a monotonic upper semicontinuous ( not necessarily Preliminaries convex valued, hence not maximal) map contained in the Main result subdifferential of a locally bounded convex function. Comments • The lack of the minus sign in (2) yields that the distance and open problem among solutions increases and typically there is no uniqueness. Bibliographie • Existence of solutions depends on arguments of convex analysis. This result has been generalized by many authors in different ways.
The dynamics On Evolution The purpose of the present talk is to study a Cauchy problem Equations Having Hypo- monotonicities of Opposite x ( t ) ∈ F ( x ( t )) − N C ( x ( t )) , ˙ F ( x ) ⊂ ∂ g ( x ) , x ( 0 ) = x 0 ∈ C , Sign (3) Tahar Haddad and Chems ⊲ in a infinite dimensional Hilbert space H , Eddine Arroud Introduction Preliminaries Main result Comments and open problem Bibliographie
The dynamics On Evolution The purpose of the present talk is to study a Cauchy problem Equations Having Hypo- monotonicities of Opposite x ( t ) ∈ F ( x ( t )) − N C ( x ( t )) , ˙ F ( x ) ⊂ ∂ g ( x ) , x ( 0 ) = x 0 ∈ C , Sign (3) Tahar Haddad and Chems ⊲ in a infinite dimensional Hilbert space H , Eddine Arroud ⊲ the closed set C is locally prox-regular at x 0 ( hence N C is Introduction hypomonotone set valued mapping), Preliminaries Main result Comments and open problem Bibliographie
The dynamics On Evolution The purpose of the present talk is to study a Cauchy problem Equations Having Hypo- monotonicities of Opposite x ( t ) ∈ F ( x ( t )) − N C ( x ( t )) , ˙ F ( x ) ⊂ ∂ g ( x ) , x ( 0 ) = x 0 ∈ C , Sign (3) Tahar Haddad and Chems ⊲ in a infinite dimensional Hilbert space H , Eddine Arroud ⊲ the closed set C is locally prox-regular at x 0 ( hence N C is Introduction hypomonotone set valued mapping), Preliminaries ⊲ the set valued mapping F is not necessarily the whole Main result subdifferential of g , and we take the plus sign , instead of Comments and open the classical minus. problem Bibliographie
The dynamics On Evolution The purpose of the present talk is to study a Cauchy problem Equations Having Hypo- monotonicities of Opposite x ( t ) ∈ F ( x ( t )) − N C ( x ( t )) , ˙ F ( x ) ⊂ ∂ g ( x ) , x ( 0 ) = x 0 ∈ C , Sign (3) Tahar Haddad and Chems ⊲ in a infinite dimensional Hilbert space H , Eddine Arroud ⊲ the closed set C is locally prox-regular at x 0 ( hence N C is Introduction hypomonotone set valued mapping), Preliminaries ⊲ the set valued mapping F is not necessarily the whole Main result subdifferential of g , and we take the plus sign , instead of Comments and open the classical minus. problem ⊲ The system (3) can be considered as a non convex cyclical Bibliographie monotone differential inclusion under control term u ( t ) ∈ − N C ( · ) which guarantees that the trajectory x ( t ) always belongs to the desired set C for all t ∈ [ 0 , T ] . We prove (local) existence of solutions.
On Evolution Equations Having Hypo- monotonicities of Opposite Sign In [Castaing, C., Syam, ’03], Tahar Haddad and Chems the authors proved the existence of solutions of problem (3), Eddine Arroud Introduction x ( t ) ∈ F ( x ( t )) − N C ( x ( t )) , ˙ F ( x ) ⊂ ∂ g ( x ) , x ( 0 ) = x 0 ∈ C , Preliminaries Main result Comments and open problem Bibliographie
On Evolution Equations Having Hypo- monotonicities of Opposite Sign In [Castaing, C., Syam, ’03], Tahar Haddad and Chems the authors proved the existence of solutions of problem (3), Eddine Arroud Introduction x ( t ) ∈ F ( x ( t )) − N C ( x ( t )) , ˙ F ( x ) ⊂ ∂ g ( x ) , x ( 0 ) = x 0 ∈ C , Preliminaries Main result Comments • the closed set C was supposed to be compact and and open problem uniformly r-prox-regular , Bibliographie
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