A Mayer Problem for A Controlled Sweeping Process Chems Eddine Arroud and A Mayer Problem for A Controlled Sweeping Giovanni Colombo Process Preliminaries The dynamics Monotonicity of The Chems Eddine Arroud and Giovanni Colombo Distance Example University Of Jijel Algeria, Università di Padova 20 septembre 2017
A Mayer Problem for A Controlled Sweeping Process Chems Eddine Preliminaries 1 Arroud and Giovanni Colombo Preliminaries The dynamics 2 The dynamics Monotonicity of The Distance Monotonicity of The Distance Example Example 3
Preliminaries A Mayer Problem for A The first differential inclusions problems have been studied in Controlled Sweeping the early 70s by H. Brezis Process Chems Eddine − ˙ x ( t ) ∈ ∂ϕ ( x ( t )) t ∈ [0 , T ] Arroud and Giovanni Colombo thanks to the theory of maximal monotone operators. Preliminaries The dynamics Monotonicity of The Distance Example
Preliminaries A Mayer Problem for A The first differential inclusions problems have been studied in Controlled Sweeping the early 70s by H. Brezis Process Chems Eddine − ˙ x ( t ) ∈ ∂ϕ ( x ( t )) t ∈ [0 , T ] Arroud and Giovanni Colombo thanks to the theory of maximal monotone operators. Preliminaries "J.J. Moreau ", Evolution problems associated with a moving The dynamics convex set in a Hilbert space Monotonicity of The Distance Example − ˙ x ( t ) ∈ ∂ I C ( t ) ( x ( t )) , x (0) ∈ C (0)
Preliminaries A Mayer Problem for A The first differential inclusions problems have been studied in Controlled Sweeping the early 70s by H. Brezis Process Chems Eddine − ˙ x ( t ) ∈ ∂ϕ ( x ( t )) t ∈ [0 , T ] Arroud and Giovanni Colombo thanks to the theory of maximal monotone operators. Preliminaries "J.J. Moreau ", Evolution problems associated with a moving The dynamics convex set in a Hilbert space Monotonicity of The Distance Example − ˙ x ( t ) ∈ ∂ I C ( t ) ( x ( t )) , x (0) ∈ C (0) where ∂ I C ( t ) is the subdifferential of the indicator function of a closed convex C (normal cone) : − ˙ x ( t ) ∈ N C ( t ) ( x ( t )) , x (0) ∈ C (0)
A Mayer Problem for A Controlled Sweeping Process Formally, the sweeping process is the differential inclusion with Chems Eddine initial condition Arroud and Giovanni Colombo − ˙ x ( t ) ∈ N C ( t ) ( x ( t )) , x (0) = x 0 ∈ C (0) Preliminaries Here N C ( x ) denotes the normal cone to C at x in C . In The dynamics Monotonicity of The particular, Distance Example
A Mayer Problem for A Controlled Sweeping Process Formally, the sweeping process is the differential inclusion with Chems Eddine initial condition Arroud and Giovanni Colombo − ˙ x ( t ) ∈ N C ( t ) ( x ( t )) , x (0) = x 0 ∈ C (0) Preliminaries Here N C ( x ) denotes the normal cone to C at x in C . In The dynamics Monotonicity of The particular, Distance Example N C ( x ) = { 0 } if x ∈ C N C ( x ) = ∅ if x / ∈ C
A Mayer Problem for A Controlled Sweeping Process Chems Eddine The perturbed sweeping process : Arroud and Giovanni Colombo − ˙ x ( t ) ∈ N C ( t ) ( x ( t )) + f ( x ( t )) , x (0) = x 0 ∈ C (0) Preliminaries The dynamics Monotonicity of The Distance Example
A Mayer Problem for A Controlled Sweeping Process Chems Eddine The perturbed sweeping process : Arroud and Giovanni Colombo − ˙ x ( t ) ∈ N C ( t ) ( x ( t )) + f ( x ( t )) , x (0) = x 0 ∈ C (0) Preliminaries Given a dynamics, it is impossible resisting to the temptation of The dynamics Monotonicity of The putting some control Distance Example − ˙ x ( t ) ∈ N C ( t ) ( x ( t )) + f ( x ( t ) , u ) , u ∈ U
Prox-regular set We say that C is ρ -prox-regular provided the inequality A Mayer Problem for A Controlled � ζ, y − x � ≤ � y − x � 2 Sweeping 2 ρ Process holds for all x , y ∈ C , for every ζ the unit external normal to C Chems Eddine Arroud and Giovanni Colombo Preliminaries The dynamics Monotonicity of The Distance Example
Prox-regular set We say that C is ρ -prox-regular provided the inequality A Mayer Problem for A Controlled � ζ, y − x � ≤ � y − x � 2 Sweeping 2 ρ Process holds for all x , y ∈ C , for every ζ the unit external normal to C Chems Eddine Arroud and Giovanni Colombo Preliminaries The dynamics Monotonicity of The Distance Example
The dynamics A Mayer Problem for A Controlled Sweeping Process Let the problem P Chems Eddine Arroud and Giovanni Colombo Minimize h ( x ( T )) Preliminaries The dynamics Monotonicity of The Distance Example
The dynamics A Mayer Problem for A Controlled Sweeping Process Let the problem P Chems Eddine Arroud and Giovanni Colombo Minimize h ( x ( T )) Preliminaries Subject to The dynamics Monotonicity of The Distance � x ( t ) ∈ − N C ( t ) ( x ( t )) + f ( x ( t ) , u ( t ))) , ˙ (1) Example x (0) = x 0 ∈ C (0) , with respect to u : [0 , T ] � U , u is measurable.
A Mayer Assumptions Problem for A Controlled H 1 : C : [0 , ∞ ) � R n is a set-valued map with the following Sweeping Process properties : Chems Eddine Arroud and Giovanni Colombo Preliminaries The dynamics Monotonicity of The Distance Example
A Mayer Assumptions Problem for A Controlled H 1 : C : [0 , ∞ ) � R n is a set-valued map with the following Sweeping Process properties : Chems Eddine Arroud and H 1 . 1 : for all t ∈ [0 , T ], C ( t ) is nonempty and compact and there Giovanni exists r > 0 such that C ( t ) is uniformly r -prox regular. Colombo H 1 . 2 : C is γ − Lipschitz and has C 3 boundary. Preliminaries The dynamics Monotonicity of The Distance Example
A Mayer Assumptions Problem for A Controlled H 1 : C : [0 , ∞ ) � R n is a set-valued map with the following Sweeping Process properties : Chems Eddine Arroud and H 1 . 1 : for all t ∈ [0 , T ], C ( t ) is nonempty and compact and there Giovanni exists r > 0 such that C ( t ) is uniformly r -prox regular. Colombo H 1 . 2 : C is γ − Lipschitz and has C 3 boundary. Preliminaries H 1 . 3 : C ( t ) = { x : g ( t , x ) ≤ 0 } with g ( ., x ) lipschitz and of class The dynamics C 2 , 1 . Monotonicity of The Distance Example
A Mayer Assumptions Problem for A Controlled H 1 : C : [0 , ∞ ) � R n is a set-valued map with the following Sweeping Process properties : Chems Eddine Arroud and H 1 . 1 : for all t ∈ [0 , T ], C ( t ) is nonempty and compact and there Giovanni exists r > 0 such that C ( t ) is uniformly r -prox regular. Colombo H 1 . 2 : C is γ − Lipschitz and has C 3 boundary. Preliminaries H 1 . 3 : C ( t ) = { x : g ( t , x ) ≤ 0 } with g ( ., x ) lipschitz and of class The dynamics C 2 , 1 . Monotonicity of The Distance H 2 : U ∈ R n is compact and convex. Example
A Mayer Assumptions Problem for A Controlled H 1 : C : [0 , ∞ ) � R n is a set-valued map with the following Sweeping Process properties : Chems Eddine Arroud and H 1 . 1 : for all t ∈ [0 , T ], C ( t ) is nonempty and compact and there Giovanni exists r > 0 such that C ( t ) is uniformly r -prox regular. Colombo H 1 . 2 : C is γ − Lipschitz and has C 3 boundary. Preliminaries H 1 . 3 : C ( t ) = { x : g ( t , x ) ≤ 0 } with g ( ., x ) lipschitz and of class The dynamics C 2 , 1 . Monotonicity of The Distance H 2 : U ∈ R n is compact and convex. Example H 3 : f : R n × U � R n such that there exist β ≥ 0 with H 3 . 1 : | f ( x , u ) | ≤ β for all ( x , u ) ; H 3 . 2 : f ( x , u ) is of class C 1 for all x and u and f ( ., . ) is Lipschitz with lipschitz constant k ; H 3 . 3 : f ( x , U ) is convex for all x ∈ R n ;
A Mayer Assumptions Problem for A Controlled H 1 : C : [0 , ∞ ) � R n is a set-valued map with the following Sweeping Process properties : Chems Eddine Arroud and H 1 . 1 : for all t ∈ [0 , T ], C ( t ) is nonempty and compact and there Giovanni exists r > 0 such that C ( t ) is uniformly r -prox regular. Colombo H 1 . 2 : C is γ − Lipschitz and has C 3 boundary. Preliminaries H 1 . 3 : C ( t ) = { x : g ( t , x ) ≤ 0 } with g ( ., x ) lipschitz and of class The dynamics C 2 , 1 . Monotonicity of The Distance H 2 : U ∈ R n is compact and convex. Example H 3 : f : R n × U � R n such that there exist β ≥ 0 with H 3 . 1 : | f ( x , u ) | ≤ β for all ( x , u ) ; H 3 . 2 : f ( x , u ) is of class C 1 for all x and u and f ( ., . ) is Lipschitz with lipschitz constant k ; H 3 . 3 : f ( x , U ) is convex for all x ∈ R n ; H 4 : h : R � R is of class C 1
Main result A Mayer Theorem Problem for A Controlled Let ( x ∗ , u ∗ ) be a global minimizer satisfying the outward (or Sweeping Process inward) pointing condition. Then there exist a BV adjoint Chems Eddine vector p : [0 , T ] → R n ,a finite signed Radon measure µ on Arroud and Giovanni [0 , T ], and measurable vectors ξ, η : [0 , T ] → R n , with ξ ( t ) ≥ 0 Colombo for µ -a.e. t and 0 ≤ η ( t ) ≤ β + γ for a.e. t , satisfying : Preliminaries The dynamics Monotonicity of The Distance Example
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