Differential inclusions J. Venel Differential inclusions and applications Sweeping process Introduction New assumption Juliette Venel 1 Theory joint work with B. Maury 2 and F. Bernicot 3 Crowd motion model Presentation New formulation 1 Université de Valenciennes et du Hainaut-Cambrésis Theoretical study 2 Université Paris-Sud XI Numerical study Numerical 3 CNRS - Université de Nantes simulations Second order differential inclusions Example General setting Workshop « Optimal Transport in the Applied Science » December 8-12 2014, Linz
Differential inclusions Outline J. Venel Sweeping Sweeping process 1 process Introduction Introduction New assumption Theory New geometrical assumption Crowd motion Theoretical result model Presentation New formulation Theoretical study Application to crowd motion modelling 2 Numerical study Numerical Presentation simulations New formulation Second order differential Theoretical study inclusions Example Numerical study General setting Numerical simulations Second order differential inclusions 3 Example : Granular flows General setting
Differential inclusions Outline J. Venel Sweeping Sweeping process 1 process Introduction Introduction New assumption Theory New geometrical assumption Crowd motion Theoretical result model Presentation New formulation Theoretical study Application to crowd motion modelling 2 Numerical study Numerical Presentation simulations New formulation Second order differential Theoretical study inclusions Example Numerical study General setting Numerical simulations Second order differential inclusions 3 Example : Granular flows General setting
Differential inclusions A simple example J. Venel Sweeping process Introduction New assumption Theory Crowd motion model Presentation New formulation Imagine a ball and a hoop... Theoretical study Numerical study Numerical simulations Second order differential Movie inclusions Example General setting
Differential inclusions Introduction J. Venel The first sweeping process was introduced by J.-J. Moreau Sweeping process in 1977 : Introduction New assumption Theory ˙ x ( t ) ∈ − ∂ I C ( t ) ( x ( t )) , x ( 0 ) ∈ C ( 0 ) Crowd motion model Presentation where ∂ I C represents the subdifferential of the indicator New formulation Theoretical study function of a closed convex set C . Numerical study Numerical simulations Second order ⇒ x ( t ) ∈ C ( t ) . differential inclusions Example General setting Important method : he creates discretized solutions with the so-called catching-up algorithm : x i + 1 = P C ( t i + 1 ) ( x i ) .
Differential inclusions Why a differential inclusion ? J. Venel Sweeping process Introduction New assumption Theory Crowd motion Why not a differential equation? model Presentation New formulation Theoretical study Numerical study Numerical because the state-variable x must remain in a (moving) set simulations Second order C . This constraint makes appear a differential inclusion. differential inclusions Example ˙ x ( t ) ∈ − ∂ I C ( t ) ( x ( t )) + u ( t ) . General setting
Differential inclusions Extension J. Venel Sweeping process Introduction New assumption Theory Crowd motion model Presentation New formulation Theoretical study Numerical study the convexity assumption can be weakened... Numerical simulations Second order differential inclusions Example General setting
Differential inclusions Notations J. Venel Sweeping process Introduction New assumption Let C be a closed subset of a Hilbert space H , Theory we define for x ∈ H Crowd motion model Presentation New formulation d C ( x ) = inf y ∈ C | y − x | Theoretical study Numerical study Numerical simulations and Second order differential P C ( x ) = { y ∈ C , | y − x | = d C ( x ) } . inclusions Example General setting The subdifferential ∂ I C will be replaced with ...
Differential inclusions Proximal normal cone J. Venel Sweeping N ( C , x 1 ) process Introduction N ( C , x 3 ) x 1 New assumption Theory Crowd motion x 3 model x 2 Presentation New formulation Theoretical study N ( C , x 0 ) Numerical study Numerical simulations x 0 Second order differential inclusions C x 4 Example General setting N ( C , x 4 ) Proximal normal cone of C at x N ( C , x ) = { v ∈ H , ∃ α > 0 , x ∈ P C ( x + α v ) } (F. Clarke, R. Stern, P . Wolenski 95)
Differential inclusions Question J. Venel Sweeping process Introduction New assumption Theory Crowd motion model Presentation New formulation Theoretical study Numerical study What is the good property of a closed convex set ? Numerical simulations Second order differential inclusions Example General setting
Differential inclusions Uniform prox-regularity J. Venel Sweeping process Introduction New assumption Uniform prox-regular set Theory Crowd motion Let C be a closed subset of H , model Presentation C is η -prox-regular if the projec- New formulation δ Theoretical study tion on C is single-valued and Numerical study x Numerical continuous at any point x satis- simulations Second order fying d C ( x ) < η . C differential inclusions Example General setting H. Federer 59, positively reached sets A. Canino 88, p-convex sets F. Clarke, R. Stern, P . Wolenski 95, proximally smooth sets R. Poliquin, R. Rockafellar, L. Thibault 00, prox-regular sets
Differential inclusions Uniform prox-regular set J. Venel Let C be a closed subset of H , Sweeping process C is η -prox-regular if for all x ∈ C and v ∈ N ( C , x ) \ { 0 } , Introduction New assumption Theory � � x + η v B ∩ C = ∅ . Crowd motion | v | model Presentation New formulation Theoretical study Numerical study Numerical simulations This is equivalent to the following property of the proximal Second order differential normal cone : inclusions Example General setting Hypomonotonicity of the proximal normal cone ∀ y ∈ C , ∀ x ∈ ∂ C , ∀ v ∈ N ( C , x ) , � y − x , v � ≤ | v | 2 η | x − y | 2 .
Differential inclusions Sweeping process J. Venel Sweeping Process (SP) Sweeping process � ˙ Introduction x ( t ) + N ( C ( t ) , x ( t )) ∋ f ( t , x ( t )) a.e.t. t ∈ I = [ 0 , T ] New assumption Theory x ( 0 ) = x 0 ∈ C ( 0 ) Crowd motion model Presentation New formulation Theoretical study Numerical study Assumptions : Numerical simulations Second order • ∀ t ∈ I , C ( t ) is a closed, nonempty and η - prox-regular differential inclusions set Example General setting • C varies in an absolutely continuous way : ∀ y ∈ R d , ∀ s , t ∈ I , | d C ( t ) ( y ) − d C ( s ) ( y ) | ≤ | a ( t ) − a ( s ) | where a : I → R is an absolutely continous map. • f is Lipschitz continuous with respect to the second variable and satisfies the following growth condition ∀ t ∈ I , | f ( t , x ) | ≤ β ( t )( 1 + | x | ) avec β ∈ L 1 ( I , R + )
Differential inclusions J. Venel Sweeping process Introduction Theorem New assumption Theory Under the previous assumptions, the (SP) problem has a Crowd motion model unique absolutely continous solution. Presentation New formulation Theoretical study Numerical study Numerical simulations J.F. E DMOND , L. T HIBAULT , Relaxation of an optimal control Second order differential problem involving a perturbed sweeping process , inclusions Example Math. Program, Ser. B 104 (2-3), 347-373, 2005. General setting
Differential inclusions Outline J. Venel Sweeping Sweeping process 1 process Introduction Introduction New assumption Theory New geometrical assumption Crowd motion Theoretical result model Presentation New formulation Theoretical study Application to crowd motion modelling 2 Numerical study Numerical Presentation simulations New formulation Second order differential Theoretical study inclusions Example Numerical study General setting Numerical simulations Second order differential inclusions 3 Example : Granular flows General setting
Differential inclusions Application J. Venel Sweeping process Introduction New assumption Theory A crowd motion model with several goals Crowd motion model Presentation New formulation Theoretical study • to deal with emergency evacuation Numerical study Numerical simulations Second order differential inclusions • to take into account direct contacts between individuals Example General setting • to determine the areas where people are crushed
Differential inclusions Two principles J. Venel Sweeping process Introduction New assumption Theory Crowd motion model Presentation New formulation Theoretical study Numerical study Numerical simulations Second order differential inclusions Example General setting Spontaneous velocity Actual velocity
Differential inclusions Notations J. Venel Sweeping e ij ( q ) q = ( q 1 , q 2 , .., q N ) ∈ R 2 N process Introduction r j New assumption Theory q j r i q j − q i Crowd motion e ij ( q ) = q i model | q j − q i | Presentation D ij ( q ) New formulation Theoretical study Numerical study Numerical simulations Second order Set of feasible configurations differential inclusions � � Example q ∈ R 2 N , ∀ i < j , D ij ( q ) = | q i − q j | − r i − r j ≥ 0 Q 0 = General setting G ij ( q ) = ∇ D ij ( q ) = ( 0 ... 0 , − e ij ( q ) , 0 ... 0 , e ij ( q ) , 0 ... 0 ) i j
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