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Differential inclusions and applications Sweeping process - - PowerPoint PPT Presentation

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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

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SLIDE 1

Differential inclusions

  • 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

Differential inclusions and applications

Juliette Venel 1 joint work with B. Maury 2 and F. Bernicot 3

1 Université de Valenciennes et du Hainaut-Cambrésis 2 Université Paris-Sud XI 3 CNRS - Université de Nantes

Workshop « Optimal Transport in the Applied Science » December 8-12 2014, Linz

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SLIDE 2

Differential inclusions

  • 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

Outline

1

Sweeping process Introduction New geometrical assumption Theoretical result

2

Application to crowd motion modelling Presentation New formulation Theoretical study Numerical study Numerical simulations

3

Second order differential inclusions Example : Granular flows General setting

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SLIDE 3

Differential inclusions

  • 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

Outline

1

Sweeping process Introduction New geometrical assumption Theoretical result

2

Application to crowd motion modelling Presentation New formulation Theoretical study Numerical study Numerical simulations

3

Second order differential inclusions Example : Granular flows General setting

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SLIDE 4

Differential inclusions

  • 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

A simple example

Imagine a ball and a hoop... Movie

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Differential inclusions

  • 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

Introduction

The first sweeping process was introduced by J.-J. Moreau in 1977 : ˙ x(t) ∈ −∂IC(t)(x(t)), x(0) ∈ C(0) where ∂IC represents the subdifferential of the indicator function of a closed convex set C. ⇒ x(t) ∈ C(t). Important method : he creates discretized solutions with the so-called catching-up algorithm : xi+1 = PC(ti+1)(xi).

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Differential inclusions

  • 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

Why a differential inclusion ?

Why not a differential equation? because the state-variable x must remain in a (moving) set

  • C. This constraint makes appear a differential inclusion.

˙ x(t) ∈ −∂IC(t)(x(t)) + u(t).

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Differential inclusions

  • 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

Extension

the convexity assumption can be weakened...

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Differential inclusions

  • 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

Notations

Let C be a closed subset of a Hilbert space H, we define for x ∈ H dC(x) = inf

y∈C |y − x|

and PC(x) = {y ∈ C , |y − x| = dC(x)} . The subdifferential ∂IC will be replaced with ...

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Differential inclusions

  • 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

Proximal normal cone

x0 x1 x2 x3 x4 N(C, x0) N(C, x1) N(C, x3) N(C, x4) C Proximal normal cone of C at x N(C, x) = {v ∈ H , ∃α > 0 , x ∈ PC(x + αv)} (F. Clarke, R. Stern, P . Wolenski 95)

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Differential inclusions

  • 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

Question

What is the good property of a closed convex set ?

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Differential inclusions

  • 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

Uniform prox-regularity

δ x C Uniform prox-regular set Let C be a closed subset of H, C is η-prox-regular if the projec- tion on C is single-valued and continuous at any point x satis- fying dC(x) < η.

  • 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
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Differential inclusions

  • 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

Uniform prox-regular set Let C be a closed subset of H, C is η-prox-regular if for all x ∈ C and v ∈ N(C, x) \ {0}, B

  • x + η v

|v|

  • ∩ C = ∅.

This is equivalent to the following property of the proximal normal cone : Hypomonotonicity of the proximal normal cone ∀y ∈ C, ∀x ∈ ∂C, ∀v ∈ N(C, x), y − x, v ≤ |v| 2η |x − y|2.

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Differential inclusions

  • 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

Sweeping process

Sweeping Process (SP) ˙ x(t) + N(C(t), x(t)) ∋ f(t, x(t)) a.e.t. t ∈ I = [0, T] x(0) = x0 ∈ C(0) Assumptions :

  • ∀t ∈ I , C(t) is a closed, nonempty and η- prox-regular

set

  • C varies in an absolutely continuous way :

∀y ∈ Rd , ∀s, t ∈ I , |dC(t)(y) − dC(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 ∈ L1(I, R+)

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Differential inclusions

  • 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

Theorem Under the previous assumptions, the (SP) problem has a unique absolutely continous solution. J.F. EDMOND, L. THIBAULT, Relaxation of an optimal control problem involving a perturbed sweeping process,

  • Math. Program, Ser. B 104 (2-3), 347-373, 2005.
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Differential inclusions

  • 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

Outline

1

Sweeping process Introduction New geometrical assumption Theoretical result

2

Application to crowd motion modelling Presentation New formulation Theoretical study Numerical study Numerical simulations

3

Second order differential inclusions Example : Granular flows General setting

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SLIDE 16

Differential inclusions

  • 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

Application

A crowd motion model with several goals

  • to deal with emergency evacuation
  • to take into account direct contacts between individuals
  • to determine the areas where people are crushed
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Differential inclusions

  • 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

Two principles

Spontaneous velocity Actual velocity

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Differential inclusions

  • 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

Notations

qi qj eij(q) Dij(q) ri rj q = (q1, q2, .., qN) ∈ R2N eij(q) = qj − qi |qj − qi| Set of feasible configurations Q0 =

  • q ∈ R2N, ∀ i < j,

Dij(q) = |qi − qj| − ri − rj ≥ 0

  • Gij(q) = ∇Dij(q) = (0 ... 0,

−eij(q) , 0 ... 0, eij(q) , 0 ... 0) i j

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Differential inclusions

  • 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

Notation : U(q) = (U1(q), U2(q), ..., UN(q)) Example : Ui(q) = −si∇D(qi), where D(x) represents the geodesic distance between x and the exit. Contour levels of D

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Differential inclusions

  • 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

Actual velocity

To handle the contacts, we define the cone of admissible velocities Cq =

  • v ∈ R2N, ∀ i < j

Dij(q) = 0 ⇒ Gij(q) · v ≥ 0

  • ,

where Gij(q) = ∇Dij(q). If u is the actual velocity of the N pedestrians, the model can be expressed as follows :      q = q0 +

  • u,

u = PCqU.

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Differential inclusions

  • 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

Cone Nq

Let us define Nq the polar cone of Cq : Definition Nq = C◦

q = {w , (w, v) ≤ 0

∀ v ∈ Cq} . D12 < 0 D13 < 0 D34 < 0 ¯ q q N¯

q

C¯

q

Nq Cq

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Differential inclusions

  • 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

Differential inclusion

Proposition Nq =

  • −
  • λijGij(q) , λij ≥ 0 , Dij(q) > 0 =

⇒ λij = 0

  • .

Since Cq and Nq are mutually polar cones, the following property holds (J.-J. Moreau 62) PCq + PNq = Id, and so the problem can be formulated as a first order differential inclusion . Model    dq dt + Nq ∋ U(q), q(0) = q0.

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Differential inclusions

  • 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

Prox-regularity of Q0

Proposition Q0 is η-prox-regular with η = η(N, ri). Sketch of the proof : One constraint’s case : Qij = {q ∈ R2N , Dij(q) = |qj − qi| − (rj + ri) ≥ 0} is ηij-prox-regular with ηij = ri + rj √ 2 . Extension to several constraints : Q0 =

i<j

Qij. n1 n2

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Differential inclusions

  • 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

Key point of the proof

A reverse triangle inequality For all q ∈ Q0, for all λij ≥ 0, there exists γ > 1 such that

  • (i,j)∈I(q)

λij|Gij(q)| ≤ γ

  • (i,j)∈I(q)

λijGij(q)

  • ,

where I(q) = {(i, j), i < j, Dij(q) = 0}. Moreover, for all q, Nq = N(Q0, q).

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Differential inclusions

  • 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

Well-posedness

Theorem Assume that U is bounded and Lipschitz continuous. Then for any q0 in Q0, there is a unique absolutely conti- nuous map q satisfying    dq dt + N(Q0, q) ∋ U(q) a.e. in [0, T], q(0) = q0.

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Differential inclusions

  • 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

Numerical scheme

Initialization : q0 = q0 Time-loop : qn is known un = PCh(qn)(U(qn)) qn+1 = qn + h un whereCh(qn) =

  • v ∈ R2N, ∀ i < j, Dij(qn) + h Gij(qn) · v ≥ 0
  • .

In terms of position, this algorithm can be formulated as follows : qn+1 = PK(qn)(qn + h U(qn)) with K(qn) =

  • q ∈ R2N, ∀ i < j, Dij(qn) + Gij(qn) · (q − qn) ≥ 0
  • .
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Differential inclusions

  • 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

Comparison between theoretical and numerical projections

qn qn + h U(qn) qn + h U(qn) qn+1 qn+1 ˜ qn+1 ˜ qn+1 Q0 K(qn)

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Differential inclusions

  • 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

Convergence

Let qh be the continous piecewise linear function associated to the numerical scheme Theorem Assume that U is bounded and Lipschitz continous. Then qh uniformly converges in [0, T] to the map q satis- fying :    dq dt + N(Q0, q) ∋ U(q) a.e. in [0, T], q(0) = q0.

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Differential inclusions

  • 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

Continuous and discrete problems

Discrete differential inclusion : un + N(K(qn), qn+1) ∋ U(qn). Continuous differential inclusion : dq dt + N(Q0, q) ∋ U(q). Proposition N(Q0, q) = N(K(q), q).

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Differential inclusions

  • 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

A second important geometrical assumption

S The set S is not suitable. C The set C is suitable. No "thin peaks".

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Differential inclusions

  • 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

Numerical simulations

  • Arches

Movie Pressure

  • With individual strategies

Movie

  • Evacuation of a building

Movie Geodesics Movie Zoom

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Differential inclusions

  • 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

Set defined by inequalities

If the moving set is defined by some inequalities : C(t) :=

  • x ∈ Rd, gi(t, x) ≥ 0
  • ,

what are the assumptions which imply

  • the well-posedness of the associated sweeping process

and

  • the convergence of the numerical scheme based on a

linear approximation of the constraints ?

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Differential inclusions

  • 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

Sufficient assumptions

So we consider C(t) :=

p

  • i=1

Ci(t) :=

  • x ∈ Rd, gi(t, x) ≥ 0
  • .

We define also Ωi := {(t, x), t ∈ I, x ∈ Ci(t)}. Assume that there exist α, β, M, κ > 0 such that gi ∈ C2 (Ω + κB(0, 1)) and satisfies in Ωi + κB(0, 1) : α ≤ |∇xgi(t, x)| ≤ β, |∂tgi(t, x)| ≤ β (1) |D2

xgi(t, x)|,

|∂2

t gi(t, x)|,

|∂t∇xgi(t, x)| ≤ M. (2)

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SLIDE 34

Differential inclusions

  • 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

For all t ∈ I, we define for ρ > 0 Iρ(t, x) := {i, gi(t, x) ≤ ρ} . We suppose that there exist constants ρ, γ > 0 such that for all x ∈ C(t) and all nonnegative reals λi

  • i∈Iρ(t,x)

λi|∇gi(t, x)| ≤ γ

  • i∈Iρ(t,x)

λi∇gi(t, x)

  • ,

(Rρ) Proposition Under the assumptions (1), (2) and (Rρ), there exists η > 0 such that the set C(t) is η-prox-regular for all t ∈ I. Moreover the set-valued map C is Lipschitz continuous with respect to the Hausdorff distance.

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SLIDE 35

Differential inclusions

  • 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

Numerical scheme xn+1 = P˜

C(tn+1,xn)(xn + h f n)

with ˜ C(t, x) =

  • y ∈ Rd,

∀i, gi(t, x) + ∇xgi(t, x) · (y − x) ≥ 0

  • .

Previous assumptions ⇒ xh converges to x solution of (SP).

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SLIDE 36

Differential inclusions

  • 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

Outline

1

Sweeping process Introduction New geometrical assumption Theoretical result

2

Application to crowd motion modelling Presentation New formulation Theoretical study Numerical study Numerical simulations

3

Second order differential inclusions Example : Granular flows General setting

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SLIDE 37

Differential inclusions

  • 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

Granular media

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SLIDE 38

Differential inclusions

  • 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

Granular flows with inelastic shocks

             ¨ q + N(Q0, q) ∋ f(t, q) ˙ q+ = PCq( ˙ q−) (inelastic shock) q(0) = q0 ˙ q(0) = u0. existence of a solution q ∈ W 1,∞(I, Rd) with ˙ q ∈ BV(I, Rd).

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Differential inclusions

  • 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

Improvements

Required assumptions : Independence of Gij(q) Gij(q) · Gkl(q) ≤ 0. Non-independent case :

  • L. PAOLI Time-stepping approximation of rigid-body

dynamics with perfect unilateral constraints. I-The inelastic impact case Arch. Rational Mech. Anal. 198, no. 2, 457-503, 2010

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SLIDE 40

Differential inclusions

  • 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

Set defined by inequalities

With the previous notations (C = Ci, gi, ...) and the previous assumptions (1), (2) and (Rρ), we obtain also the existence of a solution of              ¨ x + N(C(t), x) ∋ f(t, x) ˙ x+ = PV(t,x)( ˙ x−) x(0) = x0 ˙ x(0) = u0. where V(t, x) =

  • z ∈ Rd,

∀i, ∂tgi(t, x) + ∇xgi(t, x) · z ≥ 0

  • .
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SLIDE 41

Differential inclusions

  • 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

General set

If C is a Lipschitz set-valued map with η-prox-regular values and without "thin peaks”, we obtain the existence of a solution of              ¨ x(t) + N(C(t), x(t)) ∋ f(t, x(t)) ˙ x(t+) = PW(t,x(t))( ˙ x(t−)) x(0) = x0 ˙ x(0) = u0 with W(t, x) =

  • v = lim

ǫց0 vǫ, with vǫ ∈ C(t + ǫ) − x

ǫ

  • .
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SLIDE 42

Differential inclusions

  • 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

Thanks for your attention !