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A Mayer Problem for A Controlled Sweeping Process Chems Eddine Arroud and Giovanni Colombo Preliminaries The dynamics

Monotonicity of The Distance

Example

A Mayer Problem for A Controlled Sweeping Process

Chems Eddine Arroud and Giovanni Colombo

University Of Jijel Algeria, Università di Padova

20 septembre 2017

A Mayer Problem for A Controlled Sweeping Process Chems Eddine Arroud and Giovanni Colombo Preliminaries The dynamics

Monotonicity of The Distance

Example

1

Preliminaries

2

The dynamics Monotonicity of The Distance

3

Example

A Mayer Problem for A Controlled Sweeping Process Chems Eddine Arroud and Giovanni Colombo Preliminaries The dynamics

Monotonicity of The Distance

Example

Preliminaries

The first differential inclusions problems have been studied in the early 70s by H. Brezis −˙ x(t) ∈ ∂ϕ(x(t)) t ∈ [0, T] thanks to the theory of maximal monotone operators.

A Mayer Problem for A Controlled Sweeping Process Chems Eddine Arroud and Giovanni Colombo Preliminaries The dynamics

Monotonicity of The Distance

Example

Preliminaries

The first differential inclusions problems have been studied in the early 70s by H. Brezis −˙ x(t) ∈ ∂ϕ(x(t)) t ∈ [0, T] thanks to the theory of maximal monotone operators. "J.J. Moreau ", Evolution problems associated with a moving convex set in a Hilbert space −˙ x(t) ∈ ∂IC(t)(x(t)), x(0) ∈ C(0)

A Mayer Problem for A Controlled Sweeping Process Chems Eddine Arroud and Giovanni Colombo Preliminaries The dynamics

Monotonicity of The Distance

Example

Preliminaries

The first differential inclusions problems have been studied in the early 70s by H. Brezis −˙ x(t) ∈ ∂ϕ(x(t)) t ∈ [0, T] thanks to the theory of maximal monotone operators. "J.J. Moreau ", Evolution problems associated with a moving convex set in a Hilbert space −˙ x(t) ∈ ∂IC(t)(x(t)), x(0) ∈ C(0) where ∂IC(t) is the subdifferential of the indicator function of a closed convex C (normal cone) : −˙ x(t) ∈ NC(t)(x(t)), x(0) ∈ C(0)

A Mayer Problem for A Controlled Sweeping Process Chems Eddine Arroud and Giovanni Colombo Preliminaries The dynamics

Monotonicity of The Distance

Example

Formally, the sweeping process is the differential inclusion with initial condition −˙ x(t) ∈ NC(t)(x(t)), x(0) = x0 ∈ C(0) Here NC(x) denotes the normal cone to C at x in C. In particular,

A Mayer Problem for A Controlled Sweeping Process Chems Eddine Arroud and Giovanni Colombo Preliminaries The dynamics

Monotonicity of The Distance

Example

Formally, the sweeping process is the differential inclusion with initial condition −˙ x(t) ∈ NC(t)(x(t)), x(0) = x0 ∈ C(0) Here NC(x) denotes the normal cone to C at x in C. In particular, NC(x) = {0} if x ∈ C NC(x) = ∅ if x / ∈ C

A Mayer Problem for A Controlled Sweeping Process Chems Eddine Arroud and Giovanni Colombo Preliminaries The dynamics

Monotonicity of The Distance

Example

The perturbed sweeping process : −˙ x(t) ∈ NC(t)(x(t)) + f (x(t)), x(0) = x0 ∈ C(0)

A Mayer Problem for A Controlled Sweeping Process Chems Eddine Arroud and Giovanni Colombo Preliminaries The dynamics

Monotonicity of The Distance

Example

The perturbed sweeping process : −˙ x(t) ∈ NC(t)(x(t)) + f (x(t)), x(0) = x0 ∈ C(0) Given a dynamics, it is impossible resisting to the temptation of putting some control −˙ x(t) ∈ NC(t)(x(t)) + f (x(t), u), u ∈ U

A Mayer Problem for A Controlled Sweeping Process 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 ζ, y − x ≤ y−x2

2ρ

holds for all x, y ∈ C, for every ζ the unit external normal to C

A Mayer Problem for A Controlled Sweeping Process 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 ζ, y − x ≤ y−x2

2ρ

holds for all x, y ∈ C, for every ζ the unit external normal to C

A Mayer Problem for A Controlled Sweeping Process Chems Eddine Arroud and Giovanni Colombo Preliminaries The dynamics

Monotonicity of The Distance

Example

The dynamics

Let the problem P Minimize h(x(T))

A Mayer Problem for A Controlled Sweeping Process Chems Eddine Arroud and Giovanni Colombo Preliminaries The dynamics

Monotonicity of The Distance

Example

The dynamics

Let the problem P Minimize h(x(T)) Subject to

• ˙

x(t) ∈ −NC(t)(x(t)) + f (x(t), u(t))), x(0) = x0 ∈ C(0) , (1) with respect to u : [0, T] U, u is measurable.

A Mayer Problem for A Controlled Sweeping Process Chems Eddine Arroud and Giovanni Colombo Preliminaries The dynamics

Monotonicity of The Distance

Example

Assumptions H1 : C : [0, ∞) Rn is a set-valued map with the following properties :

A Mayer Problem for A Controlled Sweeping Process Chems Eddine Arroud and Giovanni Colombo Preliminaries The dynamics

Monotonicity of The Distance

Example

Assumptions H1 : C : [0, ∞) Rn is a set-valued map with the following properties :

H1.1 : for all t ∈ [0, T], C(t) is nonempty and compact and there exists r > 0 such that C(t) is uniformly r-prox regular. H1.2 : C is γ− Lipschitz and has C 3 boundary.

A Mayer Problem for A Controlled Sweeping Process Chems Eddine Arroud and Giovanni Colombo Preliminaries The dynamics

Monotonicity of The Distance

Example

Assumptions H1 : C : [0, ∞) Rn is a set-valued map with the following properties :

H1.1 : for all t ∈ [0, T], C(t) is nonempty and compact and there exists r > 0 such that C(t) is uniformly r-prox regular. H1.2 : C is γ− Lipschitz and has C 3 boundary. H1.3 : C(t) = {x : g(t, x) ≤ 0} with g(., x) lipschitz and of class C 2,1.

A Mayer Problem for A Controlled Sweeping Process Chems Eddine Arroud and Giovanni Colombo Preliminaries The dynamics

Monotonicity of The Distance

Example

Assumptions H1 : C : [0, ∞) Rn is a set-valued map with the following properties :

H1.1 : for all t ∈ [0, T], C(t) is nonempty and compact and there exists r > 0 such that C(t) is uniformly r-prox regular. H1.2 : C is γ− Lipschitz and has C 3 boundary. H1.3 : C(t) = {x : g(t, x) ≤ 0} with g(., x) lipschitz and of class C 2,1.

H2 : U ∈ Rn is compact and convex.

A Mayer Problem for A Controlled Sweeping Process Chems Eddine Arroud and Giovanni Colombo Preliminaries The dynamics

Monotonicity of The Distance

Example

Assumptions H1 : C : [0, ∞) Rn is a set-valued map with the following properties :

H1.1 : for all t ∈ [0, T], C(t) is nonempty and compact and there exists r > 0 such that C(t) is uniformly r-prox regular. H1.2 : C is γ− Lipschitz and has C 3 boundary. H1.3 : C(t) = {x : g(t, x) ≤ 0} with g(., x) lipschitz and of class C 2,1.

H2 : U ∈ Rn is compact and convex. H3 : f : Rn × U Rn such that there exist β ≥ 0 with

H3.1 : |f (x, u)| ≤ β for all (x, u) ; H3.2 : f (x, u) is of class C 1 for all x and u and f (., .) is Lipschitz with lipschitz constant k ; H3.3 : f (x, U) is convex for all x ∈ Rn ;

A Mayer Problem for A Controlled Sweeping Process Chems Eddine Arroud and Giovanni Colombo Preliminaries The dynamics

Monotonicity of The Distance

Example

Assumptions H1 : C : [0, ∞) Rn is a set-valued map with the following properties :

H1.1 : for all t ∈ [0, T], C(t) is nonempty and compact and there exists r > 0 such that C(t) is uniformly r-prox regular. H1.2 : C is γ− Lipschitz and has C 3 boundary. H1.3 : C(t) = {x : g(t, x) ≤ 0} with g(., x) lipschitz and of class C 2,1.

H2 : U ∈ Rn is compact and convex. H3 : f : Rn × U Rn such that there exist β ≥ 0 with

H3.1 : |f (x, u)| ≤ β for all (x, u) ; H3.2 : f (x, u) is of class C 1 for all x and u and f (., .) is Lipschitz with lipschitz constant k ; H3.3 : f (x, U) is convex for all x ∈ Rn ;

H4 : h : R R is of class C1

A Mayer Problem for A Controlled Sweeping Process Chems Eddine Arroud and Giovanni Colombo Preliminaries The dynamics

Monotonicity of The Distance

Example

Main result

Theorem Let (x∗, u∗) be a global minimizer satisfying the outward (or inward) pointing condition. Then there exist a BV adjoint vector p : [0, T] → Rn,a finite signed Radon measure µ on [0, T], and measurable vectors ξ, η : [0, T] → Rn, with ξ(t) ≥ 0 for µ-a.e. t and 0 ≤ η(t) ≤ β + γ for a.e. t, satisfying :

A Mayer Problem for A Controlled Sweeping Process Chems Eddine Arroud and Giovanni Colombo Preliminaries The dynamics

Monotonicity of The Distance

Example

Main result

Theorem Let (x∗, u∗) be a global minimizer satisfying the outward (or inward) pointing condition. Then there exist a BV adjoint vector p : [0, T] → Rn,a finite signed Radon measure µ on [0, T], and measurable vectors ξ, η : [0, T] → Rn, with ξ(t) ≥ 0 for µ-a.e. t and 0 ≤ η(t) ≤ β + γ for a.e. t, satisfying :

• (adjoint equation)

A Mayer Problem for A Controlled Sweeping Process Chems Eddine Arroud and Giovanni Colombo Preliminaries The dynamics

Monotonicity of The Distance

Example

Main result

Theorem Let (x∗, u∗) be a global minimizer satisfying the outward (or inward) pointing condition. Then there exist a BV adjoint vector p : [0, T] → Rn,a finite signed Radon measure µ on [0, T], and measurable vectors ξ, η : [0, T] → Rn, with ξ(t) ≥ 0 for µ-a.e. t and 0 ≤ η(t) ≤ β + γ for a.e. t, satisfying :

• (adjoint equation)

for all continuous functions ϕ : [0, T] → Rn −

• [0,T]

ϕ(t), dp = −

• [0,T]

ϕ(t), ∇xd(x∗(t), C(t))ξ(t) dµ −

• [0,T]

ϕ(t), η(t)∇2

xd(x∗(t), C(t))p(t) dt

+

• [0,T]

ϕ(t), ∇xf (x∗(t), u∗(t))p(t) dt, (2)

A Mayer Problem for A Controlled Sweeping Process Chems Eddine Arroud and Giovanni Colombo Preliminaries The dynamics

Monotonicity of The Distance

Example

Theorem

• (transversality condition)

−p(T) = ∇h(x∗(T)),

A Mayer Problem for A Controlled Sweeping Process Chems Eddine Arroud and Giovanni Colombo Preliminaries The dynamics

Monotonicity of The Distance

Example

Theorem

• (transversality condition)

−p(T) = ∇h(x∗(T)),

• (maximality condition)

p(t), ∇uf (x∗(t), u∗(t))u∗(t) = max

u∈U p(t), ∇uf (x∗(t), u∗(t))u

for a.e. t ∈ [0, T]. (3) .

A Mayer Problem for A Controlled Sweeping Process Chems Eddine Arroud and Giovanni Colombo Preliminaries The dynamics

Monotonicity of The Distance

Example

The Cauchy problem (P) has one and only one solution by

• M.Sene, L.Thibault, Regularization of dynamical systems associated with prox-regular moving sets,

JNCA,Vol 15, Number 4 or 5

A Mayer Problem for A Controlled Sweeping Process Chems Eddine Arroud and Giovanni Colombo Preliminaries The dynamics

Monotonicity of The Distance

Example

The Cauchy problem (P) has one and only one solution by

• M.Sene, L.Thibault, Regularization of dynamical systems associated with prox-regular moving sets,

JNCA,Vol 15, Number 4 or 5

and also that for every fixed control u∗ ∈ U ( given a minimizer u∗ ) the sequence x∗

ε of solutions to

• ˙

x(t) = −1

ε(x(t) − projC(t)(x(t))) + f (x(t), u∗(t)),

x(0) = x0 ∈ C(0) , (4)

A Mayer Problem for A Controlled Sweeping Process Chems Eddine Arroud and Giovanni Colombo Preliminaries The dynamics

Monotonicity of The Distance

Example

The Cauchy problem (P) has one and only one solution by

• M.Sene, L.Thibault, Regularization of dynamical systems associated with prox-regular moving sets,

JNCA,Vol 15, Number 4 or 5

and also that for every fixed control u∗ ∈ U ( given a minimizer u∗ ) the sequence x∗

ε of solutions to

• ˙

x(t) = −1

ε(x(t) − projC(t)(x(t))) + f (x(t), u∗(t)),

x(0) = x0 ∈ C(0) , (4) converge weakly in W 1,2([0, T], Rn) to the solution of

• ˙

x(t) ∈ −NC(t)(x(t)) + f (x(t), u∗(t))), x(0) = x0 ∈ C(0) , (5)

A Mayer Problem for A Controlled Sweeping Process Chems Eddine Arroud and Giovanni Colombo Preliminaries The dynamics

Monotonicity of The Distance

Example

Regularised Minimization Problem

A Mayer Problem for A Controlled Sweeping Process Chems Eddine Arroud and Giovanni Colombo Preliminaries The dynamics

Monotonicity of The Distance

Example

Regularised Minimization Problem

Let the problem Pε Minimize J(x, u; u∗) := h(x(T)) + 1 2

T

0 u(t) − u∗(t)2dt

A Mayer Problem for A Controlled Sweeping Process Chems Eddine Arroud and Giovanni Colombo Preliminaries The dynamics

Monotonicity of The Distance

Example

Regularised Minimization Problem

Let the problem Pε Minimize J(x, u; u∗) := h(x(T)) + 1 2

T

0 u(t) − u∗(t)2dt

Subject to ˙ x(t) = −1 ε(x(t) − projC(t)(x(t))) + f (x(t), u(t)), (6) x(0) = x0

• ver u : [0, T] U, u is measurable.

A Mayer Problem for A Controlled Sweeping Process Chems Eddine Arroud and Giovanni Colombo Preliminaries The dynamics

Monotonicity of The Distance

Example

Regularised Minimization Problem

Let the problem Pε Minimize J(x, u; u∗) := h(x(T)) + 1 2

T

0 u(t) − u∗(t)2dt

Subject to ˙ x(t) = −1 ε(x(t) − projC(t)(x(t))) + f (x(t), u(t)), (6) x(0) = x0

• ver u : [0, T] U, u is measurable.
• The corresponding solutions xε of (6) are uniformly Lipschitz,

with Lipschitz constant γ + 2β.

A Mayer Problem for A Controlled Sweeping Process Chems Eddine Arroud and Giovanni Colombo Preliminaries The dynamics

Monotonicity of The Distance

Example

Regularised Minimization Problem

Let the problem Pε Minimize J(x, u; u∗) := h(x(T)) + 1 2

T

0 u(t) − u∗(t)2dt

Subject to ˙ x(t) = −1 ε(x(t) − projC(t)(x(t))) + f (x(t), u(t)), (6) x(0) = x0

• ver u : [0, T] U, u is measurable.
• The corresponding solutions xε of (6) are uniformly Lipschitz,

with Lipschitz constant γ + 2β.

• Let (xε, uε) be solution of the problem (Pε). Then there

exists a sequence εn ↓ 0 such that xεn → x∗ weakly in W 1,2([0, T]; Rn) uεn → u∗ strongly in L2([0, T]; Rm). .

A Mayer Problem for A Controlled Sweeping Process Chems Eddine Arroud and Giovanni Colombo Preliminaries The dynamics

Monotonicity of The Distance

Example

Adjoint equation

The adjoint equation for Pε

• −˙

pε(t) = pε(t)

−1

2ε ∇2 xd2(xε(t), C(t)) + ∇xf (xε(t), uε(t))

• −pε(T)

= ∇h(xε(T)), (7)

A Mayer Problem for A Controlled Sweeping Process Chems Eddine Arroud and Giovanni Colombo Preliminaries The dynamics

Monotonicity of The Distance

Example

Adjoint equation

The adjoint equation for Pε

• −˙

pε(t) = pε(t)

−1

2ε ∇2 xd2(xε(t), C(t)) + ∇xf (xε(t), uε(t))

• −pε(T)

= ∇h(xε(T)), (7) −1 2ε ∇2

xd2(xε(t), C(t)) = −1

ε

d(xε(t), C(t))∇2

xd(xε(t), C(t))

+ ∇xd(xε(t), C(t)) ⊗ ∇xd(xε(t), C(t))

• (8)

A Mayer Problem for A Controlled Sweeping Process Chems Eddine Arroud and Giovanni Colombo Preliminaries The dynamics

Monotonicity of The Distance

Example

Monotonicity of The Distance

∇xd(xε(t), C(t)) ⊗ ∇xd(xε(t), C(t))

A Mayer Problem for A Controlled Sweeping Process Chems Eddine Arroud and Giovanni Colombo Preliminaries The dynamics

Monotonicity of The Distance

Example

Monotonicity of The Distance

∇xd(xε(t), C(t)) ⊗ ∇xd(xε(t), C(t)) I∂ := {t ∈ [0, T] : x∗(t) ∈ ∂C(t)}

A Mayer Problem for A Controlled Sweeping Process Chems Eddine Arroud and Giovanni Colombo Preliminaries The dynamics

Monotonicity of The Distance

Example

Monotonicity of The Distance

∇xd(xε(t), C(t)) ⊗ ∇xd(xε(t), C(t)) I∂ := {t ∈ [0, T] : x∗(t) ∈ ∂C(t)}

A Mayer Problem for A Controlled Sweeping Process Chems Eddine Arroud and Giovanni Colombo Preliminaries The dynamics

Monotonicity of The Distance

Example

So in this work we requiring a strongoutward pointing condition on f in order to treat the discontinuity of second derivatives of the squared distance function at the boundary of C(t).

A Mayer Problem for A Controlled Sweeping Process Chems Eddine Arroud and Giovanni Colombo Preliminaries The dynamics

Monotonicity of The Distance

Example

Problem without the strong condition

We consider the problem of minimizing the cost h(x(T)) at the endpoint of a trajectory x subject to the finite dimensional dynamics ˙ x ∈ −NC(x) + f (x, u), x(0) = x0,

A Mayer Problem for A Controlled Sweeping Process Chems Eddine Arroud and Giovanni Colombo Preliminaries The dynamics

Monotonicity of The Distance

Example

Problem without the strong condition

We consider the problem of minimizing the cost h(x(T)) at the endpoint of a trajectory x subject to the finite dimensional dynamics ˙ x ∈ −NC(x) + f (x, u), x(0) = x0, The regularized problem ˙ x(t) = −1 ε ∇Ψ(x(t)) + f (x(t), u(t)), x(0) = x0, (9) Ψ(x) = 1 3ψ3(x) 1(0,+∞)(ψ(x)).

A Mayer Problem for A Controlled Sweeping Process Chems Eddine Arroud and Giovanni Colombo Preliminaries The dynamics

Monotonicity of The Distance

Example

The adjoint equation −˙ pε(t) =

−1

ε ∇2Ψ(xε(t)) + ∇xf (xε(t), uε(t))

• pε(t)

A Mayer Problem for A Controlled Sweeping Process Chems Eddine Arroud and Giovanni Colombo Preliminaries The dynamics

Monotonicity of The Distance

Example

The adjoint equation −˙ pε(t) =

−1

ε ∇2Ψ(xε(t)) + ∇xf (xε(t), uε(t))

• pε(t)

−

• [0,T]

ϕ(t), dp(t) = −

• [0,T]

ϕ(t), n∗(t)ξ(t) dν(t) −

• [0,T]

ϕ(t), ∇2

xd(x∗(t))p(t)η(t) dt

+

• [0,T]

ϕ(t), ∇xf (x∗(t), u∗(t))p(t) dt,

A Mayer Problem for A Controlled Sweeping Process Chems Eddine Arroud and Giovanni Colombo Preliminaries The dynamics

Monotonicity of The Distance

Example

Transversality condition −pε(T) = ∇h(xε(T))

A Mayer Problem for A Controlled Sweeping Process Chems Eddine Arroud and Giovanni Colombo Preliminaries The dynamics

Monotonicity of The Distance

Example

Transversality condition −pε(T) = ∇h(xε(T)) −p(T) = ∇h(x∗(T))

A Mayer Problem for A Controlled Sweeping Process Chems Eddine Arroud and Giovanni Colombo Preliminaries The dynamics

Monotonicity of The Distance

Example

Transversality condition −pε(T) = ∇h(xε(T)) −p(T) = ∇h(x∗(T)) Maximality condition pε(t), ∇uf (xε(t), uε(t))uε(t) − uε(t) − u∗(t), uε(t) = max

u∈U {pε(t), ∇uf (xε(t), uε(t))u − uε(t) − u∗(t), u}

A Mayer Problem for A Controlled Sweeping Process Chems Eddine Arroud and Giovanni Colombo Preliminaries The dynamics

Monotonicity of The Distance

Example

Transversality condition −pε(T) = ∇h(xε(T)) −p(T) = ∇h(x∗(T)) Maximality condition pε(t), ∇uf (xε(t), uε(t))uε(t) − uε(t) − u∗(t), uε(t) = max

u∈U {pε(t), ∇uf (xε(t), uε(t))u − uε(t) − u∗(t), u}

p(t), ∇uf (x∗(t), u∗(t))u∗(t) = max

u∈U p(t), ∇uf (x∗(t), u∗(t))u

for a.e. t ∈ [0, T].

A Mayer Problem for A Controlled Sweeping Process Chems Eddine Arroud and Giovanni Colombo Preliminaries The dynamics

Monotonicity of The Distance

Example

Difference between the two results

˙ x ∈ −NC(t)(x) + f (x, u), x(0) = x0,

A Mayer Problem for A Controlled Sweeping Process Chems Eddine Arroud and Giovanni Colombo Preliminaries The dynamics

Monotonicity of The Distance

Example

Difference between the two results

˙ x ∈ −NC(t)(x) + f (x, u), x(0) = x0, C(t) is moving , we require the outward pointing condition.

A Mayer Problem for A Controlled Sweeping Process Chems Eddine Arroud and Giovanni Colombo Preliminaries The dynamics

Monotonicity of The Distance

Example

Difference between the two results

˙ x ∈ −NC(t)(x) + f (x, u), x(0) = x0, C(t) is moving , we require the outward pointing condition. ˙ x ∈ −NC(x) + f (x, u), x(0) = x0,

A Mayer Problem for A Controlled Sweeping Process Chems Eddine Arroud and Giovanni Colombo Preliminaries The dynamics

Monotonicity of The Distance

Example

Difference between the two results

˙ x ∈ −NC(t)(x) + f (x, u), x(0) = x0, C(t) is moving , we require the outward pointing condition. ˙ x ∈ −NC(x) + f (x, u), x(0) = x0, C(t) is constant, but we do not require the outward pointing condition.

A Mayer Problem for A Controlled Sweeping Process Chems Eddine Arroud and Giovanni Colombo Preliminaries The dynamics

Monotonicity of The Distance

Example

Example

Example The state space is R2 ∋ (x, y), the constraint C(t) is constant and equals C := {(x, y) : y ≥ 0}, the upper half plane. We wish to minimize x(1) + y(1) subject to

• (˙

x(t), ˙ y(t)) ∈ −NC(x(t), y(t)) + (u1(t), u2(t)) (x(0), y(0)) = (0, y0), y0 ≥ 0, (10) where the controls (u1, u2) belong to [−1, 1] × [−1, −1

2 ] =: U.

A Mayer Problem for A Controlled Sweeping Process Chems Eddine Arroud and Giovanni Colombo Preliminaries The dynamics

Monotonicity of The Distance

Example

Example

A Mayer Problem for A Controlled Sweeping Process Chems Eddine Arroud and Giovanni Colombo Preliminaries The dynamics

Monotonicity of The Distance

Example

Example

˙ px = 0, ˙ py = 0 a.e. on [0, T], px(1) = py(1) = −1

A Mayer Problem for A Controlled Sweeping Process Chems Eddine Arroud and Giovanni Colombo Preliminaries The dynamics

Monotonicity of The Distance

Example

Example

˙ px = 0, ˙ py = 0 a.e. on [0, T], px(1) = py(1) = −1 px is continous at t = 1 and py(1−) + 1 = 1, namely py(1−) = 0.

A Mayer Problem for A Controlled Sweeping Process Chems Eddine Arroud and Giovanni Colombo Preliminaries The dynamics

Monotonicity of The Distance

Example

Example

Example The adjoint vector (px, py) is :

px(t) = −1 for all t ∈ [0, 1] py(t) = 0 for all t ∈ [0, 1[ py(1) = −1.

The maximum condition

(−1, −1), (ux

∗, uy ∗) =

max

|u1|≤1,1≤u2≤ −1

2

(−1, −1), (u1, u2) t = 1 (−1, 0), (ux

∗, uy ∗) =

max

|u1|≤1,1≤u2≤ −1

2

(−1, 0), (u1, u2) 0 ≤ t < 1

which gives ux

∗ = −1 and uy ∗(1) = −1, while no information is

available for uy

∗(t) if 0 ≤ t < 1 (the optimal solution belongs

to intC). If we assume that uy

∗ is constant, then the expected

• ptimal control uy

∗ = −1 is found.

A Mayer Problem for A Controlled Sweeping Process Chems Eddine Arroud and Giovanni Colombo Preliminaries The dynamics

Monotonicity of The Distance

Example

References

• Ch. Arroud and G. Colombo.A Maximum Principle for the controlled Sweeping Process. J of SVVA,

10.1007/s11228-017-0400-4 (2017).

• S. Adly, F. Nacry, and L. Tibault, Preservation of prox-regularity of sets with application to

constrained optimization SIAM J. Optim. 26 (2016), 448-473.

• L. Ambrosio, Halil Mete Soner, Level set approach to mean curvature flow in arbitrary codimension J.

Differential Geom. 43 (1996), 693-737.

• A. Bressan and B. Piccoli, Introduction to the Mathematical Theory of Control (2007).
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