Moreaus Sweeping Process and its control Giovanni Colombo - - PowerPoint PPT Presentation

Moreau’s Sweeping Process and its control

Giovanni Colombo

Universit` a di Padova

June 15, 2016

Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 1 / 35

Joint works with – B. Mordukhovich (Wayne State), Nguyen D. Hoang (Valparaiso, Chile),

• R. Henrion (WIAS)

– Michele Palladino (Penn State) – Chems Eddine Arroud (Jijel, Algeria)

Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 2 / 35

Outline of the Talk

The dynamics of the sweeping process Control problems for the sweeping process A Hamilton-Jacobi characterization of the minimum time function Necessary optimality conditions

Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 3 / 35

The dynamics of the sweeping process

Consider a moving set C(t), depending on the time t ∈ [0, T], and an initial condition x0 ∈ C(0). In several contexts, the modelization of the displacement x(t) of the initial condition x0 subject to the dragging, or sweeping due to the displacement

• f C(t) pops up.

It is natural to think that the point x(t) remains at rest until it is caught by the boundary of C(t) and then its velocity is normal to ∂C(t). It is a kind of one sided movement.

Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 4 / 35

Formally, the sweeping process (processus de rafle) 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 ∈ C. In particular, NC(x) = {0} if x ∈ intC NC(x) = ∅ if x / ∈ C.

Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 5 / 35

Formally, the sweeping process (processus de rafle) 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 ∈ C. In particular, NC(x) = {0} if x ∈ intC NC(x) = ∅ if x / ∈ C. The simplest example is the play operator: ˙ x(t) ∈ −NC+u(t)(x(t)), namely C(t) is a translation.

Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 5 / 35

Classical assumptions are: x ∈ H, a Hilbert space t → C(t) is Lipschitz continuous C(t) is closed and convex

Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 6 / 35

Classical assumptions are: x ∈ H, a Hilbert space t → C(t) is Lipschitz continuous C(t) is closed and convex Classical results are (the inclusion is forward in time): Existence of a Lipschitz solution for the Cauchy problem Uniqueness and continuous dependence from the initial condition

Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 6 / 35

Classical assumptions are: x ∈ H, a Hilbert space t → C(t) is Lipschitz continuous C(t) is closed and convex Classical results are (the inclusion is forward in time): Existence of a Lipschitz solution for the Cauchy problem Uniqueness and continuous dependence from the initial condition Classical methods are: Moreau-Yosida approximation The catching up algorithm (both constructing a Cauchy sequence of approximate solutions). This goes back to J.-J. Moreau (early ’70s).

Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 6 / 35

Classical assumptions are: x ∈ H, a Hilbert space t → C(t) is Lipschitz continuous C(t) is closed and convex Classical results are (the inclusion is forward in time): Existence of a Lipschitz solution for the Cauchy problem Uniqueness and continuous dependence from the initial condition Classical methods are: Moreau-Yosida approximation The catching up algorithm (both constructing a Cauchy sequence of approximate solutions). This goes back to J.-J. Moreau (early ’70s). There are also (too) many generalizations.

Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 6 / 35

The perturbed sweeping process: ˙ x(t) ∈ −NC(t)(x(t)) + f (x(t)), x(0) = x0 ∈ C(0). (1)

Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 7 / 35

The perturbed sweeping process: ˙ x(t) ∈ −NC(t)(x(t)) + f (x(t)), x(0) = x0 ∈ C(0). (1) If f is Lipschitz, essentially the above results and methods are valid.

Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 7 / 35

The perturbed sweeping process: ˙ x(t) ∈ −NC(t)(x(t)) + f (x(t)), x(0) = x0 ∈ C(0). (1) If f is Lipschitz, essentially the above results and methods are valid. The important case where C is constant, namely ˙ x(t) ∈ −NC(x(t)) + f (x(t)), x(0) = x0 ∈ C(0), (2) models a state constrained evolution where the constraint appears actively in the dynamics.

Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 7 / 35

The perturbed sweeping process: ˙ x(t) ∈ −NC(t)(x(t)) + f (x(t)), x(0) = x0 ∈ C(0). (1) If f is Lipschitz, essentially the above results and methods are valid. The important case where C is constant, namely ˙ x(t) ∈ −NC(x(t)) + f (x(t)), x(0) = x0 ∈ C(0), (2) models a state constrained evolution where the constraint appears actively in the dynamics. It is different from weak flow invariance, a.k.a. viability, because the state constraint is built in the dynamics.

Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 7 / 35

The perturbed sweeping process: ˙ x(t) ∈ −NC(t)(x(t)) + f (x(t)), x(0) = x0 ∈ C(0). (1) If f is Lipschitz, essentially the above results and methods are valid. The important case where C is constant, namely ˙ x(t) ∈ −NC(x(t)) + f (x(t)), x(0) = x0 ∈ C(0), (2) models a state constrained evolution where the constraint appears actively in the dynamics. It is different from weak flow invariance, a.k.a. viability, because the state constraint is built in the dynamics. Note that the right hand side of (1), (2) is discontinuous with respect to the state, for two different reasons.

Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 7 / 35

A proof of continuous dependence from the initial condition. Two solutions: ˙ x1(t) ∈ −NC(t)(x1(t)) + f (x1(t)), x1(0) = ξ1 ˙ x2(t) ∈ −NC(t)(x2(t)) + f (x2(t)), x2(0) = ξ2

Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 8 / 35

A proof of continuous dependence from the initial condition. Two solutions: ˙ x1(t) ∈ −NC(t)(x1(t)) + f (x1(t)), x1(0) = ξ1 ˙ x2(t) ∈ −NC(t)(x2(t)) + f (x2(t)), x2(0) = ξ2 By convexity:

• − ˙

x1(t) + f (x1(t))

• ·
• x2(t) − x1(t)
• ≤ 0
• − ˙

x2(t) + f (x2(t))

• ·
• x1(t) − x2(t)
• ≤ 0

Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 8 / 35

A proof of continuous dependence from the initial condition. Two solutions: ˙ x1(t) ∈ −NC(t)(x1(t)) + f (x1(t)), x1(0) = ξ1 ˙ x2(t) ∈ −NC(t)(x2(t)) + f (x2(t)), x2(0) = ξ2 By convexity:

• − ˙

x1(t) + f (x1(t))

• ·
• x2(t) − x1(t)
• ≤ 0
• − ˙

x2(t) + f (x2(t))

• ·
• x1(t) − x2(t)
• ≤ 0

Summing and rearranging:

• ˙

x2(t) − ˙ x1(t)

• ·
• x2(t) − x1(t)
• ≤
• f (x2(t)) − f (x1(t))
• ·
• x2(t) − x1(t)
• ≤ Lx2(t) − x1(t)
• 2

Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 8 / 35

A proof of continuous dependence from the initial condition. Two solutions: ˙ x1(t) ∈ −NC(t)(x1(t)) + f (x1(t)), x1(0) = ξ1 ˙ x2(t) ∈ −NC(t)(x2(t)) + f (x2(t)), x2(0) = ξ2 By convexity:

• − ˙

x1(t) + f (x1(t))

• ·
• x2(t) − x1(t)
• ≤ 0
• − ˙

x2(t) + f (x2(t))

• ·
• x1(t) − x2(t)
• ≤ 0

Summing and rearranging:

• ˙

x2(t) − ˙ x1(t)

• ·
• x2(t) − x1(t)
• ≤
• f (x2(t)) − f (x1(t))
• ·
• x2(t) − x1(t)
• ≤ Lx2(t) − x1(t)
• 2

Conclude by Gronwall’s lemma.

Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 8 / 35

A proof of continuous dependence from the initial condition. Two solutions: ˙ x1(t) ∈ −NC(t)(x1(t)) + f (x1(t)), x1(0) = ξ1 ˙ x2(t) ∈ −NC(t)(x2(t)) + f (x2(t)), x2(0) = ξ2 By convexity:

• − ˙

x1(t) + f (x1(t))

• ·
• x2(t) − x1(t)
• ≤ 0
• − ˙

x2(t) + f (x2(t))

• ·
• x1(t) − x2(t)
• ≤ 0

Summing and rearranging:

• ˙

x2(t) − ˙ x1(t)

• ·
• x2(t) − x1(t)
• ≤
• f (x2(t)) − f (x1(t))
• ·
• x2(t) − x1(t)
• ≤ Lx2(t) − x1(t)
• 2

Conclude by Gronwall’s lemma. Also mildly nonconvex sets are allowed.

Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 8 / 35

Control problems

Given a dynamics,

Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 9 / 35

Control problems

Given a dynamics, it is impossible resisting to the temptation of putting some control.

Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 9 / 35

Control problems

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. The control may appear in the colored items: shape optimization and classical control. There are very few results on the control of the sweeping process.

Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 9 / 35

Control problems

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. The control may appear in the colored items: shape optimization and classical control. There are very few results on the control of the sweeping process. Results not presented today:

Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 9 / 35

Control problems

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. The control may appear in the colored items: shape optimization and classical control. There are very few results on the control of the sweeping process. Results not presented today: Existence of optimal solutions: not difficult, under standard convexity & coercivity assumptions, because of the graph closedness of the normal cone and Attouch’ theorem (=normals to a converging sequence of convex sets do converge).

Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 9 / 35

Control problems

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. The control may appear in the colored items: shape optimization and classical control. There are very few results on the control of the sweeping process. Results not presented today: Existence of optimal solutions: not difficult, under standard convexity & coercivity assumptions, because of the graph closedness of the normal cone and Attouch’ theorem (=normals to a converging sequence of convex sets do converge). Controllability (up to now no deep results: only a sufficient condition for the minimum time function to be finite and continuous in a neighborhood of the target, which is similar to a classical one).

Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 9 / 35

Results presented today: Characterization of the value function through Hamilton-Jacobi inequalities (i.e., it is the unique solution of a boundary value problem for a first order PDE to be interpreted in a suitable weak sense).

Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 10 / 35

Results presented today: Characterization of the value function through Hamilton-Jacobi inequalities (i.e., it is the unique solution of a boundary value problem for a first order PDE to be interpreted in a suitable weak sense). Necessary optimality conditions (i.e., PMP).

Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 10 / 35

Hamilton-Jacobi characterization of the minimum time function

Consider the Optimal Control Problem (P)    Minimize T > 0 such that ˙ x(t) ∈ G(x(t)) a.e. x(0) = x0, x(T) ∈ S Data: G : Rn Rn, Lipschitz, S ⊂ Rn is the target (closed set). Minimal Time Function: T(x0) = inf{T > 0 : ∃ a solution x(.) of (P) s.t. x(0) = x0, x(T) ∈ S}

Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 11 / 35

Dynamic Programming for T(x). Fix x0 ∈ Rn. Equivalent formulations: 1) for every solution x(.), for every t ∈ [0, T(x0)], T(x0) ≤ t + T(x(t))

Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 12 / 35

Dynamic Programming for T(x). Fix x0 ∈ Rn. Equivalent formulations: 1) for every solution x(.), for every t ∈ [0, T(x0)], T(x0) ≤ t + T(x(t)) 2) there exists ¯ x(.) such that, for every t ∈ [0, T(x0)], T(x0) = t + T(¯ x(t)) (i.e., ¯ x(.) is optimal for (P)).

Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 12 / 35

Rephrasing 1) and 2): needing some definitions. K is any closed set, with (proximal) normal cone NK. Denote as: h(x, p) := min

v∈G(x) v · p,

H(x, p) := max

v∈G(x) v · p ≤ 0.

Definition 1. (G, K) is weakly invariant if, for every x0 ∈ K, there exists a solution x such that x(0) = x0, x(t) ∈ K ∀ t ∈ [0, T].

Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 13 / 35

Rephrasing 1) and 2): needing some definitions. K is any closed set, with (proximal) normal cone NK. Denote as: h(x, p) := min

v∈G(x) v · p,

H(x, p) := max

v∈G(x) v · p ≤ 0.

Definition 1. (G, K) is weakly invariant if, for every x0 ∈ K, there exists a solution x such that x(0) = x0, x(t) ∈ K ∀ t ∈ [0, T]. Proposition 1. (G, K) is weakly invariant if and only if h(x, p) ≤ 0 ∀ x ∈ K, ∀ p ∈ NP

K(x).

Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 13 / 35

Rephrasing 1) and 2): needing some definitions. K is any closed set, with (proximal) normal cone NK. Denote as: h(x, p) := min

v∈G(x) v · p,

H(x, p) := max

v∈G(x) v · p ≤ 0.

Definition 1. (G, K) is weakly invariant if, for every x0 ∈ K, there exists a solution x such that x(0) = x0, x(t) ∈ K ∀ t ∈ [0, T]. Proposition 1. (G, K) is weakly invariant if and only if h(x, p) ≤ 0 ∀ x ∈ K, ∀ p ∈ NP

K(x).

Definition 2. (G, K) is strongly invariant if, for every x0 ∈ K and every solution x such that x(0) = x0, we have x(t) ∈ K ∀ t ∈ [0, T].

Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 13 / 35

Rephrasing 1) and 2): needing some definitions. K is any closed set, with (proximal) normal cone NK. Denote as: h(x, p) := min

v∈G(x) v · p,

H(x, p) := max

v∈G(x) v · p ≤ 0.

Definition 1. (G, K) is weakly invariant if, for every x0 ∈ K, there exists a solution x such that x(0) = x0, x(t) ∈ K ∀ t ∈ [0, T]. Proposition 1. (G, K) is weakly invariant if and only if h(x, p) ≤ 0 ∀ x ∈ K, ∀ p ∈ NP

K(x).

Definition 2. (G, K) is strongly invariant if, for every x0 ∈ K and every solution x such that x(0) = x0, we have x(t) ∈ K ∀ t ∈ [0, T]. Proposition 2. (G, K) is strongly invariant if and only if H(x, p) ≤ 0 ∀ x ∈ K, ∀ p ∈ NP

K(x).

Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 13 / 35

Rephrasing 1) and 2): needing some definitions. K is any closed set, with (proximal) normal cone NK. Denote as: h(x, p) := min

v∈G(x) v · p,

H(x, p) := max

v∈G(x) v · p ≤ 0.

Definition 1. (G, K) is weakly invariant if, for every x0 ∈ K, there exists a solution x such that x(0) = x0, x(t) ∈ K ∀ t ∈ [0, T]. Proposition 1. (G, K) is weakly invariant if and only if h(x, p) ≤ 0 ∀ x ∈ K, ∀ p ∈ NP

K(x).

Definition 2. (G, K) is strongly invariant if, for every x0 ∈ K and every solution x such that x(0) = x0, we have x(t) ∈ K ∀ t ∈ [0, T]. Proposition 2. (G, K) is strongly invariant if and only if H(x, p) ≤ 0 ∀ x ∈ K, ∀ p ∈ NP

K(x).

Remark: Lipschitz cont. of G is crucial for Prop. 2, but not for Prop. 1.

Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 13 / 35

Remark: Assume T(.) is continuous. Denote as: epiT := {(x, α) : T(x) ≤ α} (closed set) hypoT := {(x, α) : T(x) ≥ α} (closed set)

Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 14 / 35

Remark: Assume T(.) is continuous. Denote as: epiT := {(x, α) : T(x) ≤ α} (closed set) hypoT := {(x, α) : T(x) ≥ α} (closed set) Proposition: Condition 2) is equivalent to weak invariance for (G × {−1}, epiT). (h(x, p) ≤ −1).

Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 14 / 35

Remark: Assume T(.) is continuous. Denote as: epiT := {(x, α) : T(x) ≤ α} (closed set) hypoT := {(x, α) : T(x) ≥ α} (closed set) Proposition: Condition 2) is equivalent to weak invariance for (G × {−1}, epiT). (h(x, p) ≤ −1). Proposition: Condition 1) is equivalent to strong invariance for (G × {−1}, hypoT). (h(x, p) ≥ −1).

Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 14 / 35

Comparison Lemma. Suppose θ : Rn → R ∪ {∞} continuous, θ(x) = 0 if x ∈ S (target) and θ(x) > 0 otherwise . Then: If (G × {−1}, epi θ) is weakly invariant, then θ(x) ≤ T(x) ∀ x. If (G × {−1}, hypo θ) is strongly invariant, then θ(x) ≥ T(x) ∀ x.

Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 15 / 35

Comparison Lemma. Suppose θ : Rn → R ∪ {∞} continuous, θ(x) = 0 if x ∈ S (target) and θ(x) > 0 otherwise . Then: If (G × {−1}, epi θ) is weakly invariant, then θ(x) ≤ T(x) ∀ x. If (G × {−1}, hypo θ) is strongly invariant, then θ(x) ≥ T(x) ∀ x. Theorem (Wolenski-Zhuang, ’99: H-J characterization of T). There exists a unique θ : Rn → R+ ∪ {∞} such that (HJ): For each x / ∈ S and p ∈ ∂Pθ(x), h(x, p) = −1. (BC): for each x ∈ S satisfies θ(x) = 0 and h(x, p) ≥ −1, ∀ p ∈ ∂Pθ(x). (CV): T(x) = θ(x) for all x ∈ Rn.

Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 15 / 35

We want to do the same for the controlled sweeping process (now the moving set is given, i.e., not controlled).

Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 16 / 35

We want to do the same for the controlled sweeping process (now the moving set is given, i.e., not controlled). The main difficulty is the lack of Lipschitz continuity of the right-hand side

• f the dynamics. Of course dissipativity helps, but one has to figure out

which is the right Hamiltonian.

Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 16 / 35

We want to do the same for the controlled sweeping process (now the moving set is given, i.e., not controlled). The main difficulty is the lack of Lipschitz continuity of the right-hand side

• f the dynamics. Of course dissipativity helps, but one has to figure out

which is the right Hamiltonian. (SP)    Minimize T over solutions of ˙ x(t) ∈ −NC(t)(x(t)) + G(x(t)) a.e. x(t0) = x0 ∈ C(t0), x(T) ∈ S. Assumptions: the moving set C(.) is Lipschitz, closed and convex valued; G(.) is Lipschitz and compact and convex valued. Compatibility Condition: ∃ ¯ t > 0 such that C(¯ t) ∩ S = ∅.

Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 16 / 35

We want to do the same for the controlled sweeping process (now the moving set is given, i.e., not controlled). The main difficulty is the lack of Lipschitz continuity of the right-hand side

• f the dynamics. Of course dissipativity helps, but one has to figure out

which is the right Hamiltonian. (SP)    Minimize T over solutions of ˙ x(t) ∈ −NC(t)(x(t)) + G(x(t)) a.e. x(t0) = x0 ∈ C(t0), x(T) ∈ S. Assumptions: the moving set C(.) is Lipschitz, closed and convex valued; G(.) is Lipschitz and compact and convex valued. Compatibility Condition: ∃ ¯ t > 0 such that C(¯ t) ∩ S = ∅. Method: characterization through invariance of epi/hypograph.

Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 16 / 35

Characterization of strong invariance for the perturbed sweeping process (C., Palladino; 2015)

Theorem: Assume (HG), (HC) and take K ⊂ Gr C closed. ({1} × {−NC + G}, K) is strongly invariant ⇐ ⇒ for every (τ, x) ∈ K min

v∈{0}×{−NC(τ)(x)∩(LC +MG )B} v · p +

max

v∈{1}×{G(x)} v · p ≤ 0

for every p ∈ NP

K(τ, x).

Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 17 / 35

Characterization of strong invariance for the perturbed sweeping process (C., Palladino; 2015)

Theorem: Assume (HG), (HC) and take K ⊂ Gr C closed. ({1} × {−NC + G}, K) is strongly invariant ⇐ ⇒ for every (τ, x) ∈ K min

v∈{0}×{−NC(τ)(x)∩(LC +MG )B} v · p +

max

v∈{1}×{G(x)} v · p ≤ 0

for every p ∈ NP

K(τ, x).

Remark: Monotonicity of the normal cone plays a crucial role.

Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 17 / 35

HJ inequalities for SP (C.-Palladino; ’15)

Define: H−(τ, x, λ, p) := min

{0}×{−NC(τ)(x)∩(LC +MG )B}×{0} v·p+

min

v∈{1}×{G(x)}×{−1} v·p,

H+(τ, x, λ, p) := min

{0}×{−NC(τ)(x)∩(LC +MG )B}×{0} v·p+

max

v∈{1}×{G(x)}×{−1} v·p,

Theorem: Assume T(., .) continuous. Then T(., .) is the unique function satisfying: T(t, x) > 0 ∀ (t, x) ∈ Gr C for which x / ∈ S, T(t, x) = 0 ∀ (t, x) ∈ Gr C for which x ∈ S, H−(t, x, T(t, x), p) ≤ 0 ∀ (t, x) ∈ GrC, x / ∈ S, ∀ p ∈ NP

epiT(t, x, T(t, x)),

H+(t, x, T(t, x), p) ≤ 0 ∀ (t, x) ∈ GrC, ∀ p ∈ NP

hypoT(t, x, T(t, x)).

Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 18 / 35

HJ inequalities for SP (C.-Palladino; ’15)

Define: H−(τ, x, λ, p) := min

{0}×{−NC(τ)(x)∩(LC +MG )B}×{0} v·p+

min

v∈{1}×{G(x)}×{−1} v·p,

H+(τ, x, λ, p) := min

{0}×{−NC(τ)(x)∩(LC +MG )B}×{0} v·p+

max

v∈{1}×{G(x)}×{−1} v·p,

Theorem: Assume T(., .) continuous. Then T(., .) is the unique function satisfying: T(t, x) > 0 ∀ (t, x) ∈ Gr C for which x / ∈ S, T(t, x) = 0 ∀ (t, x) ∈ Gr C for which x ∈ S, H−(t, x, T(t, x), p) ≤ 0 ∀ (t, x) ∈ GrC, x / ∈ S, ∀ p ∈ NP

epiT(t, x, T(t, x)),

H+(t, x, T(t, x), p) ≤ 0 ∀ (t, x) ∈ GrC, ∀ p ∈ NP

hypoT(t, x, T(t, x)).

• Remarks. 1) If x ∈ int C(t) and x /

∈ S, then the classical HJ equality holds.

Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 18 / 35

HJ inequalities for SP (C.-Palladino; ’15)

Define: H−(τ, x, λ, p) := min

{0}×{−NC(τ)(x)∩(LC +MG )B}×{0} v·p+

min

v∈{1}×{G(x)}×{−1} v·p,

H+(τ, x, λ, p) := min

{0}×{−NC(τ)(x)∩(LC +MG )B}×{0} v·p+

max

v∈{1}×{G(x)}×{−1} v·p,

Theorem: Assume T(., .) continuous. Then T(., .) is the unique function satisfying: T(t, x) > 0 ∀ (t, x) ∈ Gr C for which x / ∈ S, T(t, x) = 0 ∀ (t, x) ∈ Gr C for which x ∈ S, H−(t, x, T(t, x), p) ≤ 0 ∀ (t, x) ∈ GrC, x / ∈ S, ∀ p ∈ NP

epiT(t, x, T(t, x)),

H+(t, x, T(t, x), p) ≤ 0 ∀ (t, x) ∈ GrC, ∀ p ∈ NP

hypoT(t, x, T(t, x)).

• Remarks. 1) If x ∈ int C(t) and x /

∈ S, then the classical HJ equality

• holds. 2) The structure of the Hamiltonians: observe the min in the

normal part.

disegna l’esempio Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 18 / 35

Necessary optimality conditions 1: shape optimization (C.,Hoang,Henrion,Mordukhovich)

Minimize a cost (final + integral, finite horizon) over trajectories of ˙ x(t) ∈ −NC(t)(x(t)), x(0) = x0 ∈ C(0), where C is assumed to be a polyhedron: C(t) :=

• x ∈ Rn

ui(t), x ≤ bi(t), i = 1, . . . , m

• with ui(t) = 1 for all t ∈ [0, T], i = 1, . . . , m,

ui, bi are the controls: they are Lipschitz and satisfy a constraint qualification. This seems to be a completely new problem.

Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 19 / 35

The cost is J[x, u, b]: = ϕ

• x(T)
• +

T

• ℓ1(t, z) + ℓ2(˙

x) + ℓ3(t, ˙ u, ˙ b)

• dt,

where u = (u1, . . . , um), b = (b1, . . . , bm) and the Lagrangians satisfy reasonable assumptions. Why a polyhedron?

Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 20 / 35

The cost is J[x, u, b]: = ϕ

• x(T)
• +

T

• ℓ1(t, z) + ℓ2(˙

x) + ℓ3(t, ˙ u, ˙ b)

• dt,

where u = (u1, . . . , um), b = (b1, . . . , bm) and the Lagrangians satisfy reasonable assumptions. Why a polyhedron? Because, as is usual in necessary conditions, one expects to use the generalized gradient of the dynamics with respect to the state, namely the generalized gradient of the normal cone, i.e., a second order object. Some explicit computations for that are available in the literature of nonsmooth analysis.

Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 20 / 35

Let ¯ z := (¯ x, ¯ u, ¯ b) be a local W 1,1 minimizer such that ¯ u, ¯ b are piecewise W 2,∞

Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 21 / 35

Let ¯ z := (¯ x, ¯ u, ¯ b) be a local W 1,1 minimizer such that ¯ u, ¯ b are piecewise W 2,∞ (a strong but not a crazy assumption, if we remember that u and b must be Lipschitz).

Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 21 / 35

Let ¯ z := (¯ x, ¯ u, ¯ b) be a local W 1,1 minimizer such that ¯ u, ¯ b are piecewise W 2,∞ (a strong but not a crazy assumption, if we remember that u and b must be Lipschitz). We use a method, designed by Mordukhovich, based on (time) discrete approximations: – we construct approximations of ¯ z and among them we minimize a discrete cost which involves a penalization of the squared distance from ¯ z;

Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 21 / 35

Let ¯ z := (¯ x, ¯ u, ¯ b) be a local W 1,1 minimizer such that ¯ u, ¯ b are piecewise W 2,∞ (a strong but not a crazy assumption, if we remember that u and b must be Lipschitz). We use a method, designed by Mordukhovich, based on (time) discrete approximations: – we construct approximations of ¯ z and among them we minimize a discrete cost which involves a penalization of the squared distance from ¯ z; – we prove that a sequence of minimizers of the discrete problems converges strongly to ¯ z;

Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 21 / 35

Let ¯ z := (¯ x, ¯ u, ¯ b) be a local W 1,1 minimizer such that ¯ u, ¯ b are piecewise W 2,∞ (a strong but not a crazy assumption, if we remember that u and b must be Lipschitz). We use a method, designed by Mordukhovich, based on (time) discrete approximations: – we construct approximations of ¯ z and among them we minimize a discrete cost which involves a penalization of the squared distance from ¯ z; – we prove that a sequence of minimizers of the discrete problems converges strongly to ¯ z; – we write down necessary conditions for each discrete minimizer;

Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 21 / 35

Let ¯ z := (¯ x, ¯ u, ¯ b) be a local W 1,1 minimizer such that ¯ u, ¯ b are piecewise W 2,∞ (a strong but not a crazy assumption, if we remember that u and b must be Lipschitz). We use a method, designed by Mordukhovich, based on (time) discrete approximations: – we construct approximations of ¯ z and among them we minimize a discrete cost which involves a penalization of the squared distance from ¯ z; – we prove that a sequence of minimizers of the discrete problems converges strongly to ¯ z; – we write down necessary conditions for each discrete minimizer; – we establish some a priori estimates, pass to the limit along such necessary conditions and obtain a set of necessary conditions for ¯ z.

Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 21 / 35

Let ¯ z := (¯ x, ¯ u, ¯ b) be a local W 1,1 minimizer such that ¯ u, ¯ b are piecewise W 2,∞ (a strong but not a crazy assumption, if we remember that u and b must be Lipschitz). We use a method, designed by Mordukhovich, based on (time) discrete approximations: – we construct approximations of ¯ z and among them we minimize a discrete cost which involves a penalization of the squared distance from ¯ z; – we prove that a sequence of minimizers of the discrete problems converges strongly to ¯ z; – we write down necessary conditions for each discrete minimizer; – we establish some a priori estimates, pass to the limit along such necessary conditions and obtain a set of necessary conditions for ¯ z. (technical)

Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 21 / 35

Necessary conditions

• Theorem. There exist a multiplier λ ≥ 0, an absolutely continuous

adjoint arc p(·), signed vector measures γ and ξ, as well as functions

• w(·), v(·)
• ∈ L2 with
• w(t), v(t)
• ∈ co ∂ℓ
• t, ¯

z(t), ˙ ¯ z(t)

• for a.e. t ∈ [0, T] such that the

following conditions are satisfied:

• The primal-dual dynamic relationships:
• ¯

ui(t), ¯ x(t)

• < ¯

bi(t) = ⇒ ηi(t) = 0 for a.e. t ∈ [0, T], i = 1, . . . , m, ηi(t) > 0 = ⇒

• λvx(t)−qx(t), ¯

ui(t)

• = 0 for a.e. t ∈ [0, T], i = 1, . . . , m,

˙ p(t) = λw(t) +

• 0,
• − η(t), λvx(t) − qx(t)
• , 0
• for a.e. t ∈ [0, T],

qu(t) = λ ∂ ∂ ˙ u ℓ2 ˙ ¯ u(t), ˙ ¯ b(t)

• , and qb(t) = λ ∂

∂ ˙ b ℓ2 ˙ ¯ u(t), ˙ ¯ b(t)

• for a.e. t,

where η(·) = (η1(·), . . . , ηm(·)) with nonnegative components is a uniquely defined vector function determined by the representation ˙ ¯ x(t) = −

m

• i=1

ηi(t)¯ ui(t) for a.e. t ∈ [0, T],

Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 22 / 35

and where q is a function of bounded variation on [0, T] with its left-continuous representative given, for all t ∈ [0, T] except at most a countable subset, by q(t) = p(t) −

• [t,T]

m

• i=1

¯ ui(s)dγi(s),

• [t,T]

¯ x(s)), dγ(s) + 2

• [t,T]

¯ u(s), dξ(s), −

• [t,T]

dγ(s)

• .
• The transversality conditions at the right endpoint:

−px(T) ∈ λ∂ϕ

• ¯

x(T)

• +

m

• i=1

pb

i (T)¯

ui(T), pu

i (T)+pb i (T)¯

x(T) =

• pu

i (T)+pb i (T)¯

x(T), ¯ ui(T)

• ¯

ui(T), i = 1, . . . , m, pb(T) ∈ Rm

+ and

• ¯

ui(T), ¯ x(T)

• < ¯

bi(T) = ⇒ pb

i (T) = 0,

i = 1, . . . , m.

Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 23 / 35

• The measure nonatomicity condition:

If t ∈ [0, T) and ¯ ui(t), ¯ x(t) < ¯ bi(t) for all i = 1, . . . , m, then there is a neighborhood Vt of t in [0, T] such that γ(V ) = 0 for any Borel subset V of Vt.

• Nontriviality condition: We have

λ + q(0) + p(T) = 0.

Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 24 / 35

• The measure nonatomicity condition:

If t ∈ [0, T) and ¯ ui(t), ¯ x(t) < ¯ bi(t) for all i = 1, . . . , m, then there is a neighborhood Vt of t in [0, T] such that γ(V ) = 0 for any Borel subset V of Vt.

• Nontriviality condition: We have

λ + q(0) + p(T) = 0. What is missing?

Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 24 / 35

• The measure nonatomicity condition:

If t ∈ [0, T) and ¯ ui(t), ¯ x(t) < ¯ bi(t) for all i = 1, . . . , m, then there is a neighborhood Vt of t in [0, T] such that γ(V ) = 0 for any Borel subset V of Vt.

• Nontriviality condition: We have

λ + q(0) + p(T) = 0. What is missing? The maximization of the Hamiltonian!

Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 24 / 35

• The measure nonatomicity condition:

If t ∈ [0, T) and ¯ ui(t), ¯ x(t) < ¯ bi(t) for all i = 1, . . . , m, then there is a neighborhood Vt of t in [0, T] such that γ(V ) = 0 for any Borel subset V of Vt.

• Nontriviality condition: We have

λ + q(0) + p(T) = 0. What is missing? The maximization of the Hamiltonian! However our necessary conditions can be applied, and do give information, to some academic examples.

Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 24 / 35

• The measure nonatomicity condition:

If t ∈ [0, T) and ¯ ui(t), ¯ x(t) < ¯ bi(t) for all i = 1, . . . , m, then there is a neighborhood Vt of t in [0, T] such that γ(V ) = 0 for any Borel subset V of Vt.

• Nontriviality condition: We have

λ + q(0) + p(T) = 0. What is missing? The maximization of the Hamiltonian! However our necessary conditions can be applied, and do give information, to some academic examples. Generalizations (Tan Cao & Mordukhovich, 2016): allow a controlled dynamics like ˙ x ∈ −NC(x) + f (x, u), with applications to a model of movement of population in a crowded environment.

Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 24 / 35

A maximum principle for the controlled sweeping process (Arroud-C., 2016)

Minimize h(x(T)) subject to ˙ x(t) ∈ −NC(t)(x(t)) + f (x(t), u(t))), x(0) = x0 ∈ C(0) , with respect to u : [0, T] → U, u measurable (now C is given, smooth).

Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 25 / 35

A maximum principle for the controlled sweeping process (Arroud-C., 2016)

Minimize h(x(T)) subject to ˙ x(t) ∈ −NC(t)(x(t)) + f (x(t), u(t))), x(0) = x0 ∈ C(0) , with respect to u : [0, T] → U, u measurable (now C is given, smooth). What should one expect?

Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 25 / 35

A maximum principle for the controlled sweeping process (Arroud-C., 2016)

Minimize h(x(T)) subject to ˙ x(t) ∈ −NC(t)(x(t)) + f (x(t), u(t))), x(0) = x0 ∈ C(0) , with respect to u : [0, T] → U, u measurable (now C is given, smooth). What should one expect? An example: in R2, minimize x(1) + y(1) subject to (˙ x, ˙ y) ∈ −NC(x, y) + (ux, uy), |ux|, |uy| ≤ 1 where C = {(x, y) : y ≥ 0}.

disegna Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 25 / 35

A maximum principle for the controlled sweeping process (Arroud-C., 2016)

Minimize h(x(T)) subject to ˙ x(t) ∈ −NC(t)(x(t)) + f (x(t), u(t))), x(0) = x0 ∈ C(0) , with respect to u : [0, T] → U, u measurable (now C is given, smooth). What should one expect? An example: in R2, minimize x(1) + y(1) subject to (˙ x, ˙ y) ∈ −NC(x, y) + (ux, uy), |ux|, |uy| ≤ 1 where C = {(x, y) : y ≥ 0}.

disegna

Some degeneracy of the y-component of the adjoint vector is to be expected (is not a pathology).

Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 25 / 35

Method: Moreau-Yosida approximation (inspired by Brokate & Kreiˇ c´ ı, DCDS-B 2013). Given ε > 0 and a global minimizer (¯ x, ¯ u), one considers the problem of minimizing h(x(T)) + 1 2 T |u − ¯ u|2 dt subject to ˙ xε(t) = −1 ε

• xε(t) − projC(t)(xε(t))
• + f (xε(t), u),

x(0) = x0

Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 26 / 35

Method: Moreau-Yosida approximation (inspired by Brokate & Kreiˇ c´ ı, DCDS-B 2013). Given ε > 0 and a global minimizer (¯ x, ¯ u), one considers the problem of minimizing h(x(T)) + 1 2 T |u − ¯ u|2 dt subject to ˙ xε(t) = −1 ε

• xε(t) − projC(t)(xε(t))
• + f (xε(t), u),

x(0) = x0 Existence and uniqueness of a solution and estimate d(xε(t), C(t)) ≤ const ε for all t ∈ [0, T]. Strong L2 convergence of a sequence of minimizers uε to ¯ u.

Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 26 / 35

This is a classical control problem: – write down necessary conditions. – try to pass to the limit.

Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 27 / 35

This is a classical control problem: – write down necessary conditions. – try to pass to the limit.

• Remark. xε(t) − projC(t)(xε(t)) = ∇x 1

2d2(xε(t), C(t))

(∇xd2(., C(t)) is Lipschitz). Then necessary conditions read as

• −˙

pε(t) = −1

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

• pε(t), t ∈ [0, T]

−pε(T) = ∇h(xε(T)) + PMP.

Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 27 / 35

Computation (valid for all x / ∈ ∂C(t)) ∇2

xd2(x, C(t)) = d(x, C(t))∇2 xd(x, C(t)) + ∇xd(x, C(t)) ⊗ ∇xd(x, C(t))

continuous discontinuous at ∂C(t) 1 ε∇2

xd2(x, C(t)) =

= 1 εd(x, C(t))∇2

xd(x, C(t)) + 1

ε∇xd(x, C(t)) ⊗ ∇xd(x, C(t)) bounded ??

Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 28 / 35

Computation (valid for all x / ∈ ∂C(t)) ∇2

xd2(x, C(t)) = d(x, C(t))∇2 xd(x, C(t)) + ∇xd(x, C(t)) ⊗ ∇xd(x, C(t))

continuous discontinuous at ∂C(t) 1 ε∇2

xd2(x, C(t)) =

= 1 εd(x, C(t))∇2

xd(x, C(t)) + 1

ε∇xd(x, C(t)) ⊗ ∇xd(x, C(t)) bounded ?? full bottle vs. drunk mathematician

Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 28 / 35

Computation (valid for all x / ∈ ∂C(t)) ∇2

xd2(x, C(t)) = d(x, C(t))∇2 xd(x, C(t)) + ∇xd(x, C(t)) ⊗ ∇xd(x, C(t))

continuous discontinuous at ∂C(t) 1 ε∇2

xd2(x, C(t)) =

= 1 εd(x, C(t))∇2

xd(x, C(t)) + 1

ε∇xd(x, C(t)) ⊗ ∇xd(x, C(t)) bounded ?? full bottle vs. drunk mathematician Brokate & Kreiˇ c´ ı smooth out the distance.

(disegna) This simplifies the

estimate of the red part, while complicates the estimate of the blue part: they need to assume uniform strict convexity of C.

Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 28 / 35

We avoid strict convexity of C(t) by adopting another strategy: an

• utward/inward pointing condition, which transforms the state

discontinuity into a time discontinuity.

(disegna)

This way we can get an a priori estimate also on the red part.

Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 29 / 35

We avoid strict convexity of C(t) by adopting another strategy: an

• utward/inward pointing condition, which transforms the state

discontinuity into a time discontinuity.

(disegna)

This way we can get an a priori estimate also on the red part. The difficulty (namely the discontinuity of ∇xd(., C(t)) at points in ∂C(t)) is brutalized.

Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 29 / 35

Theorem (Arroud-C). Let (x∗, u∗) be a global minimizer satisfying the

• utward (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 the following properties:

• (adjoint equation)

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

• [0,T]

ϕ(t), dp(t) = −

• [0,T]

ϕ(t), ∇xd(x∗(t), C(t))ξ(t) dµ(t) −

• [0,T]

ϕ(t), ∇2

xd(x∗(t), C(t))p(t)η(t) dt

+

• [0,T]

ϕ(t), ∇xf (x∗(t), u∗(t))p(t) dt,

• (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.

Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 30 / 35

When ¯ x(t) ∈ int C(t), the usual adjoint equation holds true.

Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 31 / 35

When ¯ x(t) ∈ int C(t), the usual adjoint equation holds true. Under the outward pointing condition, the optimal trajectory is forced to slide on the boundary if hits it. The adjoint vector is absolutely continuous in the open interval where ¯ x(t) ∈ ∂C(t). The measure µ may admit a Dirac mass only at the final time, as it happens in the example.

Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 31 / 35

Minimize x(1) + y(1) subject to (˙ x, ˙ y) ∈ −NC(x, y) + (ux, uy), |ux|, |uy| ≤ 1 where C = {(x, y) : y ≥ 0}. The adjoint vector (px, py) is absolutely continuous on (0, 1), ˙ px = 0, ˙ py = 0 a.e. on [0, T], px(1) = py(1) = −1, px is continuous at t = 1 and py(1−) + 1 = 1, namely py(1−) = 0. Thus 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 ⇒ µ = −δ1. The maximum condition reads as (−1, 0), (¯ ux, ¯ uy) = max

|ux|,|uy|≤1(−1, 0), (ux, uy)

for 0 ≤ t < 1, i.e., −¯ ux = max

|ux|≤1{−ux} ⇒ ¯

ux ≡ 1, no information on ¯ uy.

Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 32 / 35

Some information on ¯ uy in the interval where the optimal solution belongs to int C are lost. Work in progress (with Michele Palladino) to find another method to

• btain necessary optimality conditions which allow a more general behavior
• f minimizers.

Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 33 / 35

References

Colombo, Palladino The Minimum Time Function for the Controlled Moreau’s Sweeping Process (SICON, in print) Colombo, Hoang, Henrion, Mordukhovich Optimal control of the sweeping process: the polyhedral case

• J. Differential Eqs. 260 (2016), 3397-3447.

Colombo, Hoang, Henrion, Mordukhovich Discrete approximations of a controlled sweeping process, Set-Valued Variat. Anal. 23 (2015), 69-86. Arroud, Colombo A maximum principle for the controlled sweeping process preprint (2016).

Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 34 / 35

THANK YOU FOR YOUR ATTENTION.

Giovanni Colombo (Universit` a di Padova) Moreau’s Sweeping Process and its control June 15, 2016 35 / 35