Journal of Convex Analysis Volume 2 (1995), No.1/2, 19–40

Partial Differential Inclusions Governing Feedback Controls

Jean-Pierre Aubin, Hélène Frankowska

CEREMADE, Université de Paris-Dauphine, 75775 Paris Cedex 16, France.

Received 10 June 1994 Revised manuscript received 7 December 1994

Dedicated to R. T. Rockafellar on his 60th Birthday

We derive partial differential inclusions of hyperbolic type solutions of which are feedbacks governing viable (controlled invariant) solutions of a control system. We also show that the tracking property leads to such partial differential inclusions. We state a variational principle and an existence theorem of a (single-valued contingent) solution to such an inclusion.

1. Introduction

Let X, Y, Z denote finite dimensional vector-spaces. We studied in [14] the existence of dynamical closed-loop controls regulating state-control solutions of a control system:

\{ \begin{matrix}x′(t) & = f(x(t), u(t)) \\ u(t) & ∈ U(x(t)) \end{matrix}  (1.1)

where U : X ∼>Y is a closed set-valued map and f : Graph(U) ↦ X a continuous (single-valued) map with linear growth (in the sense that ∥f(x, u)∥ ≤ c(∥x∥ + ∥u∥ + 1)). Consider a nonnegative continuous function φ : Graph(U) ↦ IR_{+} with linear growth and set K := Dom(U). We address in this paper the problem of finding feedback (or closed-loop) controls r : K ↦ Y satisfying the constraint

∀ x ∈ K, r(x) ∈ U(x)

and the regulation property: for any x_{0} ∈ K, there exists a solution to the differential equation

x′(t) = f(x(t), r(x(t))) & x(0) = x_{0}

such that u(t) := r(x(t)) ∈ U(x(t)) is absolutely continuous and fulfils the growth condition

∥u′(t)∥ ≤ φ(x(t), u(t))

ISSN 0944-6532 / $ 2.50 ◯^{c} Heldermann Verlag

20 J-P. Aubin, H. Frankowska / Partial differential inclusions governing feedback controls

We observe that the graphs of such feedback controls are viability domains (whose definition is recalled in the appendix) of the system of differential inclusions

\{ \begin{matrix}x′(t) & = f(x(t), u(t)) \\ u′(t) & ∈ φ(x(t), u(t))B \end{matrix}  (1.2)

contained in the graph of U. By the Viability Theorem such feedback controls are solutions to the following contingent differential inclusion

∀ x ∈ K, 0 ∈ Dr(x)(f(x, r(x))) − φ(x, r(x))B

satisfying the constraints

∀ x ∈ K, r(x) ∈ U(x)

where Dr(x) is the contingent derivative^{1} of r at x. We shall study this partial differential inclusion, provide a variational principle and an existence theorem. We observe that the existence of a dynamical closed loop is a particular case of the tracking problem, which is studied under several names in many fields, and specially, arises in engineering (see for instance [18]). Indeed, consider two set-valued maps F : X × Y ∼ >X, G : X × Y ∼ >Y and the system of differential inclusions

\{ \begin{matrix}x′(t) & ∈ F (x(t), y(t)) \\ y′(t) & ∈ G(x(t), y(t)) \end{matrix}

We would like to characterize a set-valued map H : X ∼>Y, regarded as an observation map satisfying the following tracking property: for every x_{0} ∈ Dom(H) and every y_{0} ∈ H(x_{0}), there exists a solution (x(·), y(·)) to this system starting at (x_{0}, y_{0}) and satisfying

∀ t ≥ 0, y(t) ∈ H(x(t))

The answer to this question is again a solution to a viability problem, since we actually look for (x(·), y(·)) which remains viable in the graph of H. By the Viability Theorem the tracking property is equivalent to the fact that H is a solution to the contingent differential inclusion

∀ (x, y) ∈ Graph(H), 0 ∈ DH(x, y)(F (x, y)) − G(x, y)

When F and G are single-valued maps f and g and H is a differentiable single-valued map h, the contingent differential inclusion boils down to the quasi-linear hyperbolic system of

1 The contingent derivative DH(x, y) of a set-valued map H : X ∼ >Y at (x, y) ∈ Graph(H)

is defined by

Graph(DH(x, y)) := T_{Graph(H)}(x, y)

where T_{Graph(H)}(x, y) denotes the contingent cone to Graph(H) at (x, y) (whose definition is recalled in the appendix). When H = h is single-valued, we set Dh(x) := Dh(x, h(x)). See [10, Chapter 5] for more details on differential calculus of set-valued maps. first-order partial differential equations^{2}:

∀ j = 1, …, m, ∑_{i=1}^{n} \frac{∂h_{j}}{∂x_{i}} f_{i}(x, h(x)) − g_{j}(x, h(x)) = 0

Knowing F and G, we have to find observation maps H satisfying the tracking property, i.e., to solve the above contingent differential inclusion. Furthermore, we can address other questions such as : Find single-valued solutions h to the contingent differential inclusion which then becomes

∀ x ∈ K, 0 ∈ Dh(x)(F (x, h(x))) − G(x, h(x))  (1.3)

In this case, the tracking property states that there exists a solution to the “reduced” differential inclusion

x′(t) ∈ F (x(t), h(x(t)))

so that (x(·), y(·) := h(x(·))) is a solution to the initial system of differential inclusions starting at (x_{0}, h(x_{0})). Knowing h allows to divide the system by half, so to speak. Let us mention right now that looking for “weak” solutions to this contingent differential inclusion in Sobolev spaces or other spaces of distributions does not help since we require solutions h to be defined through their graph, and thus, solutions which are defined everywhere. It may seem strange to accept set-valued maps as solutions to an hyperbolic system of partial differential inclusions. But this may offer a way to describe shocks by the setvalued character of the solution. Derivatives in the sense of distributions do not provide the unique way to describe weak or generalized solutions. Contingent derivatives offer another possibility to weaken the required properties of a derivative, losing the linear character of the differential operator, but allowing a pointwise definition. It may be useful for tackling hyperbolic problems. This has been already noticed in [16, 17, 24, 25] for conservation laws. Indeed, let us consider a single-valued map f : X ↦ Y and its differential quotients

∇_{h}f(x)(v) := \frac{f(x + hv) − f(x)}{h}. If f is Gâteaux differentiable, then these differential

quotients converge for the pointwise convergence topology (when h → 0). This strong requirement can be weakened in (at least) two ways, each way sacrificing different groups of properties of the usual derivatives.

• The distributional derivative is the limit of the difference-quotients x ↦ ∇_{h}f(x)(v) :=

\frac{f(x + hv) − f(x)}{h} (when h → 0) in the space of distributions, and the limit

is a vectorial distribution D_{v}f ∈ D′(X; Y) (i.e. no longer necessarily a singlevalued function). Furthermore, one can define differential quotients of any vectorial distribution T ∈ D′(X; Y) and the derivative of a distribution as their limit (when h → 0) in the space of distributions.

2 For several special types of systems of differential equations, the graph of such a map h (satisfying some additional properties) is called a center manifold. Existence of local center manifolds have been widely used for the study of stability near an equilibrium and in control theory. See [7, 8, 20, 23] for instance.

• The contingent derivative is the upper graphical limit of the difference-quotients

v ↦ ∇_{h}f(x)(v) := \frac{f(x + hv) − f(x)}{h} (when h → 0+), and the limit is a set-valued

map Df(x) : X ∼>Y (and no longer necessarily a single-valued function). Furthermore, one can define differential quotients for any set-valued map F : X ∼>Y at (x, y) ∈ Graph(F) by v ∼>\frac{F (x+hv)−y}{h}, h > 0 and the contingent derivative of a set-valued map as their upper graphical limit (when h → 0+).

In both cases, the approaches are similar: they use (different) convergencies weaker than the pointwise convergence for increasing the possibility for the difference-quotients to converge, at the price of losing some properties by passing to these weaker limits (the pointwise character for distributional derivatives, the linearity of the differential operator for graphical derivatives). The use of contingent derivatives (for instance of the value function for optimal control problems) is by no means new (see for instance [1], [6, Chapter 6], [28], [29]). It has been shown in [28], [29] that “contingent solutions” are related by duality to the “viscosity solutions” introduced in the context of Hamilton-Jacobi equations by Crandall & Lions in [22] (see also [21] and the literature following these papers). In the context of this paper (quasi-linear but set-valued hyperbolic differential inclusions), Proposition 4.4 makes explicit the duality relations between contingent solutions and solutions very closed in spirit to the viscosity solutions in the case when Y = R. The variational principle we prove below (Theorem 4.1) states that for systems of partial differential equations or inclusions, the contingent solutions are adaptations to the vector-valued case of viscosity solutions. Solutions to the contingent differential inclusion are defined in section 2. We then devote section 3 to the study of the transpose of contingent derivatives, and in particular, to a series of new convergence results, which play here as well as elsewhere a crucial role. For instance Proposition 3.5 below implies that a functional involving such transposes of contingent derivatives is lower semicontinuous for the ... pointwise convergence topology. The variational principle is the topic of section 4 and the existence of solutions the object of section 5. These results are applied to characterize and find feedback controls regulating solutions in section 6.

2. Contingent Differential Inclusion

Consider two finite dimensional vector-spaces X and Y, two set-valued maps F : X ×Y ∼ >X, G : X × Y ∼ >Y and a set-valued map H : X ∼ >Y. Throughout the whole paper we assume that F, G are upper semicontinuous, have nonempty convex compact images and the linear growth :

∃ c > 0, ∀ x ∈ X, y ∈ Y, ∥F (x, y)∥ + ∥G(x, y)∥ ≤ c(∥x∥ + ∥y∥ + 1)

where ∥F (x, y)∥ := sup_{v∈F (x,y)} ∥v∥. We associate with these data the contingent differential inclusion

∀ (x, y) ∈ Graph(H), 0 ∈ DH(x, y)(F (x, y)) − G(x, y)  (2.1)

Definition 2.1. A set-valued map H : X ∼ >Y satisfying (2.1) is a solution to the above contingent differential inclusion if its graph is a closed subset of Dom(F) ∩ Dom(G).

When H = h : Dom(h) ↦ Y is a single-valued map with closed graph contained in Dom(F) ∩ Dom(G), the partial contingent differential inclusion (2.1) becomes

∀ x ∈ Dom(h), 0 ∈ Dh(x)(F (x, h(x))) − G(x, h(x))  (2.2)

Actually solutions to (2.1) enjoy some stability. Recall that the graph of the graphical upper limit H^{♯} of a sequence of set-valued maps H_{n} : X ∼>Y is, by definition, the graph of the upper limit of the graphs of the maps H_{n}. (See [10, Chapter 7].) Then results from [5] imply

Theorem 2.2. [Stability] Let us consider a sequence of upper semicontinuous set-valued maps F_{n} : X × Y ∼>X, G_{n} : X × Y ∼ >Y with nonempty convex compact images and uniform linear growth in the sense that there exists a constant c > 0 such that

sup_{n≥0} (∥F_{n}(x, y)∥ + ∥G_{n}(x, y)∥) ≤ c(∥x∥ + ∥y∥ + 1)

and their graphical upper limit F^{♯} and G^{♯}. Consider also a sequence H_{n} : X ∼>Y of solutions to the contingent differential inclusions

∀ (x, y) ∈ Graph(H_{n}), 0 ∈ DH_{n}(x, y)(F_{n}(x, y)) − G_{n}(x, y)  (2.3)

Then the graphical upper limit H^{♯} of the solutions H_{n} is a solution to

∀ (x, y) ∈ Graph(H^{♯}), 0 ∈ DH^{♯}(x, y)(\overline{co}F^{♯}(x, y)) − \overline{co}(G^{♯}(x, y))  (2.4)

In particular, if the set-valued maps F_{n} and G_{n} converge graphically to F and G respectively, then the graphical upper limit H^{♯} of the solutions H_{n} is a solution to (2.1).

Since the graphical convergence of single-valued maps is weaker than pointwise convergence, the graphical limits of single-valued maps which are converging pointwise may well be set-valued. In the next section we provide some convergence results implying a dual characterization of solutions to partial differential inclusions.

3. Codifferentials

A set-valued map whose graph is a closed cone is called a closed process. It is a closed convex process if its graph is furthermore convex. They were introduced by Rockafellar in [32, 33]. Closed convex processes enjoy most of the properties of continuous linear operators, as it was shown in [32, 33] and [10, Chapter 2]. The transpose of a closed process A : X ∼>Y is the closed convex process A^{∗} : Y^{∗} ∼>X^{∗} defined by Rockafellar in [32] by

p ∈ A^{∗}(q) if and only if ∀ (x, y) ∈ Graph(A), ⟨p, x⟩ ≤ ⟨q, y⟩

We define in a symmetric way the bitranspose A^{∗∗} : X ∼>Y of A, the graph of which is the closed convex cone spanned by the graph of A:

Graph(A^{∗∗}) = (Graph(A))^{− −}

Definition 3.1. Let H : X ∼ >Y be a set-valued map and (x, y) belong to its graph. The transpose DH(x, y)^{∗} : Y^{∗} ∼ >X^{∗} of the contingent derivative DH(x, y) is called the codifferential of H at (x, y). When H := h is single-valued, we set Dh(x)^{∗} := Dh(x, h(x))^{∗}.

Recall that whenever h is Lipschitz around x, Dh(x)(u) ≠ ∅ for every u ∈ X (see [10, Proposition 5.1.4]).

Lemma 3.2. Let K ⊂ X and h : K ↦ Z be a single-valued map Lipschitz around x ∈ K. Then p ∈ Dh(x)^{∗}(q) if and only if for any ε > 0, there exists δ > 0 such that

∀ y ∈ B(x, δ) ∩ K, ⟨p, y − x⟩ − ⟨q, h(y) − h(x)⟩ ≤ ε∥x − y∥  (3.1)

Proof. The sufficient condition being straightforward, let us prove the necessary one. Assume the contrary: there exists ε > 0 and a sequence of elements x_{n} ∈ K converging to x such that

⟨p, x_{n} − x⟩ − ⟨q, h(x_{n}) − h(x)⟩ > ε∥x − x_{n}∥

We set ε_{n} := ∥x_{n} − x∥ which converges to 0 and u_{n} := (x_{n} − x)/ε_{n}, a subsequence of which converges to some u of the unit sphere. Since h is Lipschitz around x, there exists a cluster point v ∈ Dh(x)(u) of the sequence

(h(x + ε_{n}u_{n}) − h(x))/ε_{n}

We thus derived the following contradiction:

⟨p, u⟩ − ⟨q, v⟩ ≤ 0 & ⟨p, u⟩ − ⟨q, v⟩ ≥ ε

The contingent epiderivative of an extended function V : X ↦ R ∪ {+∞} at a point x of its domain is defined by

D_{↑}V (x)(u) := lim inf_{h→0+, u′→u} \frac{V (x + hu′) − V (x)}{h}

See [10, Chapter 6] for more details on this topic. The following result characterizes the transpose of the contingent derivative of a set-valued map H in terms of the contingent epiderivatives of its support function:

Proposition 3.3. Assume that H : X ∼>Y has compact convex values. We associate with any q ∈ Y^{∗} the functions H_{q}^{♭} : X ↦ IR and H_{q}^{♯} : X ↦ IR defined by

∀ x ∈ X, H_{q}^{♭}(x) := inf_{y∈H(x)}⟨q, y⟩ & H_{q}^{♯}(x) := sup_{y∈H(x)}⟨q, y⟩

Let y_{q}^{♭}, y_{q}^{♯} ∈ H(x) satisfy ⟨q, y_{q}^{♭}⟩ = H_{q}^{♭}(x) and ⟨q, y_{q}^{♯}⟩ = H_{q}^{♯}(x). Then

{p ∈ X^{∗} | ∀ u ∈ X, ⟨p, u⟩ ≤ D_{↑}H_{q}^{♭}(x)(u) } ⊂ DH(x, y_{q}^{♭})^{∗}(q)

If H is Lipschitz at x (in the sense that there exists l > 0 such that H(x) ⊂ H(y) + l∥x − y∥B for every y in a neighborhood of x), then

DH(x, y_{q}^{♯})^{∗}(q) ⊂ {p ∈ X^{∗} | ∀ u ∈ X, ⟨p, u⟩ ≤ D_{↑}H_{q}^{♯}(x)(u) }

Consequently, when H = h is single-valued and Lipschitz at x, setting h_{q}^{∗}(x) := ⟨q, h(x)⟩ =

h_{q}^{♯}(x) = h_{q}^{♭}(x) we obtain the equality

Dh(x)^{∗}(q) = {p ∈ X^{∗} | ∀ u ∈ X, ⟨p, u⟩ ≤ D_{↑}h_{q}^{∗}(x)(u) }

Proof. Assume first that p ∈ X^{∗} satisfies

∀ u ∈ X, ⟨p, u⟩ ≤ D_{↑}H_{q}^{♭}(x)(u)

We prove that for every v ∈ DH (x, y_{q}^{♭}) (u),

D_{↑}H_{q}^{♭}(x)(u) ≤ ⟨q, v⟩

Indeed, there exist sequences t_{n} > 0, u_{n} ∈ X and v_{n} ∈ Y converging to 0, u and v respectively such that y_{q}^{♭} + t_{n}v_{n} ∈ H(x + t_{n}u_{n}). Therefore,

D_{↑}H_{q}^{♭}(x)(u) ≤ lim inf_{n→∞} \frac{H_{q}^{♭}(x + t_{n}u_{n}) − H_{q}^{♭}(x)}{t_{n}} ≤ lim inf_{n→∞} ⟨q, v_{n}⟩

Consequently, ⟨p, u⟩ ≤ ⟨q, v⟩ for every (u, v) ∈ Graph(DH(x, y_{q}^{♭})), so that

p ∈ DH(x, y_{q}^{♭})^{∗}(q).

Conversely, assume that H is Lipschitz at x, p ∈ DH (x, y_{q}^{♯})^{∗} (q) and fix u ∈ X. Then

there exist sequences t_{n} > 0 and u_{n} converging to 0 and u such that

D_{↑}H_{q}^{♯}(x)(u) = lim_{n→∞} \frac{H_{q}^{♯}(x + t_{n}u_{n}) − H_{q}^{♯}(x)}{t_{n}}

Since H is Lipschitz at x, there exists l > 0 such that, for n large enough, y_{q}^{♯} belongs to H(x + t_{n}u_{n}) + lt_{n}∥u_{n}∥B, so that it can be written y_{q}^{♯} = y_{n} − t_{n}v_{n} where y_{n} ∈ H(x + t_{n}u_{n}) and ∥v_{n}∥ ≤ l∥u_{n}∥. Therefore a subsequence (again denoted by) v_{n} converges to some v,

which belongs to DH(x, y_{q}^{♯})(u). Since ⟨q, y_{n}⟩ ≤ H_{q}^{♯}(x + t_{n}u_{n}) and ⟨q, y_{q}^{♯}⟩ = H_{q}^{♯}(x), we

infer that

D_{↑}H_{q}^{♯}(x)(u) ≥ ⟨q, v⟩ ≥ ⟨p, u⟩

When h is real-valued, we need only to know the values of Dh(x)^{∗} at the points 0, +1 and −1 to reconstruct the whole set-valued map Dh(x)^{∗}.

We observe that for q = +1, D_{↑}h_{q}^{∗}(x)(u) = D_{↑}h(x)(u) and that for q = −1, D_{↑}h_{q}^{∗}(x)(u) = D_{↑}(−h)(x)(u) and that for q = 0, Dh(x)^{∗}(0) = (Dom(Dh(x)))^{−}. Recall (see for instance [10, Definition 6.4.7 and Proposition 6.4.8] or [21]) that:

{ p ∈ X^{∗} | ∀ u ∈ X, ⟨p, u⟩ ≤ D_{↑}h(x)(u) } = ∂_{−}h(x)

is the local subdifferential and

{ p ∈ X^{∗} | ∀ u ∈ X, ⟨p, u⟩ ≤ D_{↑}(−h)(x)(u) } = −∂_{+}h(x)

is the local superdifferential of h at x. The above characterization then becomes Proposition 3.4. Let h : X ↦ R be a continuous at x function. Then

Dh(x)^{∗}(+1) = ∂_{−}h(x) & Dh(x)^{∗}(−1) = −∂_{+}h(x)

Proof. We already know that

∂_{−}h(x) ⊂ Dh(x)^{∗}(+1)

Assume now that p ∈ Dh(x)^{∗}(+1). We have to show that for all u, ⟨p, u⟩ ≤ D_{↑}h(x)(u). There is nothing to prove if D_{↑}h(x)(u) = +∞. If D_{↑}h(x)(u) is finite, then v := D_{↑}h(x)(u) belongs to Dh(x)(u), so that ⟨p, u⟩ ≤ D_{↑}h(x)(u). We finally claim that the continuity of h at x implies that D_{↑}h(x)(u) > −∞ for any u ∈ X, which is equivalent to D_{↑}h(x)(0) = 0 thanks to [10, Propositions 6.1.3]. Indeed if not, by [10, Propositions 6.1.4 and Lemma 6.1.1], the pair (0, −1) belongs to the contingent cone to the epigraph of h at (x, h(x)). Then there exist sequences t_{n} > 0 converging to 0, u_{n} converging to 0 and a sequence of v_{n} > 0 going to 1 such that h(x + t_{n}u_{n}) ≤ h(x) − t_{n}v_{n}. On the other hand, h being continuous at x, the continuous function φ defined by φ(t) := h(x + tu_{n}) satisfies for all large n

φ(t_{n}) ≤ h(x) − t_{n}v_{n} ≤ φ(0)

and therefore, there exist s_{n} ∈ [0, t_{n}] such that φ(s_{n}) = h(x) − t_{n}v_{n}. Setting ũ_{n} := \frac{s_{n}}{t_{n}} u_{n}, which also converges to 0, we observe that h(x + t_{n}ũ_{n}) = h(x) − t_{n}v_{n}. This means that −1 ∈ Dh(x)(0). But p ∈ Dh(x)^{∗}(1), and thus, we derived the contradiction:

0 = ⟨p, 0⟩ ≤ ⟨1, −1⟩ = −1

Remark. The above proposition allows to reformulate the notion of viscosity solution of a scalar Hamilton-Jacobi equation Ψ(x, h′(x)) = 0 in the following way: h is a viscosity solution if and only if

\{ \begin{matrix}i) & ∀ p ∈ Dh(x)^{∗}(+1), Ψ(x, p) ≥ 0 \\ ii) & ∀ p ∈ Dh(x)^{∗}(−1), Ψ(x, −p) ≤ 0 \end{matrix}  (3.2)

The variational principle given in section 4 is based on the following convergence result:

Proposition 3.5. Let K ⊂ X be a closed subset. Assume that h is the pointwise limit of an equicontinuous family of maps h_{n} : K ↦ Y. Let x ∈ K and p ∈ Dh(x)^{∗}(q) be fixed. Then there exist subsequences of elements x_{n_{k}} ∈ K converging to x, q_{n_{k}} converging to q and p_{n_{k}} ∈ Dh_{n_{k}}(x_{n_{k}})^{∗}(q_{n_{k}}) converging to p. If the functions h_{n} are differentiable, we deduce that there exist subsequences of elements x_{n_{k}} ∈ K converging to x and q_{n_{k}} converging to q such that h′_{n_{k}}(x_{n_{k}})^{∗}(q_{n_{k}}) converges to p.

Proof. We have to prove that there exist subsequences x_{n_{k}} ∈ K and

(p_{n_{k}}, −q_{n_{k}}) ∈ (T_{Graph(h_{n_{k}})}(x_{n_{k}}, h_{n_{k}}(x_{n_{k}})))^{−}

converging to x and (p, −q) respectively. So our proposition follows from

Theorem 3.6. Let us consider a sequence of closed subsets K_{n} and an element x ∈ Liminf_{n→∞}K_{n}. Set K^{♯} := Limsup_{n→∞}K_{n}. Then, for any p ∈ (T_{K^{♯}}(x))^{−}, there exist subsequences of elements x_{n_{k}} ∈ K_{n_{k}} and p_{n_{k}} ∈

(T_{K_{n_{k}}} (x_{n_{k}}))^{−} converging to p and x respectively, i.e.

(T_{K^{♯}}(x))^{−} ⊂ Limsup_{n→∞, x_{n}→_{Kn} x} (T_{K_{n}}(x_{n}))^{−}

Proof. It is sufficient to consider the case when x ∈ ⋂_{n=1}^{∞} K_{n}. If not, we set K̂_{n} := K_{n} +x−u_{n} where u_{n} ∈ K_{n} converges to x and observe that x ∈ ⋂_{n=1}^{∞} K̂_{n} and T_{K_{n}}(x_{n}) = T_{K_{n}}(x_{n} − x + u_{n}). Let p ∈ (T_{K^{♯}}(x))^{−} be given with norm 1. We associate with any positive λ the projection x_{n}^{λ} of x + λp onto K_{n}:

∥x + λp − x_{n}^{λ}∥ = min_{x_{n}∈K_{n}} ∥x + λp − x_{n}∥  (3.3)

and set

v_{n}^{λ} := \frac{x_{n}^{λ} − x}{λ} & p_{n}^{λ} := p − v_{n}^{λ} ∈ (T_{K_{n}}(x_{n}^{λ}))^{−}

Let us fix for the time λ > 0. By taking x_{n} = x ∈ K_{n} in (3.3), we infer that ∥v_{n}^{λ}∥ ≤ 2.

Therefore some subsequences x_{n′}^{λ} and v_{n′}^{λ} converge to elements x^{λ} ∈ K^{♯} and v^{λ} = \frac{x^{λ}−x}{λ} respectively. Furthermore, there exists a sequence λ_{i} → 0+ such that v^{λ_{i}} converge to some v ∈ T_{K^{♯}}(x) because ∥v^{λ}∥ ≤ 2 and x^{λ} = x + λv^{λ} ∈ K^{♯}. Therefore ⟨p, v⟩ ≤ 0. On the other hand, we deduce from (3.3) the inequalities

∥p − v_{n}^{λ}∥^{2} = ∥p∥^{2} + ∥v_{n}^{λ}∥^{2} − 2⟨p, v_{n}^{λ}⟩ ≤ ∥p∥^{2}

which imply, by passing to the limit, that ∥v∥^{2} ≤ 2⟨p, v⟩ ≤ 0. Thus v = 0. We deduce that a subsequence v_{n_{k}}^{λ_{k}} = p − p_{n_{k}}^{λ_{k}} converges also to 0. The lemma ensues.

We shall need stronger convergence results, where in the conclusion of Proposition 3.5 we require that q_{n} and/or x_{n} remain constant. We have to pay some price for that: stronger convergence assumptions and the use of graphical derivatives D_{δ}h(x) contained in the graph of Dh(x) which are closed convex processes. For instance, the circatangent derivative Ch(x), defined in the following way from the Clarke tangent cone:

Graph(Ch(x)) := C_{Graph(h)}(x, h(x))

is a closed convex process contained in the contingent derivative Dh(x). They coincide whenever h is sleek at x. We can also use the asymptotic derivative D_{∞}h(x), whose graph is the asymptotic cone to the graph of h at (x, h(x)). (See [10, Chapters 4,5] for further details.) We prove for instance the following

Proposition 3.7. Let K ⊂ X be a closed subset. Assume that h is Lipschitz around x on K and consider a sequence of continuous maps h_{n} converging to h uniformly on compact subsets of K. Let x ∈ K and p ∈ Dh(x)^{∗}(q) be fixed. Then there exist a sequence x_{n} ∈ K converging to x and a sequence p_{n} ∈ D_{δ}h_{n}(x_{n})^{∗}(q) converging to p. If the functions h_{n} are differentiable, we infer that there exists a sequence x_{n} ∈ K converging to x such that h′_{n}(x_{n})^{∗}(q) converges to p.

Proof. Let μ > 0, L := K ∩ B(x, μ) be a compact neighborhood of x on which the maps h_{n} converge uniformly to h. We apply Ekeland’s Theorem to the functions y ↦ ⟨q, h_{n}(y)⟩−⟨p, y⟩ defined on this subset. Fix ε ∈]0, μ[. Then there exists x_{n} ∈ L satisfying

\{ \begin{matrix}⟨q, h_{n}(x_{n})⟩ − ⟨p, x_{n}⟩ + ε∥x_{n} − x∥ & ≤ ⟨q, h_{n}(x)⟩ − ⟨p, x⟩ \\ ∀ y ∈ L, ⟨q, h_{n}(x_{n})⟩ − ⟨p, x_{n}⟩ & ≤ ⟨q, h_{n}(y)⟩ − ⟨p, y⟩ + ε∥y − x_{n}∥ \end{matrix}

The first inequality implies that

\begin{aligned}ε∥x − x_{n}∥ & ≤ ⟨q, h_{n}(x) − h(x)⟩ + ⟨q, h(x_{n}) − h_{n}(x_{n})⟩ + ⟨p, x_{n} − x⟩ \\ −⟨q, h(x_{n}) − h(x)⟩ & ≤ 2∥q∥∥h_{n} − h∥ + ⟨p, x_{n} − x⟩ − ⟨q, h(x_{n}) − h(x)⟩ \end{aligned}

By Lemma 3.2, there exists 0 < δ ≤ μ such that

∀ y ∈ B(x, δ) ∩ K, ⟨p, x_{n} − x⟩ − ⟨q, h(x_{n}) − h(x)⟩ ≤ ε∥x_{n} − x∥/2

Hence, ∥x_{n} − x∥ ≤ 4∥q∥∥h_{n} − h∥/ε < μ for n large enough. On the other hand, consider any v ∈ Dh_{n}(x_{n})(u): There exist ε_{p} > 0 converging to 0, u_{p} converging to u and v_{p} converging to v such that h_{n}(x_{n} + ε_{p}u_{p}) = h_{n}(x_{n}) + ε_{p}v_{p} for all p. Taking y = x_{n} + ε_{p}u_{p} for p large enough in the second inequality, we infer that

0 ≤ ⟨q, v_{p}⟩ − ⟨p, u_{p}⟩ + ε∥u_{p}∥

and thus, by passing to the limit,

∀ (u, v) ∈ Graph(Dh_{n}(x)), 0 ≤ ⟨q, v⟩ − ⟨p, u⟩ + ε∥u∥

In particular, taking the restriction to Graph(D_{δ}h_{n}(x_{n})) and noticing that ∥u∥ = sup_{e∈B_{∗}}⟨u, e⟩, this inequality can be written in the form:

0 ≤ inf_{(u,v)∈Graph(D_{δ}h_{n}(x))} sup_{e∈B_{∗}}(⟨q, v⟩ − ⟨p, u⟩ + ε⟨e, u⟩)

Since B_{∗} is convex compact and since the graph of D_{δ}h_{n}(x) is convex, the lop-sided minimax theorem (see for instance [9]) implies the existence of e_{0} ∈ B_{∗} such that

0 ≤ inf_{(u,v)∈Graph(D_{δ}h_{n}(x))}(⟨q, v⟩ − ⟨p, u⟩ + ε⟨e_{0}, u⟩)

Consequently, (p − εe_{0}, −q) belongs to the polar cone to Graph(D_{δ}h_{n}(x_{n})), so that p_{n} := p − εe_{0} ∈ D_{δ}h_{n}(x_{n})^{∗}(q). Summarizing, for any ε > 0 and for any n such that ∥h_{n} − h∥ ≤ ε^{2}/4∥q∥, we have proved the existence of x_{n} ∈ K and p_{n} ∈ D_{δ}h_{n}(x_{n})^{∗}(q) such that ∥x_{n} − x∥ ≤ ε, ∥p_{n} − p∥ ≤ ε.

Let K ⊂ X be a closed subset and C_{Λ}(K, Z) denote the space of Lipschitz (single-valued) bounded maps from K to a finite dimensional vector-space Z,

∥h∥_{Λ} := sup_{x≠y∈K} \frac{∥h(x) − h(y)∥}{∥x − y∥} & ∥h∥_{∞} := sup_{x∈K} ∥h(x)∥

denote the Lipschitz semi-norm and the sup-norm. Define the norm of the Banach space C_{Λ}(K, Z) by ∥h∥ = ∥h∥_{Λ} + ∥h∥_{∞}. We observe the following continuity properties of the contingent derivative:

Lemma 3.8. Let x ∈ K be fixed. Then the map

(h, u) ∈ C_{Λ}(K, Y) × X ∼ > Dh(x)(u)

is Lipschitz:

∀ h, g ∈ C_{Λ}(K, Y), Dh(x)(u) ⊂ Dg(x)(v) + ∥h − g∥_{Λ}∥u∥ + ∥g∥_{Λ}∥u − v∥

Proof. The proof is straightforward from the inequality

∥\frac{h(x + tu) − h(x)}{t} − \frac{g(x + tv) − g(x)}{t}∥ ≤ ∥h − g∥_{Λ}∥u∥ + ∥g∥_{Λ}∥u − v∥

We shall need the following stronger statement than Proposition 3.7 :

Proposition 3.9. Let K ⊂ X be a closed subset. Assume that h is Lipschitz and consider a sequence of Lipschitz maps h_{n} converging to h in C_{Λ}(K, Y). Let x ∈ K and p ∈ Dh(x)^{∗}(q) be fixed. Then there exists a sequence p_{n} ∈ D_{δ}h_{n}(x)^{∗}(q) converging to p. In particular, if the maps h_{n} are differentiable, h′_{n}(x)^{∗}q converges to p.

Proof. Set ε_{n} := 2∥q∥∥h_{n} − h∥_{Λ}. By Lemma 3.2 , there exists μ > 0 such that

⟨p, y − x⟩ − ⟨q, h(y) − h(x)⟩ ≤ ε_{n}∥y − x∥/2

whenever y ∈ K ∩ B(x, μ). Therefore

⟨p, y − x⟩ − ⟨q, h_{n}(y) − h_{n}(x)⟩

≤ ε_{n}∥y − x∥/2 + ∥q∥∥(h_{n} − h)(y) − (h_{n} − h)(x)∥

≤ (ε_{n}/2 + ∥q∥∥h_{n} − h∥_{Λ})∥y − x∥ ≤ ε_{n}∥y − x∥

On the other hand, consider any v ∈ Dh_{n}(x)(u) and t_{p} → 0+, u_{p} → u and v_{p} → v be such that h_{n}(x + t_{p}u_{p}) = h_{n}(x) + t_{p}v_{p} for all p. Taking y = x + t_{p}u_{p} ∈ K ∩ B(x, μ) for n large enough, we infer that

0 ≤ ⟨q, v_{p}⟩ − ⟨p, u_{p}⟩ + ε_{n}∥u_{p}∥

and thus, by letting u_{p} and v_{p} converge to u and v,

∀ (u, v) ∈ Graph(Dh_{n}(x)), 0 ≤ ⟨q, v⟩ − ⟨p, u⟩ + ε_{n}∥u∥

In particular, taking the restriction to Graph(D_{δ}h_{n}(x)) which is convex,

0 ≤ inf_{(u,v)∈Graph(D_{δ}h_{n}(x))} sup_{e∈B_{∗}}(⟨q, v⟩ − ⟨p, u⟩ + ε_{n}⟨e, u⟩)

The lop-sided minimax theorem implies the existence of e_{n} ∈ B_{∗} such that (p − ε_{n}e_{n}, −q) belongs to the negative polar cone to Graph(D_{δ}h_{n}(x)), i.e., p_{n} := p−ε_{n}e_{n} ∈ D_{δ}h_{n}(x)^{∗}(q)

4. The Variational Principle

We characterize in this section solutions to the contingent differential inclusion (2.2) through a variational principle. For that purpose, we denote by

σ(M, p) := sup_{z∈M}⟨p, z⟩ & σ^{♭}(M, p) := inf_{z∈M}⟨p, z⟩

the support functions of M ⊂ X and by B_{∗} the unit ball of Y^{∗}. Consider a closed subset K ⊂ X. We introduce the nonnegative functional Φ defined on the space C(K, Y) of continuous maps by

Φ(h) = inf{c ≥ 0 | ∀ x, q, sup_{p∈Dh(x)^{∗}(q)}(σ^{♭}(F (x, h(x)), p) − σ(G(x, h(x)), q)) ≤ c ∥q∥}

Observe that c is finite if and only if

sup_{p∈Dh(x)^{∗}(0)} σ^{♭}(F (x, h(x)), p) ≤ 0

and

Φ(h) = sup_{q∈B_{∗}} sup_{x∈K} sup_{p∈Dh(x)^{∗}(q)} (σ^{♭}(F (x, h(x)), p) − σ(G(x, h(x)), q))

Theorem 4.1. [Variational Principle] Let c ≥ 0. Then a single-valued map h : K ↦ Y is a solution to the contingent differential inclusion

∀ x ∈ K, 0 ∈ Dh(x)(F (x, h(x))) − G(x, h(x)) + cB

if and only if Φ(h) ≤ c. Consequently, h is a solution to the contingent differential inclusion (2.2) if and only if Φ(h) = 0.

Proof. Let u ∈ F (x, h(x)), v ∈ G(x, h(x)) and e ∈ cB be such that v − e ∈ Dh(x)(u). Then, for any q ∈ Y^{∗} and p ∈ Dh(x)^{∗}(q),

⟨p, u⟩ − ⟨q, v − e⟩ ≤ 0

so that

σ^{♭}(F (x, h(x)), p) − σ(G(x, h(x)), q) ≤ ⟨p, u⟩ − ⟨q, v⟩ ≤ ⟨q, e⟩ ≤ c ∥q∥

Taking the supremum with respect to p ∈ Dh(x)^{∗}(q), we infer that Φ(h) ≤ c. Conversely, we can write inequality Φ(h) ≤ c in the form of the minimax inequality: for any x ∈ K, q ∈ Y^{∗},

sup_{p∈Dh(x)^{∗}(q)} inf_{u∈F (x,h(x))} inf_{v∈G(x,h(x))}(⟨p, u⟩ − ⟨q, v⟩) ≤ c∥q∥

Setting

β(p, q; u, v, e) := ⟨p, u⟩ − ⟨q, v − e⟩

this inequality can be written in the form: for every x ∈ K,

sup_{(p,−q)∈Graph(Dh(x))^{−}} inf_{(u,v,e)∈F (x,h(x))×G(x,h(x))×cB} β(p, q; u, v, e) ≤ 0

The lop-sided minimax theorem (see for instance [9]) implies the existence of u_{0} ∈ F (x, h(x)), v_{0} ∈ G(x, h(x)) and e_{0} ∈ cB such that

sup_{(p,−q)∈Graph(Dh(x))^{−}}(⟨p, u_{0}⟩ − ⟨q, v_{0} − e_{0}⟩) =

sup_{(p,−q)∈Graph(Dh(x))^{−}} inf_{(u,v,e)∈F (x,h(x))×G(x,h(x))×cB} β(p, q; u, v, e) ≤ 0

This means that (u_{0}, v_{0} − e_{0}) belongs to \overline{co}(Graph(Dh(x))). In other words, we have proved that

(F (x, h(x)) × (G(x, h(x)) + cB)) ∩ \overline{co} (T_{Graph(h)}(x, h(x))) ≠ ∅

But by Proposition 7.1 of the Appendix, this is equivalent to the condition

(F (x, h(x)) × (G(x, h(x)) + cB)) ∩ T_{Graph(h)}(x, h(x)) ≠ ∅

i.e., h is a solution to our contingent differential inclusion.

Remark. Since the bipolar cone of Graph(Dh(x)) is the graph of the bitranspose Dh(x)^{∗∗}, we have actually proved that h is a solution to the contingent differential inclusion if and only if it is a solution to the “relaxed” contingent differential inclusion

0 ∈ Dh(x)^{∗∗}(F (x, h(x))) − G(x, h(x)) + cB

Theorem 4.2. Let H ⊂ C(K, Y) be a compact subset for the compact convergence topology. Assume that c := inf_{h∈H} Φ(h) < +∞. Then there exists a solution h ∈ H to the contingent differential inclusion

0 ∈ Dh(x)(F (x, h(x))) − G(x, h(x)) + cB

Thanks to Theorem 4.1 it is sufficient to prove that the functional Φ is lower semicontinuous on the space C(K, Y) for the compact convergence topology.

Proposition 4.3. The functional Φ is lower semicontinuous on equicontinuous subsets of the space C(K, Y) for the compact convergence topology.

Proof. We may assume that Φ is proper. Let h_{n} be a sequence of equicontinuous maps satisfying for any n, Φ(h_{n}) ≤ c and converging uniformly to some h. We have to show that Φ(h) ≤ c. Indeed, fix x ∈ K, q ∈ B_{∗} and p ∈ Dh(x)^{∗}(q). By Proposition 3.5, there exist subsequences (again denoted by) x_{n} ∈ K converging to x, q_{n} converging to q and p_{n} ∈ Dh_{n}(x_{n})^{∗}(q_{n}) converging to p such that h_{n}(x_{n}) converges to h(x). Since F and G are upper semicontinuous with compact values, we know that for any (p, q) and ε > 0, we have

\{ \begin{matrix}σ^{♭}(F (x, h(x)), p) − σ(G(x, h(x)), q) & ≤ \\ σ^{♭}(F (x_{n}, h_{n}(x_{n})), p_{n}) − σ(G(x_{n}, h_{n}(x_{n})), q_{n}) + ε & ≤ (Φ(h_{n}) + ε) ∥q_{n}∥ + ε \end{matrix}

for n large enough. Hence, by letting n go to ∞, we infer that for any ε > 0,

σ^{♭}(F (x, h(x)), p) − σ(G(x, h(x)), q) ≤ (c + ε) ∥q∥ + ε

Letting ε converge to 0 and taking the supremum on p ∈ Dh(x)^{∗}(q), we infer that Φ(h) ≤ c.

In the case when Y = R, the contingent solutions are very closed in spirit to the viscosity solutions:

Proposition 4.4. Assume that Y = R. Then a continuous function h : X ↦ R is a solution to (2.2) on a closed set K ⊂ X if and only if for every x ∈ K,

i) sup_{p∈∂_{−}h(x)} (σ^{♭}(F (x, h(x)), p) − sup(G(x, h(x)))) ≤ 0

ii) inf_{p∈∂_{+}h(x)} (σ(F (x, h(x)), p) − inf(G(x, h(x)))) ≥ 0

Proof. Set

Φ_{+}(h, x) = sup_{p∈Dh(x)^{∗}(+1)} (σ^{♭}(F (x, h(x)), p) − σ(G(x, h(x)), +1))

Φ_{−}(h, x) = sup_{p∈Dh(x)^{∗}(−1)} (σ^{♭}(F (x, h(x)), p) − σ(G(x, h(x)), −1))

Φ_{0}(h, x) = sup_{p∈Dh(x)^{∗}(0)} (σ^{♭}(F (x, h(x)), p))

If h is a solution, then properties i), ii) follow from Proposition 3.4 and Theorem 4.1 with c = 0. Conversely if i), ii) hold true, then sup_{x∈K} max (Φ_{+}(h, x), Φ_{−}(h, x)) ≤ 0. It remains to show that Φ_{0}(h, x) ≤ 0. Fix any p ∈ Dh(x)^{∗}(0). Then (p, 0) ∈ (T_{Ep(h)}(x, h(x)))^{−}. By Rockafellar’s result from [35] there exist x_{n} → x, (p_{n}, q_{n}) → (p, 0) such that q_{n} ≠ 0, \frac{p_{n}}{|q_{n}|} ∈ ∂_{−}h(x_{n}). Hence, by i),

σ^{♭}(F (x_{n}, h(x_{n})), p_{n}) − ∥q_{n}∥ σ(G(x_{n}, h(x_{n})), +1) ≤ 0

Taking the limit we get σ^{♭}(F (x, h(x)), p) ≤ 0.

We can relate solutions to the contingent differential inclusion (2.2) to viscosity solutions when the set-valued map F : X ∼>X does not depend on y and when G is equal to 0. The above proposition implies that both h and −h are viscosity subsolutions to the Hamilton-Jacobi equation

−σ(F (x), h′(x)) = 0

The apparent discrepancy comes from the fact that solutions h of the contingent partial differential inclusion are energy functions and not the value function of an optimal control problem.

5. Single-Valued Solutions to Partial Differential Inclusions

We shall look for solutions in a compact convex subset H of the space C_{Λ}(K, Y) of Lipschitz maps from K to Y.

Theorem 5.1. Let K be a closed subset of X. Consider a compact convex subset H of C_{Λ}(K, Y). When h ∈ H, we denote by T_{H}(h(·)) ⊂ C(K, Y) the tangent cone to H at h for the pointwise convergence topology. Let D_{δ}h(x) be a family of closed convex processes satisfying D_{δ}h(x) ⊂ Dh(x) and assume that for every h ∈ H, there exist v, w ∈ C(K, Y) such that ∀ x ∈ K,

w(x) ∈ D_{δ}h(x)(F (x, h(x))), v(x) ∈ G(x, h(x)) & w(·) − v(·) ∈ T_{H}(h(·))

Then there exists a solution h ∈ H to the contingent differential inclusion

∀ x ∈ K, 0 ∈ Dh(x)(F (x, h(x))) − G(x, h(x))

Proof. We assume that there is no solution to the contingent differential inclusion and we shall derive a contradiction. Indeed, thanks to Proposition 7.1, this amounts to assume that for any h ∈ H, there exists x ∈ K such that

0 ∉ \overline{co} (T_{Graph(h)}(x, h(x))) − F (x, h(x)) × G(x, h(x))

Since the images of F and G are compact and convex, the separation theorem implies that there exists also (p, −q) ∈ X^{∗} × Y^{∗} such that

0 < σ^{♭}(F (x, h(x)), p) − σ(G(x, h(x)), q) & p ∈ Dh(x)^{∗}(q)

Set

a(h; x, q) := sup_{p∈D_{δ}h(x)^{∗}(q)} σ^{♭}(F (x, h(x)), p) − σ(G(x, h(x)), q)

Since Dh(x)^{∗}(q) ⊂ D_{δ}h(x)^{∗}(q), we observe that

0 < a(h; x, q)

On the other hand, the function (y, p) ↦ σ^{♭}(F (x, y), p) − σ(G(x, y), q) being lower semicontinuous (because F and G are upper semicontinuous with compact values), there exist neighborhoods N_{1}(h(x)) and N_{2}(p) such that

∀ y ∈ N_{1}(h(x)), ∀ p′ ∈ N_{2}(p), 0 < σ^{♭}(F (x, y), p′) − σ(G(x, y), q)

By Proposition 3.9, there exists η(x) > 0 such that whenever ∥l−h∥_{Λ} ≤ η(x), there exists p′ ∈ D_{δ}l(x)^{∗}(q) satisfying l(x) ∈ N_{1}(h(x)) and p′ ∈ N_{2}(p). Hence

0 < σ^{♭}(F (x, l(x)), p′) − σ(G(x, l(x)), q) ≤ a(l; x, q)

Consequently, h belongs to the subset N(x, q) defined by

N(x, q) := {l ∈ C_{Λ}(K, Y) | 0 < a(l; x, q)}

which is open in C_{Λ}(K, Y) by Proposition 3.9. Summing up, we just have proved that if there is no solution to the contingent differential inclusion, then H can be covered by the open subsets N(x, q). Being compact, it can be covered by a finite number m of such neighborhoods N(x_{i}, q_{i}). Let α_{i}(·) be a continuous partition of unity associated with this covering. We introduce now the function φ : H × H ↦ R defined by

φ(h, l) := ∑_{i=1}^{m} α_{i}(h)⟨q_{i}, l(x_{i}) − h(x_{i})⟩

It is continuous with respect to h on C_{Λ}(K, Y) (because the α_{i}’s are so and h ↦ ⟨q_{i}, h(x_{i})⟩ are continuous for the pointwise topology), affine with respect to l and satisfies φ(l, l) = 0. Hence, H being convex and compact, Ky Fan Inequality (see [10, Theorem 3.1.1]) implies the existence of h̄ ∈ H such that for every l ∈ H, φ(h̄, l) ≤ 0. This means that the discrete measure ∑_{i=1}^{m} α_{i}(h̄)q_{i} ⊗ δ(x_{i}) belongs to the normal cone to H at h̄ since for every l ∈ H,

⟨∑_{i=1}^{m} α_{i}(h̄)q_{i} ⊗ δ(x_{i}), l − h̄⟩ = ∑_{i=1}^{m} α_{i}(h̄)⟨q_{i}, l(x_{i}) − h̄(x_{i})⟩ ≤ 0

We then deduce from the assumption of theorem that ∑_{i=1}^{m} α_{i}(h̄)a(h̄; x_{i}, q_{i}) ≤ 0. Indeed, there exist continuous functions v(·) and w(·) such that

∀ x ∈ K, v(x) ∈ G(x, h̄(x)) & w(x) ∈ D_{δ}h̄(x)(F (x, h̄(x)))

and w(·) − v(·) ∈ T_{H}(h̄(·)). Therefore, for any p_{i} ∈ D_{δ}h̄(x_{i})^{∗}(q_{i}), there exists u_{i} ∈ F (x_{i}, h̄(x_{i})) such that

\{ \begin{matrix}σ^{♭}(F (x_{i}, h̄(x_{i})), p_{i}) − σ(G(x_{i}, h̄(x_{i})), q_{i}) \\ ≤ ⟨p_{i}, u_{i}⟩ − ⟨q_{i}, v(x_{i})⟩ ≤ ⟨q_{i}, w(x_{i}) − v(x_{i})⟩ \end{matrix}

So that, by taking the supremum on p_{i} ∈ D_{δ}h̄(x_{i})^{∗}(q_{i}), we obtain

a(h̄; x_{i}, q_{i}) ≤ ⟨q_{i}, w(x_{i}) − v(x_{i})⟩

Multiplying by α_{i}(h̄) ≥ 0 and summing from i = 1 to m, we obtain

∑_{i=1}^{m} α_{i}(h̄)a(h̄; x_{i}, q_{i}) ≤ ⟨∑_{i=1}^{m} α_{i}(h̄)q_{i} ⊗ δ(x_{i}), w(·) − v(·)⟩ ≤ 0

because w(·) − v(·) ∈ T_{H}(h̄(·)) and ∑_{i=1}^{m} α_{i}(h̄)q_{i} ⊗ δ(x_{i}) belongs to the normal cone to H at h̄. We claim that in the same time ∑_{i=1}^{m} α_{i}(h̄)a(h̄; x_{i}, q_{i}) > 0. Indeed, whenever α_{i}(h̄) > 0, then h̄ belongs to N(x_{i}, q_{i}), which implies that 0 < a(h̄, x_{i}, q_{i}). We have therefore obtained a contradiction.

Lemma 5.2. Let H : K ∼>Y be a set-valued map and let H be a subset of continuous selections of H. Then

T_{H}(h(·)) ⊂ {v ∈ C(K, Y) | ∀ x ∈ K, v(x) ∈ T_{H(x)}(h(x))}

If in addition for any finite sequence (x_{i}, y_{i}) ∈ Graph(H) (i = 1, …, m) such that x_{i} ≠ x_{j} when i ≠ j, there exists a selection s ∈ H interpolating it:

∀ i = 1, …, m, s(x_{i}) = y_{i}

then equality holds true:

T_{H}(h(·)) = {v ∈ C(K, Y) | ∀ x ∈ K, v(x) ∈ T_{H(x)}(h(x))}

Proof. Indeed, let v ∈ C(K, Y) such that v(x) ∈ T_{H(x)}(h(x)) for all x ∈ K. Then there exists ε_{λ}(·) converging to 0 with λ for the pointwise convergence topology such that h(x) + λv(x) + λε_{λ}(x) ∈ H(x). Let us consider any neighborhood of 0 for the pointwise topology

V := {l ∈ C(K, Y) | sup_{i=1,...,n} ∥l(x_{i})∥ ≤ ε}

associated with a finite subset {x_{1}, …, x_{n}} and λ small enough for ε_{λ}(·) to belong to it. By the interpolation assumption there exists l_{λ} ∈ H such that

∀ x_{i}, l_{λ}(x_{i}) = h(x_{i}) + λv(x_{i}) + λε_{λ}(x_{i}) ∈ H(x_{i})

Thus the continuous function u_{λ} := (l_{λ} − h)/λ is such that h + λu_{λ} ∈ H and belongs to the neighborhood v + V of v for the topology of the pointwise convergence. In other words, v belongs to the tangent cone to H at h for the pointwise topology.

6. Feedback Controls Regulating Smooth Evolutions

Consider a control system (U, f):

\{ \begin{matrix}x′(t) = & f(x(t), u(t)) \\ u(t) ∈ & U(x(t)) \end{matrix}  (6.1)

Let (x, u) → φ(x, u) be a non negative continuous function with linear growth. We have proved in [14] that there exists a closed regulation map R^{φ} ⊂ U larger than any closed regulation map R : K ∼ >Z contained in U and enjoying the following viability property: For any initial state x_{0} ∈ Dom(R) and any initial control u_{0} ∈ R(x_{0}), there exists a solution (x(·), u(·)) to the control system (6.1) starting at (x_{0}, u_{0}) such that

∀ t ≥ 0, u(t) ∈ R(x(t))

and

for almost all t ≥ 0, ∥u′(t)∥ ≤ φ(x(t), u(t))

Let K ⊂ Dom(U) be a closed subset. We recall that a closed set-valued map R : K ∼ >Z is a feedback control regulating viable solutions to the control problem satisfying the above growth condition if and only if R is a solution to the contingent differential inclusion

∀ x ∈ K, 0 ∈ DR(x, u)(f(x, u)) − φ(x, u)B

satisfying the constraint

∀ x ∈ K, R(x) ⊂ U(x)

In particular, a closed graph single-valued regulation map r : K ↦ Z is a solution to the contingent differential inclusion

∀ x ∈ K, 0 ∈ Dr(x)(f(x, r(x))) − φ(x, r(x))B  (6.2)

satisfying the constraint

∀ x ∈ K, r(x) ∈ U(x)

Such solution can be obtained by a variational principle. We introduce the functional Φ defined by

Φ(r) := inf{c ≥ 0 | ∀ x, q, sup_{p∈Dr(x)^{∗}(q)}(⟨p, f(x, r(x))⟩ ≤ (φ(x, r(x)) + c)∥q∥)}

Theorem 6.1. Let R ⊂ C(K, Y) be a nonempty compact subset of selections of the set-valued map U (for the compact convergence topology). Suppose that the functions f and φ are continuous and that c := inf_{r∈R} Φ(r) < +∞. Then there exists a solution r(·) ∈ R to the contingent differential inclusion

∀ x ∈ K, 0 ∈ Dr(x)(f(x, r(x))) − (φ(x, r(x)) + c)B

As for the existence of such a feedback, we deduce from Theorem 5.1 the following consequence:

Theorem 6.2. Consider a nonempty convex subset R ⊂ C_{Λ}(K, Z) of selections of the set-valued map U which is compact in C_{Λ}(K, Z). Suppose that the functions f and φ are continuous and fix any family of closed convex processes D_{δ}r(x) ⊂ Dr(x). If for every r ∈ R, there exist v, w ∈ C(K, Y) such that ∀ x ∈ K,

w(x) ∈ D_{δ}r(x)(f(x, r(x))), v(x) ∈ φ(x, r(x))B & w(·) − v(·) ∈ T_{R}(r(·))

then there exists a solution r ∈ R to the contingent differential inclusion (6.2).

7. Appendix: Dual Characterization of the Viability Domain

Let F : X ∼ >X be a set-valued map and K ⊂ Dom(F) be a nonempty subset. We denote by

T_{K}(x) := \{v ∈ X | lim inf_{h→0+} \frac{d(x + hv; K)}{h} = 0\}

the contingent cone to K at x ∈ K. The set K is called a viability domain of F if

∀ x ∈ K, F (x) ∩ T_{K}(x) ≠ ∅

For every x ∈ K denote by

N_{K}^{P}(x) := {y − x | dist(y, K) = ∥y − x∥}

The following result provides a very useful duality characterization of viability domains: Proposition 7.1. Assume that the set-valued map F : K ∼ >X is upper semicontinuous with convex compact values. Then the four following properties are equivalent:

i) ∀ x ∈ K, F (x) ∩ T_{K}(x) ≠ ∅
ii) ∀ x ∈ K, F (x) ∩ \overline{co} (T_{K}(x)) ≠ ∅

iii) ∀ x ∈ K, ∀ p ∈ (T_{K}(x))^{−} , σ(F (x), −p) ≥ 0

iv) ∀ x ∈ K, ∀ p ∈ N_{K}^{P}(x), σ(F (x), −p) ≥ 0

Remark. The equivalence between i) and ii) was first noticed in a different context in [26]. A more simple proof of this fact was given in [13]. The equivalence of i) and iv) was proved in [36] and [19]. However the proof from [13] also implied that i) ⟺ iv). We provide it below for the reader convenience.

Proof. Clearly i) yields ii). The equivalence between ii) and iii) follows obviously from the Separation Theorem. Since y − x ∈ N_{K}^{P}(x) implies that y − x ∈ (T_{K}(x))^{−} we have iii) ⟹ iv). Assume that iv) holds true and fix x ∈ K. Let u ∈ F (x) and v ∈ T_{K}(x) achieve the distance between F (x) and T_{K}(x):

∥u − v∥ = inf_{y∈F (x), z∈T_{K}(x)} ∥y − z∥

and set w := \frac{u+v}{2}. We have to prove that u = v. Assume the contrary. Since v is contingent to K at x, there exist sequences h_{n} > 0 converging to 0 and v_{n} converging to v such that x + h_{n}v_{n} belongs to K for every n ≥ 0. We also introduce a projection of best approximation

x_{n} ∈ Π_{K}(x + h_{n}w) of x + h_{n}w onto K and we set z_{n} := \frac{x_{n} − x}{h_{n}}

so that h_{n}(w − z_{n}) ∈ N_{K}^{P}(x_{n}). By assumption iv),

∃ y_{n} ∈ F (x_{n}), such that ⟨w − z_{n}, y_{n}⟩ ≤ 0  (7.1)

Since x_{n} converges to x, the upper semicontinuity of F implies that for any ε > 0, there exists N_{ε} such that for n ≥ N_{ε}, y_{n} belongs to the neighborhood F (x) + εB, which is compact. Thus a subsequence (again denoted by) y_{n} converges to some element y ∈ F (x). We shall now prove that z_{n} converges to v. Indeed, inequality

∥w − z_{n}∥ = \frac{1}{h_{n}} ∥x + h_{n}w − x_{n}∥ ≤ \frac{1}{h_{n}} ∥x + h_{n}w − x − h_{n}v_{n}∥ = ∥w − v_{n}∥

implies that the sequence z_{n} has a cluster point and that every cluster point z of the sequence z_{n} belongs to T_{K}(x). Furthermore, every such z satisfies ∥w − z∥ ≤ ∥w − v∥. We now observe that v is the unique best approximation of w by elements of T_{K}(x). If not, there would exist p ∈ T_{K}(x) satisfying either ∥w − p∥ < ∥w − v∥ or p ≠ v and ∥w−p∥ = ∥w−v∥ = ∥w−u∥. In the latter case, we have ⟨u−w, w−p⟩ < ∥u−w∥∥w−p∥, since the equality holds true only for p = v. Each of these conditions together with the estimates

∥u − p∥^{2} = ∥u − w∥^{2} + ∥w − p∥^{2} + 2⟨u − w, w − p⟩ ≤ ∥u − v∥^{2}

imply the strict inequality ∥u − p∥ < ∥u − v∥, which is impossible since v is the projection of u onto T_{K}(x). Hence z = v. Consequently, all the cluster points being equal to v, we conclude that z_{n} converges to v. Therefore, we can pass to the limit in inequalities (7.1) and obtain, observing that w − v = (u − v)/2,

⟨u − v, y⟩ = 2⟨w − v, y⟩ ≤ 0 where y ∈ F (x)  (7.2)

Since F (x) is closed and convex and since u ∈ F (x) is the projection of v onto F (x), we infer that

⟨u − v, u − y⟩ ≤ 0  (7.3)

Finally, T_{K}(x) being a cone and v ∈ T_{K}(x) being the projection of u onto this cone, and in particular, onto the half-line vR_{+}, we deduce that

⟨u − v, v⟩ = 0  (7.4)

Therefore, properties (7.2, 7.3, 7.4) imply that

∥u − v∥^{2} = ⟨u − v, −v⟩ + ⟨u − v, u − y⟩ + ⟨u − v, y⟩ ≤ 0

and thus, that u = v.

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