Semigeodesics and the minimal time function

ESAIM: Control, Optimisation and Calculus of Variations

January 2006, Vol. 12, 120–138 DOI: 10.1051/cocv:2005032

SEMIGEODESICS AND THE MINIMAL TIME FUNCTION

Chadi Nour

Abstract. We study the Hamilton-Jacobi equation of the minimal time function in a domain which contains the target set. We generalize the results of Clarke and Nour [J. Convex Anal., 2004], where the target set is taken to be a single point. As an application, we give necessary and sufficient conditions for the existence of solutions to eikonal equations.

Mathematics Subject Classification. 49J52, 49L20, 49L25.

Received June 29, 2004. Accepted February 25, 2005.

Introduction

Let F be a multivalued function mapping IRⁿ to the subsets of IRⁿ. We assume throughout this paper that F satisfies the standing hypotheses; that is, F takes nonempty compact convex values, has closed graph, and satisfies a linear growth condition: for some positive constantsγandc, and for allx∈IRⁿ,

v∈F(x) =⇒ v ≤γx+c.

The multivalued functionF is also taken to be locally Lipschitz: everyx∈IRⁿadmits a neighborhoodU =U(x) and a positive constantK=K(x) such that

x₁, x₂∈U =⇒F(x₂)⊆F(x₁) +Kx₁−x₂B.¯ We associate withF the following functionh, thelower Hamiltonian:

h(x, p) := min{p, v:v∈F(x)}.

Now let S be a nonempty compact subset of IRⁿ, we denote byT(·, S) the well-known minimal time function associated to S (for the dynamicF). We recall that this function is defined as follows:

T(α, S) :=

⎧⎪

⎪⎨

⎪⎪

⎩

infT ≥0,

˙

x(t)∈F(x(t)) a.e. t∈[0, T], x(0) =α,

x(T)∈S.

Keywords and phrases. Minimal time function, Hamilton-Jacobi equations, viscosity solutions, minimal trajectories, eikonal equations, monotonicity of trajectories, proximal analysis, nonsmooth analysis.

1Computer Science and Mathematics Division, Lebanese American University, Byblos Campus, P.O. Box 36, Byblos, Lebanon;

cnour@lau.edu.lbThis paper was written while the author was an ATER at Institut Girard Desargues, Universit´e Lyon I.

c EDP Sciences, SMAI 2006

Article published by EDP Sciences and available at http://www.edpsciences.org/cocvor http://dx.doi.org/10.1051/cocv:2005032

If no trajectory between αandS exists, then T(α, S) := +∞. The minimal time function associated to S for the dynamic−F is denoted by T(S,·). We set

R^S₊:={α∈IRⁿ:T(S,α)<+∞},

the set of points attainable (in finite time) by trajectories beginning from S. For more information about the minimal time function, see for example [3, 5, 20, 22]. We also define the bilateral minimal time function, see [16, 17],T : IRⁿ×IRⁿ−→[0,+∞] as follows:

T(α, β) :=

⎧⎨

⎩

InfT ≥0,

˙

x(t)∈F(x(t))a.e. t∈[0, T], x(0) =α and x(T) =β,

(if no trajectory between α from β exists, then T(α, β) is taken to be +∞). We denote by R the effective domain ofT(·,·); that is,

R:={(α, β)∈IRⁿ×IRⁿ:T(α, β)<+∞}.

In this paper we study the following Hamilton-Jacobi equation:

1 +h(x, ∂_Pϕ(x)) = 0 ∀x∈ R^S₊, ϕ(S) = 0, (HJ_S)

where∂_Pis theproximal subdifferential. We recall that for a lower semicontinuous functionf : IRⁿ−→IR∪{+∞}

and a pointx∈domf :={x:f(x)<+∞}, ζ∈∂_Pf(x) if and only if there existsσ=σ(x, ζ)≥0 such that f(y)−f(x) +σy−x²≥ ζ, y−x,

for all y in a neighborhood of x. A solution of (HJ_S) means a lower semicontinuous function ϕ : R^S₊ −→

IR∪ {+∞}such thatϕ(S) = 0 and for everyx∈ R^S₊, for everyζ∈∂_Pϕ(x) (if any), we haveh(x, ζ) + 1 = 0 (we say thatϕ(x) is aproximal solution, see [8]). This is equivalent to the statement thatϕis a lower semicontinuous viscosity solution of the following Hamilton-Jacobi equation:

H(x,−ϕ(x))−1 = 0 ∀x∈ R^S₊, ϕ(S) = 0, whereH is theupper Hamiltonianassociated toF defined byH(x, p) := max

v∈F(x)p, v¹.

It is well-known that the minimal time function T(·, S) is a solution of (HJ_S) if we replace R^S₊ by R^S₊\S (see for example [1, 3, 7, 18, 22]), but it is never a solution onR^S₊ since for allα∈ S we have 0∈∂_PT(·, S)(α) and h(α,0) = 0. In [11], Clarke and Nour study the Hamilton-Jacobi equation (HJ_S) in the case S = {α0} (we denote this equation by (HJ_α

0))². Let us recall the principal result of [11]. We say that−F is α₀-STLC (α₀-small-time locally controllable) if and only if the minimal time functionT(α₀,·) is continuous atα₀. This is equivalent to: R^α₊⁰ is open,T(α₀,·) is continuous inR^α₊⁰ and for anyβ ∈∂R^α₊⁰ we have

α−→βlim T(α₀, α) = +∞.

We define G_α0 := {Γ ⊂ R^α₊⁰ : there exists a sequence β_i ∈ Γ such that T(α₀, β_i) −→ +∞}. The following theorem is proven in [11] (see [11], Ths. 5.2 and 5.3).

1See [4, 15, 18] for the definition of lower semicontinuous viscosity solutions. For an historical references about viscosity solution, see [12–14].

2WhenS={β}then in all our notations, we replaceSbyβ.

Theorem 0.1 (existence of solutions of (HJ_α0)). Assume that−F isα₀-STLC. ThenGα0 is nonempty and for Γ∈ Gα0 the function ϕ_Γ:R^S₊−→IR∪ {−∞,+∞} defined by:

ϕ_Γ(α) := lim inf

α−→α, β∈Γ T(α0,β)−→+∞

[T(α, β)−T(α₀, β)],

is a solution of (HJ_α0). Moreover, if we denote by ϕ_α0 the function ϕ_Γ corresponding to the choice R^α₊⁰ of Γ, thenϕ_α0 is the minimal solution of (HJ_α0).

After Theorem 0.1, Clarke and Nour studied in [11] the regularity of solutions and the linear case. They also proved an important relationship between these solutions and global geodesics trajectories. As in [22], the methods used by Clarke and Nour in [11] are based upon nonsmooth proximal monotonicity-invariance considerations developed in [10].

The purpose of this paper is the generalization of the results of [11] for a general compact target set S.

First, we show by an example (see Ex. 2.7) that if we only assume that −F is S-STLC (that is, T(S,·) is continuous) then we cannot guarantee the existence of solutions of (HJ_S) as in the case S = {α₀}. Then by adding another hypothesis (which is always satisfied if S is a single point), we show an existence theorem for (HJ_S) which generalizes Theorem 0.1. We also prove that this hypothesis is a necessary condition for the existence of solutions and then we generalize all the results of [11]. A number of examples (see Ex. 2.7 in which we prove the existence of solutions of (HJ_α0) not of the form presented in Th. 0.1)³ and some applications concerning the globaleikonal equations are also given throughout this paper.

The plan of the paper is as follows. In the next section we present some preliminaries. Several results concerning the existence and the regularity of solutions, and their relations with semigeodesics trajectories are presented in Section 2. In Section 3, we study the minimal solution of (HJ_S). The results of Sections 2 and 3 are applied to eikonal equations in Section 4. Finally, Section 5 is devoted to the existence of geodesics passing throughS.

1. Preliminaries 1.1. Notation and hypotheses

Forρ >0 we denote byB(0;ρ) :={x∈IRⁿ:x< ρ} and ¯B(0;ρ) :={x∈IRⁿ:x ≤ρ}. The open (resp.

closed) unit ball in IRⁿ is denoted byB (resp. ¯B). For a set A⊂IRⁿ, intA,∂A and coA are the interior, the boundary and the convex hull ofA, respectively.

Definition 1.1. Letλ∈]0,1]. We say that:

• F isS-LC (S-locally controllable), ifS⊂intR^S₊.

• F isS-STLC (S-small-time locally controllable), ifT(·, S) is continuous at every point of∂S.

• F satisfies the hypothesis (H_λ) atβ∈IRⁿ, if there existr >0 andδ >0 such that for anyβ∈B(β;r) and for any unit vectorγwe have

h(β, γ)< −δβ−β^1−λ λ

4·

We denote byR^S₋ :={α∈IRⁿ:T(α,S)<+∞},the set of points which can be steered toS in finite time. It is well-known (see [3, 17, 19]) that:

• F isS-LC⇐⇒ R^S₋ is open.

3This is an important question not treated in [11].

4The hypothesis (Hλ) is known by thePetrovλ-H¨older modulus condition.

• F isS-STLC⇐⇒ R^S₋ is open,T(·, S) is continuous inR^S₋ and for anyβ∈∂R^S₋ we have

α−→βlim T(α, S) = +∞.

• Forβ∈IRⁿ we have:

• F satisfies (H_λ) at β (λ∈]0,1]) =⇒F isβ-STLC.

• F satisfies (H₁) atβ ⇐⇒0∈intF(β) =⇒F isβ-STLC.

The basic hypotheses in force throughout the article are the following:

• S is a nonempty compact set.

• intS=∅.

• −F isS-LC (which is equivalent toR^S₊ is open).

We note that we must assume that intS =∅. This assumption is related to the nature of the semicontinuous solution chosen here. Indeed, if intS=∅, then a solutionϕof (HJ_S) vanishes on intS and then 0∈∂_Pϕ(x) for allx∈intS, which gives a contradiction sinceh(·,0) = 0.

1.2. Monotonicity of trajectories

Let Ω be an open subset of IRⁿand letϕ: IRⁿ−→IR∪ {+∞}be an extended real-valued function which is lower semicontinuous on Ω with domϕ∩Ω=∅. We say that the system (ϕ, F) isstrongly increasingon Ω if for any trajectoryxon an interval [a, b] for whichx([a, b])⊂Ω, we have

ϕ(x(s))≤ϕ(x(t)) ∀s, t∈[a, b], s≤t.

The system (ϕ, F) is said to beweakly decreasingon Ω if for everyα∈Ω there is a trajectoryxon a nontrivial interval [a, b] satisfying

x(a) =α, ϕ(x(t))≤ϕ(α) ∀t∈[a, b];

by reducing b if necessary we may also arrange to havex([a, b])⊂Ω. In each case, one obtains an equivalent definition by requiring the inequality to hold on [a, τ[, where τ ∈]a,+∞] is theexit time of the trajectoryx from Ω: the supremum of allT >0 having the property thatx([a, T])⊂Ω. The following proposition is proven in [10], Chapter 4, Section 6. See also [2] and [9].

Proposition 1.2. The system (ϕ, F) is strongly increasing on Ωif and only if h(x, ∂_Pϕ(x))≥0 ∀x∈Ω,

and weakly decreasing on Ωif and only if

h(x, ∂_Pϕ(x))≤0 ∀x∈Ω.

2. Existence of solutions

We begin this section by the following proposition which gives some properties of a solution of (HJ_S). The proof follows using Proposition 1.2. The details is left to the reader (see the proof of [11], Prop. 4.4).

Proposition 2.1. Let ϕa solution of(HJ_S). Then we have:

(i) T(α, β) +ϕ(β)≥ϕ(α), for allα,β ∈ R^S₊.

(ii) T(α, S)≥ϕ(α)≥ −T(S, α), for allα∈ R^S₊ (and thenR^S₊∩ R^S₋ ⊂domϕ).

(iii) For every α∈ domϕthere exists a trajectoryxof F such thatx(0) =αand ϕ(x(t)) +t=ϕ(α) ∀t≥0.

Now let α∈IRⁿ. A trajectoryx: [0,+∞[−→IRⁿ ofF is a semigeodesic fromα if and only if x(0) = αand T(x(s), x(t)) = t−s for all s ≤ t ∈ [0,+∞[. If α ∈ S, then we say that x is a semigeodesic from S. The following proposition presents the relationship between a solution of (HJ_S) and semigeodesic trajectories.

Proposition 2.2. Let ϕbe a solution of(HJ_S). Then for every α∈domϕthere exists a semigeodesic xfrom αsuch that

ϕ(x(t)) +t=ϕ(α) ∀t≥0.

Moreover, if α∈S then we have T(S, x(t)) =t∀t≥0.

Proof. Let ϕ be a solution of (HJ_S) and let α ∈ domϕ. By Proposition 2.1, there exists a trajectory x : [0,+∞[−→IRⁿ ofF in R^S₊ such thatx(0) =αand

ϕ(x(t)) +t= 0 ∀t≥0. (1)

We claim thatxis a semigeodesic fromα. Indeed, lets≤t∈[0,+∞[, then by (1) and Proposition 2.1 we have T(x(s), x(t))≥ϕ(x(s))−ϕ(x(t)) =t−s,

but

T(x(s), x(t))≤t−s, thereforeT(x(s), x(t)) =t−s.

Now we assume thatα∈S. Then by (1) and Proposition 2.1 we get that t=T(α, x(t))≥T(S, x(t))≥ −ϕ(x(t)) =t ∀t≥0.

Hence T(S, x(t)) =t ∀t≥0.

As mentioned in the introduction, Clarke and Nour proved in [11] that ifS is a single point, then under the hypothesis “−F isS-STLC” the equation (HJ_S) always admits a solutions. In the following example we prove, using Proposition 2.2, that this fails in general ifSis not a single point. Thus to prove the existence of solutions we must add another hypothesis.

Example 2.3. We consider the same data of [16], Example 7.8; that is forn= 2, we defineF as the following:

• If(x, y) ≥2, thenF(x, y) :=F₁(x, y) where

F₁(x, y) :=

⎧⎪

⎨

⎪⎩

x²−y² x²+y², 2xy

x²+y²

ify= 0, {(1,0)} ify= 0.

• If(x, y) ≤1, thenF(x, y) :=F₂(x, y) whereF₂(x, y) := ¯B for all (x, y)∈IR².

• If 1<(x, y):=r <2, then

F(x, y) :={(2−r)v₂+ (r−1)v₁:v₁∈F₁(x, y) and v₂∈F₂(x, y)}.

We take S := [−¹₂,0]× {0}. Then we can easily verify that all our hypotheses are satisfied and that −F is S-STLC. Moreover, we have the following⁵:

• R^S₊= IR²\]− ∞,−2]× {0}.

• For all (a,0)∈Sthere exists only one semigeodesicx: [0,+∞[−→IR²from (a,0) which is the following trajectory: x(t) = (t+a,0).

5The proof of these claims follows using the same ideas as in [11], Example 7.8.

Now we assume that the Hamilton-Jacobi equation (HJ_S) admits a solutionϕ; we shall derive a contradiction.

By Proposition 2.2 there exists a semigeodesic (it must be the trajectoryx defined above) from (−¹₂,0) such that

T(S, x(t)) =t ∀t≥0.

But this contradicts the fact that

x(t)∈S ∀t∈ 0,1 2

· Now we define the following set of subsets ofR^S₊:

GS:={Γ⊂ R^S₊ : for allα∈S there exists a sequenceβ_i∈Γ such thatT(α, β_i) =T(S, β_i)−→+∞}. Then we have the following theorem which generalizes Theorem 0.1.

Theorem 2.4 (existence of solutions of (HJ_S)). Assume that −F isS-STLC and thatGS is nonempty. Then for Γ∈ GS the functionϕ_Γ:R^S₊−→IR∪ {−∞,+∞}defined by:

ϕ_Γ(α) := lim inf

α−→α, β∈Γ T(S,β)−→+∞

[T(α, β)−T(S, β)],

is a solution of (HJ_S). Moreover, if we denote by ϕ_S the function ϕ_Γ corresponding to the choice R^S₊ of Γ, thenϕ_S is the minimal solution of (HJ_S).

Proof. We proceed exactly as in the proofs of [11], Theorems 5.2 and 5.3. The only difference here is that we

must use the definition of Γ∈ G_S to prove thatϕ_Γ(S)≤0.

Remark 2.5. IfS={α₀}and−F isα₀-STLC then the hypothesis “G_S is nonempty” is always satisfied since in this case we have thatT(S,·) =T(α₀,·).

In the following corollary we prove that the hypothesis “GS is nonempty” is also a necessary condition for the existence of solutions.

Corollary 2.6. Assume that−F isS-STLC. Then the following statements are equivalent:

(i) The Hamilton-Jacobi equation (HJ_S)admits a solution.

(ii) For every α∈S there exists a semigeodesicxfromαsuch that T(S, x(t)) =t∀t≥0.

(iii) The setGS is nonempty.

(iv) The Hamilton-Jacobi equation (HJ_S)admits a minimal solution.

Proof. Follows from Proposition 2.2 and Theorem 2.4.

Now we give an example in which we prove, using Theorem 2.4, that (HJ_S) may admit a solution not of the formϕ_Γ presented in Theorem 0.1.

Example 2.7. Forn= 1, letF(x) =−x+ [−1,1]. It is easy to prove thatR=R₁∪ R₂ where

• R₁={(x, y)∈IR×IR : −1< y≤x};

• R₂={(x, y)∈IR×IR : x≤y <1}.

We calculateT(·,·) and we find that:

T(x, y) =

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩ ln

1 +x 1 +y

if (x, y)∈ R₁, ln

1−x 1−y

if (x, y)∈ R₂.

In this example we haveh(x, p) =−x, p − p and then the Hamilton-Jacobi equation (HJ_S) is x, ∂_Pϕ(x)+∂_Pϕ(x)= 1 ∀x∈IR, ϕ(S) = 0.

We consider four cases ofS (in all these cases, we have that−F isS-STLC):

• S₁= 1

2

:

Clearly we have R₊¹² =]−1,1[. Moreover, there exist only two semigeodesics from ¹₂ which are:

(i) x(t) = 1−¹₂e^−t,t≥0.

(ii) y(t) =³₂e^−t−1,t≥0.

Let us calculate all the solutions of (HJ1

2) of the formϕ_Γ. Let Γ∈ G¹₂, then there exist three cases:

(a) Γ contains a sequence which converges to 1 and does not contain a sequence which converges to−1.

Then we find the following solution of (HJ1 2):

ϕ_Γ1(x) = ln(2) + ln(1−x).

(b) Γ contains a sequence which converges to -1 and does not contain a sequence which converges to 1.

Then we find the following solution of (HJ1 2):

ϕ_Γ2(x) =−ln 3

2

+ ln(1 +x).

(c) Γ contains a sequence which converges to −1 and a sequence which converges to 1. Then we find the minimal solution of (HJ1

2):

ϕ1

2(x) =−T 1

2, x

=

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

ln(2) + ln(1−x) ifx∈ 1 2,1 ,

−ln 3

2

+ ln(1 +x) ifx∈

−1,1 2

·

• S₂=

−1 2

:

As in the preceding case we haveR⁻₊¹² =]−1,1[ and there exist only two semigeodesics from−1 2 which are:

(i) x(t) = 1−3

2e^−t, t≥0.

(ii) y(t) =1

2e^−t−1,t≥0.

The solutions of (HJ₋1

2) of the formϕ_Γ are the following:

(a) ϕ_Γ3(x) = ln(1−x)−ln₃

2

.

(b) ϕ_Γ4(x) = ln(2) + ln(1 +x).

(c) ϕ₋1

2(x) =−T

−₂¹, x

=

−ln₃

2

−ln(1−x) ifx∈

−¹₂,1 , ln(2) + ln(1 +x) ifx∈

−1,−¹₂ .

• S₃=

−1 2,1

2

:

Clearly we have R^S₊³ =]−1,1[. We claim that G_S3 =∅. Indeed, forα = ¹₂ the semigeodesicx(t) = 1−¹₂e^−t satisfiesT(S₃, x(t)) = t ∀t ≥0, and for α =−¹₂ the semigeodesicy(t) = ¹₂e^−t−1 satisfies T(S₃, y(t)) =t∀t≥0. Then by Theorem 2.4, the Hamilton-Jacobi equation (HJ_S3) admits a minimal

solutionϕ_S3. Using Theorem 2.4, we calculate this solution and we find that

ϕ_S3(x) =

ln(2) + ln(1−x) ifx∈ [0,1[, ln(2) + ln(1 +x) ifx∈]−1,0].

Butϕ_S3 is also a solution of (HJ1

2), then (HJ1

2) admits a solution not of the formϕ_Γ (all the solutions of the formϕ_Γ calculated above are different ofϕ_S3).

• S₄=

−1 2,0,1

2

:

In this case we also have thatR^S₊⁴ = ]−1,1[. We claim thatG_S4 =∅(and then (HJ_S4) does not admit any solution). Indeed, we cannot find any semigeodesicz from 0 which satisfiesT(S₄, z(t)) =t for all t≥0.

The following proposition presents how we can calculate a solution of (HJ_S) using the solutions of (HJ_α) for allα∈S.

Proposition 2.8. We have the following statements:

(a) Let α∈S and assume that S⊂ R^α₊ and that−F isα-LC. We consider the following statements:

(i) There exists a solutionψ_α of (HJ_α)such that ψ_α(S)≥0.

(ii) There exists a semigeodesic xfromαsuch that T(S, x(t)) =t ∀t≥0.

Then (i) =⇒(ii)and the reverse implication is true if we add that−F isβ-STLCfor all β∈S.

(b) Assume thatS×S⊂ Rand that−F isS-STLC. Assume further that for all α∈S there exists a solution ψ_α of (HJ_α) such that ψ_α(S) ≥ 0. Then G is a nonempty set. Moreover, the function ϕ¯ : R^S₊ −→

IR∪ {−∞,+∞} defined by

¯

ϕ(x) := lim inf

x−→xϕ(x), where ϕ(x) := inf

α∈Sϕ_α(x), is a solution of (HJ_S).

Proof. (a) (i) =⇒(ii): We apply Proposition 2.2 (for S ={α}) and we get that for allα∈ S there exists a semigeodesicx_α fromαsuch that

ψ_α(x_α(t)) +t= 0 ∀t≥0.

We claim thatT(S, x_α(t)) =t for allt≥0. Indeed, letβ ∈S, then by Proposition 2.1 we have T(β, x_α(t)) +ψ_α(x_α(t))≥ψ_α(β)≥0 ∀t≥0.

Hence

T(β, x_α(t))≥ −ψ_α(x_α(t)) =t ∀t≥0, which gives (sinceT(α, x_α(t)) =tfor allt≥0) thatT(S, x_α(t)) =tfor allt≥0.

Now we assume further that −F is β-STLC for all β ∈ S. Let us prove that (ii) =⇒ (i). We consider Γ :={x(t) :t≥0}and we denote byψ_αthe solution of (HJ_α) corresponding to Γ. Letβ∈S. Then we have

0≤T(β, x(t))−T(α, x(t))≤T(β, β) +T(β, x(t))−T(α, x(t)) ∀β ∈ R^α₊, hence

T(β, x(t))−T(α, x(t))≥ −T(β, β) ∀β∈ R^α₊, which gives by the definition of ψ_α and by the continuity ofT(β,·) thatψ_α(β)≥0.

(b) Since S×S ∈ Rwe have that the open set R^S₊ =R^α₊ for allα∈S. Then by (a) we get thatG =∅. Now let α∈S and let β ∈ R^S₊. We considerθ ∈S such that T(θ, β) = T(S, β). By Proposition 2.1 (applied for S={α}) we have

T(S, β) +ψ_α(β) =T(θ, β) +ψ_α(β)≥ψ_α(θ)≥0, then

ψ_α(β)≥ −T(S, β).

Hence for allα∈S we haveψ_α(·)≥ −T(S,·) which gives (by the continuity ofT(S,·)) that

−T(S,·)≥ϕ(¯ ·)≥ϕ(·).

Therefore ¯ϕ(·)>−∞and it is lower semicontinuous. Moreover, ¯ϕ(S) = 0 since for allβ∈S we have 0 =−T(S, β)≥ϕ(β¯ )≥ϕ(β) = inf

α∈Sϕ_α(β) = 0.

Now we prove that 1 +h(x, ∂_Pϕ(x)) = 0 for all¯ x∈ R^S₊. It is sufficient to prove thatt+ ¯ϕis weakly decreasing and strongly increasing. We begin by strongly increasing.

Letx: [a, b]−→IRⁿ be a trajectory ofF such thatx([a, b])⊂ R^S₊ and lett∈[a, b]. We need to prove that t+ ¯ϕ(x(t))≤b+ ¯ϕ(b).

We can assume thatx(b)∈dom ¯ϕ. Then there exists a sequenceβ_i−→β such that

¯

ϕ(x(b)) = lim

i−→+∞ϕ(β_i).

By the continuous dependence on the initial condition (see [10], Th. 4.3.11) there exists a sequence of trajectories x_i on [t, b] such thatx_i(b) =β_i and

i−→+∞lim x_i(t) =x(t).

Clearly the trajectory x_i remains in R^S₊ (since it begins in R^S₊), then since ψ_α is a solution of (HJ_α) for all α∈S, we have

t+ψ_α(x_i(t))≤b+ψ_α(x_i(b)) ∀α∈S.

Hence

t+ϕ(x_i(t))≤b+ϕ(x_i(b))≤b+ ¯ϕ(x_i(b)).

Takingi−→+∞we get

t+ ¯ϕ(x(t))≤b+ ¯ϕ(x(b)).

The strong increase follows.

Now we prove the weak decrease property. Letβ ∈dom ¯ϕ, then there exists a sequenceβ_i −→β such that

¯

ϕ(x(b)) = lim

i−→+∞ϕ(β_i).

By the definition ofϕwe have that for allithere exists α_i∈S such that ψ_αi(β_i)≤ϕ(β_i) +1

i· (2)

Sinceψ_αiis a solution of (HJ_αi) for alli, there exists a sequence of trajectoriesx_ion [0,+∞[ such thatx_i(0) =β_i and

ψ_αi(β_i)≥t+ψ_αi(x_i(t)) ∀t≥0. (3)

By the compactness property of trajectories, we can assume that there exists a trajectory xon [0,+∞[ such that x(0) =β andx_i−→xuniformly on compact interval. By (2) and (3) we have

ϕ(β_i) +1

i ≥ψ_αi(β_i)≥t+ψ_αi(x_i(t))≥t+ϕ(x_i(t)) ∀t≥0.

Takingi−→+∞we get

¯

ϕ(β)≥t+ ¯ϕ(x(t)) ∀t≥0.

The weak decrease follows.

Example 2.9. We return to Example 2.7 and we remark that

• ϕ_Γ1

−1 2

= ln(3)>0,

• ϕ_Γ4 1

2

= ln(3)>0.

Then by Proposition 2.8 (clearly forS₃=

−¹₂,¹₂

we haveS₃×S₃⊂ R) we get that the function min{ϕ_Γ1, ϕ_Γ4} is a solution of (HJ_S3). If we calculate this function we find exactly the minimal solutionϕ_S3.

Now we study the regularity of solutions.

Proposition 2.10. Let ϕbe a solution of (HJ_S)and letα∈domϕ. Then we have:

(i) F isα-STLC =⇒ϕis continuous at α.

(ii) F satisfies (H_λ)atα(λ∈]0,1]) =⇒ϕisλ-H¨older continuous nearα.⁶ Proof.

(i) Follows exactly as the proof of (1) of [11], Proposition 6.1.

(ii) We proceed as in the proof of (2) of [11], Proposition 6.1, and we use [17], Proposition 4.4, which asserts the existence of ρ >0 such thatT(·,·) isλ-H¨older continuous inB(α, ρ)×B(α, ρ).

Proposition 2.11. We have the following statements:

(i) Assume that F isβ-STLC for allβ∈ R^S₊∩ R^S₋. Then all solutions of (HJ_S)are continuous in the open set R^S₊∩ R^S₋.

(ii) Assume that F satisfies (H_λ) atβ for all β ∈ R^S₊∩ R^S₋ ((λ ∈]0,1])). Then all solutions of (HJ_S) are locally λ-H¨older continuous in the open setR^S₊∩ R^S₋.

Proof. SinceS⊂ R^S₊∩ R^S₋, we have that F isS-STLC in 1) and 2). ThenR^S₋ is open and henceR^S₊∩ R^S₋ is open. By Proposition 2.1 we haveR^S₊∩ R^S₋⊂domϕfor allϕa solution of (HJ_S). Then by Proposition 2.10

we find the two results.

3. The minimal solution

In this section we present the relationship between the minimal solution ϕ_S and the function−T(S,·). The semigeodesicsxfromS which satisfiesT(S, x(t)) =tfor allt≥0, will play an important role in this study. We note that in all this section, we assume that−F isS-STLC and thatGS =∅.

Theorem 3.1. Letx: [0,+∞[−→IRⁿbe a trajectory ofF fromS. Then the following statements are equivalent:

(i) The trajectory xis a semigeodesic fromS which satisfiesT(S, x(t)) =t for allt≥0.

(ii) For all t≥0, we haveϕ_S(x(t)) +t= 0.

6Ifλ= 1 then we obtain that 0∈intF(α) =⇒ϕis Lipschitz nearα.

Proof. To prove the implication (i) =⇒(ii) and the first part of the implication (ii) =⇒(i) (xis a semigeodesic from S) we proceed exactly as in the proof of [11], Theorem 7.4. Let us prove the second part of (ii) =⇒(i), that is,T(S, x(t)) =tfor allt≥0. Letα∈S, then by Proposition 2.1 we have

T(α, x(t)) +ϕ_S(x(t))≥ϕ_S(α) = 0 ∀t≥0, hence

T(α, x(t))≥ −ϕS(x(t)) =t ∀t≥0.

This gives that

t≤T(S, x(t))≤T(x(0), x(t)) =t ∀t≥0.

The proof is completed.

The following theorem gives a necessary and sufficient condition forϕ_S(α) to be −T(S, α).

Theorem 3.2. Let α∈ R^S₊. Then the following statements are equivalent:

(i) The point αlies on a semigeodesic fromS which satisfies T(S, x(t)) =t for allt≥0.

(ii) ϕ_S(α) =−T(S, α).

Proof. See the proof of [11], Theorem 7.6.

Remark 3.3. In Example 2.7 if we calculate the function −T(S₃,·) (S₃ = {−1 2,1

2}) then we obtain the following:

−T(S₃, x) =

⎧⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎩

ln(2) + ln(1−x) ifx∈ 1 2,1 ,

−ln 3

2

−ln(1 +x) ifx∈ 0,1 2

,

−ln 3

2

+ ln(1−x) ifx∈ −1 2,0

, ln(2) + ln(1 +x) ifx∈

−1,−1 2

·

We remark that ϕ_S3(·) and −T(S₃,·) coincide only on ]−1,−¹₂]∪[¹₂,1[. We can easily deduce this from Theorem 3.2. Indeed, for all α ∈]−¹₂,¹₂[ we can not find any semigeodesic from S₃ passing through α and which satisfiesT(S₃, x(t)) =tfor allt≥0, but if we takeα∈[¹₂,1[ (resp. α∈]−1,−¹₂]) then the semigeodesic x(t) = 1−¹₂e^−t(resp. x(t) = ¹₂e^−t−1) begins fromS₃, passes throughαand satisfiesT(S₃, x(t)) =tfor allt≥0.

We also note that in this example we haveϕ1

2(·) =−T(¹₂,·) andϕ₋1

2(·) =−T(−¹₂,·) butϕ_S3(·)=−T(S₃,·).

We recall (see [11]) that a continuous function ϕ is said to be mildly regular at a point x if it satisfies

∂^Pϕ(x)⊂∂_Lϕ(x), where∂^Pϕ(x) is theproximal superdifferentialofϕatxdefined (for an upper semicontinuous function) by ∂^Pϕ(x) := −∂P(−ϕ)(x) and∂_Lϕ(x) is the limiting subdifferential of ϕ at xdefined (for a lower semicontinuous function) by

∂_Lϕ(x) :={limξ_i :ξ_i ∈∂_Pϕ(x_i), x_i−→xandϕ(x_i)−→ϕ(x)}.

For more informations about these definitions, see [10]. The following proposition gives a sufficient conditions for a continuous function to be mildly regular at a pointx. For the proof, see [11], Proposition 6.3.

Proposition 3.4. Let ϕ be continuous in a neighborhood of a point x and assume that one of the following condition holds:

(1) ϕis regular atx⁷; (2) ϕis differentiable atx;

(3) ∂_Pϕ(x)is nonempty.

Then ϕis mildly regular atx.

Before giving some applications for the mildly regular property, let us give some geometric definitions. LetA be a nonempty closed subset of IRⁿ. Forα∈IRⁿ, we denote by proj_A(α) the set of closest points ofαontoA (that is, the set of pointsβ ∈A such thatd_A(α) =α−β). Now letβ ∈A. We define theproximal normal conetoA atβ by

N_A^P(β) :={t(α−β) :t≥0 andβ∈proj_A(α)}.

We also define thegeneralized exterior normalto Aatβ by N_A(β) :=

ξ

ξ :ξ∈N_A^P(β)\ {0}

.

For more information about these definitions, see [10, 21]. The following theorem gives a necessary condition for−T(S,·) to be the minimal solution of (HJS).

Theorem 3.5. Assume that the functionT(S,·)is mildly regular onR^S₊\S. Assume further thatdomN_S^P :=

{s∈S :N_S^P(s)={0}} =∂S ⁸. Then the minimal solution ϕ_S coincides with −T(S,·) and then every point of R^S₊ lies on a semigeodesic xfromS which satisfies T(S, x(t)) =t for allt≥0.

Proof. It is sufficient to prove that−T(S,·) is a solution of (HJ_S). By the definition, we have−T(S, x) = 0 for allx∈S. Let us show that −T(S,·) satisfies the Hamilton-Jacobi equation of (HJ_S). Letα∈ R^S₊, then there exist two cases.

Case 1. α∈S.

Letζ∈∂_P(−T(S,·))(α). Then−ζ∈∂^PT(S,·)(α)⊂∂_LT(S,·)(α), since T(S,·) is mildly regular atα. But it is well-known that we have

1 +h_−F(α, ∂_PT(S,·)(α)) = 0⁹.

Hence since h_−F is continuous and ∂_L is constructed from ∂_P by a limiting process we get that 1 +h_−F(α,−ζ) = 0, and then 1 +h(α, ζ) = 0.

Case 2. α∈S.

We claim that ∂_P(−T(S,·))(α) = ∅. Indeed, if not then since 0∈∂_PT(S,·)(α) we get thatT(S,·) is differen- tiable at α and ∂_PT(S,·)(α) ={0}. But by [22], Theorem 5.1, we have∂_PT(S,·)(α) = N_S^P(α)∩ {ζ ∈ IRⁿ :

h(x, ζ)≥ −1},then sinceN_S^P(α)={0} we find a contradiction.

Remark 3.6. We remark that in the preceding proof, we proved directly that−T(S,·) is a (minimal) solution of (HJ_S) and then we can eliminate the hypothesisG_S =∅in Theorem 3.5.

Corollary 3.7. Let F admit a representation of the form F(x) = {Ax+u : u∈ U}, where A is an n×n matrix andU is a compact and convex set. Assume further that we have the following hypotheses:

• S is a convex set;

• for allα∈S and for all unit vectorγ∈N_S^P(α), we haveh(α, γ)<0¹⁰.

7ϕis regular in the sense of Clarke, see [10].

8A sufficient condition for domN_S^P =∂S(=Ssince intS=∅) is for exampleS=∂A, whereAis a compact and convex set.

9This is a well-known characterization of the minimal time function but here applied for the dynamic−F. We note thath−F

is the lower Hamiltonian for the dynamic−F.

10This condition reduces to 0∈intF(α) ifS={α}. For more informations about this hypothesis see [3].

Then ϕ_S is semiconcave and coincides with −T(S,·), and then every point of R^S₊ lies on a semigeodesic x fromS which satisfiesT(S, x(t)) =t for allt≥0.

Proof. Follows from Theorem 3.5 and using [5], Theorem 4.1, which asserts that in this case the functionT(S,·) is semiconvex onR^S₊and then mildly regular onR^S₊(for more informations about semiconcave and semiconvex

functions, see [6]).

4. Eikonal equations

In this section we apply the results of the preceding sections to eikonal equations. We consider the following globaleikonal equation:

∂_Pϕ(x)= 1 ∀x∈IRⁿ, ϕ(S) = 0. (e_S)

We remark that (e_S) is exactly the Hamilton-Jacobi equation (HJ_S) corresponding to the choice ¯B ofF (that is,F(x) = ¯B for allx∈IRⁿ). Clearly in this case we have:

• F and−F areβ-STLC for allβ ∈IRⁿ (since 0∈intF(β)).

• R= IRⁿ×IRⁿandT(α, β) =α−βfor all (α, β)∈IRⁿ×IRⁿ.

• T(S,·) =T(·, S) =d_S(·), whered_S(·) is the distance function associated toS.

• The minimal trajectory between two points αandβ is exactly the segment [α, β].

• Ifϕis a solution of (e_S) then domϕ= IRⁿ. Then we have the following corollary.

Corollary 4.1. The following statements are equivalent:

(i) The equation(e_S)admits a solution.

(ii) For all α∈S there exists a unit vectorλ∈IRⁿ such that d_S(α+tλ) =t ∀t≥0.

(iii) S⊂∂(coS).

Proof. The equivalence (i)⇐⇒ (ii) follows from Corollary 2.6 and the equivalence (ii)⇐⇒ (iii) follows from the following lemma (we omit the proof):

Lemma 4.2. Let α∈S. Then the following assertions are equivalent.

• There exists a unit vectorλ∈IRⁿ such that d_S(α+tλ) =t ∀t≥0.

• α∈∂(coS).

Example 4.3. We taken= 2, and we consider the following data:

• F(x, y) := ¯B for all (x, y)∈IR².

• α:= (0,1),β:= (1,0) and γ:= (−1,0).

• S:= [α, β]∪[α, γ], where [α, β] (resp. [α, γ]) is the segment joiningαtoβ (resp. αtoγ).

Clearly we haveS⊂∂(coS). Then by Corollary 4.1, the Hamilton-Jacobi equation (HJ_S) admits a solutions.

Let us calculate a solution using Proposition 2.8. Let (a, b)∈[α, γ]. We denote byz the exterior perpendicular to [α, γ] at (a, b) (see Fig. 1). Clearly z is a semigeodesic from (a, b) which satisfiesT(S, z(t)) =t ∀t≥0. We calculatez and we find that

z(t) =

−

√2 2 t+a,

√2 2 t+b

t≥0.

Now let Γ₁:={z(t) :t≥0}, then by Proposition 2.8 we have thatϕ_Γ1 is a solution of (HJ_(a,b)) which satisfies ϕ_Γ1(S)≥0. We calculate this function (using its definition) and we obtain that

ϕ_Γ1(x, y) =

√2

2 (x−y+ 1) ∀(x, y)∈IR².

½

½

´ µ

½

¼

´µ

Figure 1. Example 4.3.

We remark that this function does not depend on (a, b) and then it is a solution of (HJ_(a,b)) for all (a, b)∈[α, γ].

Using the same idea, the following function

ϕ_Γ2(x, y) :=

√2

2 (−x−y+ 1),

is a solution of (HJ_(a,b)) for all (a, b)∈[α, β]. Moreover as forϕ_Γ1, we haveϕ_Γ2(S)≥0. Then by Proposition 2.8, the functionϕ:= min{ϕΓ1, ϕ_Γ2}is a solution of (HJ_S).

We note that at the pointα, there exist several semigeodesicsz(t) satisfyingT(S, z(t)) =t∀t≥0, and then we can construct (using the preceding algorithm) several solutions of (HJ_S).

Now we characterize, using the result of Section 3, the minimal solution of (e_S).

Proposition 4.4. Assume that S ⊂ ∂(coS) and let P_S := {α ∈ IRⁿ\S : proj_(coS)(α)∩S = ∅}. Then ϕ_S(α) =−d_S(α)for all α∈P_S. Moreover, for α∈P_S andβ ∈proj_(coS)(α)∩S (β is unique) if N_(coS)(β)is a singleton then for all ϕa solution of(e_S), we haveϕ(α) =−d_S(α).

Proof. Let α ∈ P_S. Then there exists (a unique) β ∈ S such that d_(coS)(α) = d_S(α) = α−β. We set λ:= _α−β^α−β and we consider the trajectoryx(t) :=β+λtfor allt≥0. Clearly the trajectoryxis a semigeodesic fromS passing throughα. Moreover, we can easily prove that

d_(coS)(x(t)) =d_S(x(t)) =t ∀t≥0.

ButT(S,·) =T(·, S) =d_S(·), then by Theorem 3.2 we get that ϕ_S(α) =−T(S, α) =−d_S(α).

Now we take α∈ P_S and we assume that for (the unique)β ∈proj_(coS)(α)∩S we have that N_(coS)(β) is a singleton. Let ϕbe a solution of (e_S). We considerλand x(·) as above. Sinceβ ∈S and by Proposition 2.2 there exists a semigeodesicy(t) =β+λt(λ= 1) formβ such that

ϕ(y(t)) +t= 0 and d_S(y(t)) =T(S, y(t)) =t ∀t≥0.

Using the fact that d_S(y(t)) = t for all t ≥0, we can prove that λ ∈ N_{co S}(α). Then λ = λand this gives that ycoincides with x. Hence

ϕ(β+λt) +t= 0 ∀t≥0.

Takingt=α−β, we get that

ϕ(α) =−α−β=−dS(α).

Theorem 4.5. Assume that S=∂(coS). Then we have the following:

(i) If int (coS) =∅then ϕ_S(·) =−d_S(·).

(ii) If int (coS)=∅then

ϕ_S(α) =

⎧⎨

⎩

−dS(α) if α∈IRⁿ\coS ,

0 if α∈S, (∗)

d_S(α) if α∈int (coS).

Proof.

(i) Follows from Corollary 3.7.

(ii) By Proposition 4.4, we have that ϕ_S(·) =−d_S(·) on IRⁿ\coS. Let us prove that ϕ_S(·) = d_S(·) on int (coS). Letα∈int (coS), then by Proposition 2.2 there exists a semigeodesicx(t) =α+tλ(λ= 1) fromαsuch that

ϕ_S(x(t)) +t=ϕ_S(α) ∀t≥0.

Clearly there existst₀>0 such thatx(t₀) =β ∈S. Then

ϕ_S(β) +t₀=ϕ_S(α).

But we can easily prove thatt₀=α−β, hence

α−β=ϕ_S(α).

On the other hand and by Proposition 2.2, we have thatϕ_S(α)≤d_S(α)≤ α−β.This gives thatϕ_S(α) =d_S(α)

which completes the proof.

Corollary 4.6. Assume thatS=∂(coS)and thatN_(coS)(β)is a singleton for allβ ∈S. Then the functionϕ_S defined in (∗)is the unique solution of (e_S).

Proof. This follows from Theorem 4.5 and from the fact that if N_(coS)(β) is a singleton for all β ∈ S, then

int (coS)=∅.

Remark 4.7. In Theorem 4.5, we can prove that ϕ_S(·) =d_S(·) on int (coS) using the following known result (see [3]):

Theorem 4.8. Let Ωbe a nonempty bounded open set ofIRⁿ. Then the functiond_∂Ω(·)is the unique solution of the following eikonal equation

∂_Pϕ(α)= 1 ∀α∈Ω, ϕ= 0 on ∂Ω.