SIAM J. CONTROL OPTIM.
Vol. 35, No. 5, pp. 1638–1652, September 1997
c 1997 Society for Industrial and Applied Mathematics
011
ON THE REGULARITY OF SEMIPERMEABLE SURFACES
IN CONTROL THEORY WITH APPLICATION
TO THE OPTIMAL EXIT-TIME PROBLEM (PART I)∗
PIERRE CARDALIAGUET†
Abstract. In control theory, a semipermeable surface is an (in general nonsmooth) oriented
surface that, on one hand, contains solutions (the so-called barrier solutions) of the controlled system
and, on the other hand, may be crossed by the solutions of this system in only one direction. Without
making any assumption on the regularity of the boundary of the semipermeable surface, we show
that the barrier solutions contained in this semipermeable surface satisfy the Pontryagin principle,
that this surface is a Lipschitz manifold, and that it is, locally, the graph of a semiconcave function.
Applying these results to the optimal exit-time function from a given open set yields, without any
controllability assumption at the boundary of the open set, that this function is semiconcave on an
open dense subset of its domain.
Key words. semipermeable surfaces, differential inclusion, viability theory, minimal time function
AMS subject classifications. 49J24, 49J52, 49N60
PII. S0363012995287295
Introduction. Let
(1)
x′ (t) = f (x(t), u(t)), u(t) ∈ U,
x(0) = x0
be a controlled system with a hamiltonian defined by
H(x, p) :=
inf hf (x, u), pi.
u∈U (x)
A smooth, semipermeable surface is an oriented hypersurface S such that the outward
normal p at each point x ∈ S satisfies H(x, p) = 0. Such a surface S is called
semipermeable because
(α) S can be crossed in only one direction by the trajectories of the controlled
system.
(β) From any initial position x0 ∈ S at least one solution x(·) of (1) starts, and
remains locally on S (namely, ∃τ > 0 such that, ∀t ∈ [0, τ ], x(t) ∈ S).
A solution satisfying condition (β) is called a barrier solution.
In many problems, one encounters closed sets (which are not necessarily smooth)
with a boundary enjoying properties (α) and (β). We still say that their boundary
is “a semipermeable surface.” The aim of this work is to show that a closed set with
semipermeable boundary enjoys some regularity properties. Namely, under suitable
assumptions on f , the boundary of such a set is a Lipschitz (and even semiconcave)
manifold, and the barrier solutions satisfy the Pontryagin principle.
To our knowledge, this problem has never been treated, although it is of great
interest for qualitative and quantitative control problems (see the examples below).
However, our work is related to several studies on the regularity of the value function
∗ Received
by the editors June 5, 1995; accepted for publication (in revised form) July 1, 1996.
http://www.siam.org/journals/sicon/35-5/28729.html
† CEREMADE, URA CNRS 749, Université Paris-Dauphine, Place du Maréchal de Lattre de
Tassigny, 75775 Paris cedex 16, France ([email protected]).
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SEMIPERMEABLE SURFACES I
1639
of optimal control problems (see, for instance, [1], [8], [9], [11], [14], [19], [20] [21]). It
is not easy to compare our results (which are of geometric nature) with those given
in the previous references (which are concerned with the regularity of functions). For
this reason, we illustrate our results through the study of the regularity of the optimal
exit-time function.
(1) The optimal exit-time function θΩ from the open subset Ω ⊂ RN is defined
by, ∀x0 ∈ Ω,
/ Ω} .
θΩ (x0 ) := inf {t ≥ 0 | ∃x(·) solution to (1) such that x(t) ∈
Roughly speaking, θΩ (x0 ) is the minimal time any solution of the controlled system (1)
starting from x0 needs to leave Ω. The regularity of the optimal exit-time function is
the aim of several works [23], [24], [6], [7], [9], [10]. In [9] Cannarsa and Sinestrari prove
that θΩ is semiconcave on its (open) domain under the following assumptions: (a) f is
smooth; (b) ∂Ω enjoys some regularity (roughly speaking, its curvature is bounded);
(c) a “controllability condition” on the boundary of Ω is required, which ensures that
θΩ is Lipschitz continuous. Thanks to conditions (b) and (c), θΩ is “smooth” in a
neighborhood of ∂Ω. Then condition (a) ensures, by using the Pontryagin principle,
that this “smoothness” propagates along the (smooth) optimal trajectories.1 So, in
this method, the crucial points are, on one hand, the smoothness of θΩ at the boundary
of Ω and, on the other hand, the propagation of this regularity.
Our method is, on the contrary, based on the local study of the epigraph of θΩ .
Combining the results of [10] and of [25] yields that this epigraph has a semipermeable
boundary for some dynamics Φf constructed from f . Thanks to the regularity results
of semipermeable surfaces given in this paper, we prove, without conditions (b) and
(c), that θΩ is (locally) Lipschitz and semiconcave on an open dense subset of its
domain.
(2) Boundary of the viability kernel: The first definition of (nonsmooth) semipermeability is due to Quincampoix and appeared in the framework of viability theory
(Aubin [3]). If K ⊂ RN is a closed set, the viability kernel Viabf (K) of K for f is
∃x(·) solution to (1)
Viabf (K) := x0 ∈ K |
.
such that x(t) ∈ K ∀t ≥ 0
Under suitable assumptions, the viability kernel of K for f is a closed subset of K (see
also [5], [15]). In [25], Quincampoix proves that the boundary of Viabf (K) enjoys the
semipermeability property in the interior of K.
(3) Boundary of the reachable set: The reachable set for f starting from a point
x0 ∈ RN is the set of points y for which there exists a solution x(·) of (1) and a time
t ≥ 0 such that x(t) = y.
S
If, for instance, 0 belongs to the interior of u f (x0 , u) and f is Lipschitz continuous, then the reachable set is open. Moreover, its boundary is semipermeable for
−f (see Quincampoix [26]).
This research is presented as follows. In the present paper, Part I, we give two
equivalent definitions of the semipermeability, and we also prove that semipermeable
boundaries are Lipschitz manifolds. Then we show that semipermeable surfaces are
“smooth” along barrier solutions. We also explain how to recover the Pontryagin
principle.
1 A similar method based on the propagation of the regularity of the final data along optimal
trajectories is also applied to Mayer’s problem in [14] and to the Bolza problem in [11].
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PIERRE CARDALIAGUET
Part II, also in this issue, is devoted to the regularity results for the closed sets
with semipermeable boundaries. We first show that the contingent cone to such
closed sets is a union of half-spaces and enjoys some upper semicontinuity property.
We then show, under a stronger assumption on the dynamics, that the boundaries of
such closed sets are locally graphs of semiconcave functions. We complete this paper
by applying these results to the case of the optimal exit-time functions.
1. Semipermeable boundaries.
1.1. Definition of semipermeability. Let us from now on replace the controlled system (1) by the differential inclusion
(2)
x′ (t) ∈ F (x(t)),
x(0) = x0 ,
S
where F (x) := u∈U f (x, u). The advantage of using differential inclusions instead
of controlled systems lies in the fact that the regularity properties explained below
depend on the geometrical properties of the sets F (x) and not on its representation
as a controlled system. Moreover, the formulation as differential inclusions simplifies
the statements and the proofs of the results.
It is well known that, under conditions that we impose here, controlled system
(1) has the same solutions as differential inclusion (2). We denote by SF (x0 ) the
set of (Carathéodory) solutions of differential inclusion (2). With a set-valued map
RN , we associate the hamiltonian HF defined by
F : RN
(3)
∀(x, p) ∈ RN × RN ,
HF (x, p) := inf hv, pi.
v∈F (x)
Note that the hamiltonian HF is concave with respect to p.
Let us now recall two basic definitions of nonsmooth analysis.
If K is a closed subset of RN and x belongs to K, the contingent cone to K at x
is the set of vectors v ∈ RN such that
lim inf dK (x + hv)/h = 0
h→0+
(dK (y) denotes the distance from the point y to the set K). The contingent cone is a
closed cone. It is denoted by TK (x).
We also denote by TK (x)− the polar cone of TK (x), i.e.,
TK (x)− := {p ∈ RN | ∀v ∈ TK (x), hp, vi ≤ 0}.
The polar cone is a closed convex cone. The contingent cone plays the role of tangent
half-space, while the polar cone plays the role of exterior normal for nonsmooth sets.
If K is a subset of RN and x belongs to K, the Dubovitsky–Miljutin cone to K
at x is the set of vectors v ∈ RN for which there exists some α > 0 such that
x+]0, α](v + αB) ⊂ K.
The Dubovitsky–Miljutin cone to K at x is denoted by DK (x). It is an open cone. If
K is a closed subset of RN , then DRN \K (x) = RN \TK (x) for x ∈ K [25]. Moreover,
DK (x) ⊂ Int (TK (x)), but there is no equality in general even if the set K is a Lipschitz
manifold.
SEMIPERMEABLE SURFACES I
1641
Notation. Below, BN denotes the closed unit ball of RN (endowed with the
o
euclidean topology). If there is no ambiguity, we write only B. In the same way, B N
denotes the open unit ball of RN .
DEFINITION 1.1 (semipermeability). A closed set M has a semipermeable boundary for the set-valued map F : RN
RN (or enjoys the semipermeability property)
in a neighborhood of x0 ∈ ∂M if there is some positive radius r such that
∀x ∈ M ∩ (x0 + rB), ∀p ∈ TM (x)− , HF (x, p) = 0.
An equivalent definition of semipermeability in terms of trajectories is the
following.
PROPOSITION 1.1 (semipermeability). Assume that the set-valued map F satisfies
the following conditions:

N

RN has convex compact values;
 (a) F : R
(b) F is ℓ-Lipschitz, i.e.,
(4)


∀x, y ∈ RN × RN , F (y) ⊂ F (x) + ℓB.
Then a closed set M ⊂ RN has a semipermeable boundary in a neighborhood of x0 ∈
∂M if and only if there are open subsets O and O′ of RN with x0 ∈ O ⊂ O′ and a
time T > 0 such that
(i) ∀x ∈ M ∩ O, there is at least one solution x(·) ∈ SF (x) which remains in
M ∩ O′ on [0, T ];
(ii) ∀x ∈ M ∩ O′ , any solution of the differential inclusion for −F remains in M
on [0, T ];
(iii) ∀x ∈ ∂M ∩ O′ , any solution of the differential inclusion for F remains in
c := RN \M on [0, T ].
M
The notations of this definition are kept throughout this paper.
Before proving that result, let us point out an important consequence.
COROLLARY 1.1 (barrier solutions). Assume that F and M are as in Proposition 1.1.
If M enjoys the semipermeability property, then any solution x(·) of the differential inclusion for F starting from ∂M ∩ O which remains in M on [0, T ] remains in
∂M on [0, T ].
Such a solution is “a barrier solution.”
Remarks.
(1) Thanks to Proposition 1.1(i), at least one barrier solution starts from any
initial position of ∂M ∩ O.
(2) Combining Proposition 1.1 and Corollary 1.1, we recover the definition of
semipermeability given at the beginning of this paper. Property (α) holds true thanks
to (iii), and Corollary 1.1 is exactly the same as property (β).
c from Proposition 1.1(iii), and in M from
Proof. The solution x(·) remains in M
c = ∂M , the corollary holds true.
assumption. Since M ∩ M
Proof of Proposition 1.1. Assume that the set M satisfies the described property
in a neighborhood of a point x0 . Then M is (locally) viable2 for F in O, so that the
2 If K ⊂ RN is locally compact, the viability theorem [3], [4], [17] gives the equivalence among
the following statements.
(i) K is a viability domain for F ; i.e.,∀x ∈ K, F (x) ∩ TK (x) 6= ∅.
(ii) K is viable for F ; i.e., ∀x ∈ K, ∃x(·) ∈ SF (x) and t > 0 such that x(s) ∈ K ∀s ∈ [0, t].
(iii) K satisfies the following: ∀x ∈ K, ∀p ∈ TK (x)− , HF (x, p) ≤ 0.
1642
PIERRE CARDALIAGUET
viability theorem, applied to the locally compact set M ∩ O, states that HF (x, p) ≤ 0
for x ∈ O ∩ M and p ∈ TM (x)− . Moreover, M is locally invariant3 for −F , so that
the invariance theorem, applied to the locally compact set M ∩ O again, states that
H−F (x, −p) ≥ 0 for x ∈ O ∩ M and p ∈ TM (x)− . Since
H−F (x, −p) = inf h−v, −pi = HF (x, p),
v∈F (x)
we have finally proved that HF (x, p) = 0 for any x ∈ O ∩ M and p ∈ TM (x)− .
Conversely, if the boundary of M is semipermeable, there is some radius r > 0
such that
∀x ∈ (x0 + rB) ∩ M, ∀p ∈ TM (x)− , HF (x, p) = 0.
Set ρ := maxx∈(x0 +rB) maxv∈F (x) kvk and T :=
Oi := x0 +
r
4ρ .
Define, for any i = 1, . . . , 4,
ir o
B.
4
Note that any solution of the differential inclusion for F (or for −F ) starting from Oi
(i = 1, . . . , 3) remains in Oi+1 on [0, T ].
Since the tangential condition
∀x ∈ (x0 + rB) ∩ M, ∀p ∈ TM (x)− , HF (x, p) ≤ 0
is satisfied, the viability theorem states that for any initial position x ∈ O1 ∩ M , there
is an x(·) ∈ SF (x) such that x(t) ∈ M as long as x(t) ∈ x0 + rB, i.e., at least on
[0, T ]. In particular, such a solution remains in M ∩ O2 on [0, T ]. Thus (i) holds true
with O := O1 and O′ := O2 .
Since HF (x, p) = 0 implies that H−F (x, −p) ≥ 0, the tangential condition
∀x ∈ (x0 + rB) ∩ M, ∀p ∈ TM (x)− , H−F (x, −p) ≥ 0
is fulfilled. Thus M is locally invariant for −F , and any solution of the differential
inclusion for −F starting from M ∩ O2 (and also from M ∩ O3 ) remains in M as long
as it remains in x0 + rB, and in particular, on [0, T ]. Thus (ii) holds true.
Assume, contrary to our claim, that (iii) is false. There is a solution x(·) of the
c on
differential inclusion for F starting from ∂M ∩ O2 which does not remain in M
[0, T ]. We already know that x(·) remains in O3 on [0, T ]. There is some time t ∈]0, T ]
such that x(t) belongs to the interior of M and to O3 . From Filippov’s theorem [12],
the set-valued map x
SF (x), endowed with the uniform topology, is continuous.
Thus there is a neighborhood W of x(0) such that, from any initial position y ∈ W ,
at least one solution y(·) ∈ SF (y) sufficiently close to x(·) on [0, T ] starts so that y(t)
belongs to the interior of M and to O3 . Since x belongs to ∂M , there is some ȳ ∈ W
which does not belong to M . Let us denote by ȳ(·) the associated solution. We now
consider the function z(·) defined by z(s) := ȳ(t − s) for s ∈ [0, T ]. Then z(·) is a
solution of the differential inclusion for −F starting from M ∩ O3 , which leaves M
before T . This is in contradiction to the proof of (ii). So (iii) holds true.
3 The invariance theorem [3] states that, for F satisfying (4) and for K ⊂ RN locally compact,
there is an equivalence among the following statements.
(i) K is an invariance domain for F , i.e.,∀x ∈ K, F (x) ⊂ TK (x).
(ii) K is invariant for F , i.e., ∀x ∈ K, ∃t > 0 such that ∀x(·) ∈ SF (x), ∀s ∈ [0, t], x(s) ∈ K.
(iii) K satisfies the following: ∀x ∈ K, ∀p ∈ TK (x)− , HF (x, −p) ≥ 0.
SEMIPERMEABLE SURFACES I
1643
1.2. Semipermeable boundaries are Lipschitz manifolds.
PROPOSITION 1.2. Assume that the boundary of M is semipermeable in a neighRN satisfying (4). If the values of F
borhood of x0 for a set-valued map F : RN
have a nonempty interior, then ∂M is a Lipschitz manifold in a neighborhood of x0 .
To prove Proposition 1.2, let us recall a sufficient condition for a closed set to be
a Lipschitz manifold (see [16, Thm. 1.2.2.2, p. 12]).
LEMMA 1.1. Let K be a closed subset of RN and let x belong to ∂K. Assume that
there exist some open set C, some ρ > 0, and some neighborhood U of x such that
(
y + [0, ρ]C ⊂ K,
∀y ∈ ∂M ∩ U,
b
y − [0, ρ]C ⊂ K,
b := RN \K.
where K
Then ∂K is a Lipschitz manifold in a neighborhood of x.
Proof of Proposition 1.2. It is enough to combine Lemma 1.1 with the following
lemma.
LEMMA 1.2. Let M be as in Proposition 1.1 and let F satisfy (4). Also let x
belong to ∂M ∩ O and v ∈ Int(F (x)). Then
c,
x + [0, t](v + a2 B) ⊂ M
x + [0, t](−v + a2 B) ⊂ M,
a
}.
where a := d∂F (x) (v) and t := min{T, ℓ(2kvk+a)
Proof of Lemma 1.2. Note that v + aB ⊂ F (x).
LEMMA 1.3. If C1 , C2 , and C3 are compact convex subsets of RN ,
[C1 + C3 ⊂ C2 + C3 ] ⇒ [C1 ⊂ C2 ].
Since the set-valued map F is ℓ-Lipschitz, use Lemma 1.3 to obtain
∀y ∈ x +
a
B,
2ℓ
a
v + B ⊂ F (y).
2
The map s → x + s(v + a2 u) is a solution of the differential inclusion for F on [0, t]
a
a
(for any u ∈ B) because ks(v + 2ℓ
u)k ≤ 2ℓ
for s ∈ [0, t]. Since x ∈ ∂M ∩ O and M
c (see
is semipermeable, any solution of the differential inclusion for F remains in M
Proposition 1.1(iii)). Thus
a c
x + [0, t] v + B ⊂ M
.
2
We can prove in a similar way (using the fact that M is locally invariant by −F from
Proposition 1.1(ii)) that
a x + [0, t] −v + B ⊂ M.
2
To complete the proof of Proposition 1.2, let v belong to the interior of F (x) and
set a := d∂F (x) (v). Then
∀y ∈ x +
a
,
2ℓ
a
v + B ⊂ F (y).
2
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PIERRE CARDALIAGUET
In particular, v belongs to the interior of F (y) and d∂F (y) (v) ≥
Lemma 1.2,

o
a
a  y + [0, t](−v + 4 B ) ⊂ M,
∀y ∈ x + ,
2ℓ  y + [0, t](v + a o ) ⊂ M
c,
4 B
a
2.
Thus, from
a
}.
where t := min{T, ℓ(4kvk+a)
N
Recall that equality R \TM (x) = DM
c (x) is always fulfilled. Moreover, we have
the following corollary.
COROLLARY 1.2. Under the assumptions and notations of Proposition 1.2, we
have, for any x ∈ ∂M ∩ O,
N
DM
c (x) = DM (x) = DRN \M
c (x).
c (x) = DRN \M (x) and R \TM
Note, moreover, that
Int(F (x)) ∩ TM (x) = ∅ and
− Int(F (x)) ∩ TM
c (x) = ∅.
Proof. We prove only the first equality, the proof of the second one being essentially the same. Since RN \TM (x) = DRN \M (x) ⊂ DM
c (x), it remains to prove
that
DM
c (x) ⊂ DRN \M (x).
Let v belong to DM
c (x). Since ∂M is a Lipschitz manifold, there is a Lipschitz function
φ : W → R (W ⊂ RN −1 open) and an open neighborhood W ′ ⊂ RN of x such that
∂M ∩ W ′ = {(y, φ(y)) | y ∈ W }.
c is
We can assume, without loss of generality, that M is the epigraph of φ, while M
the hypograph of φ and
RN \M ∩ W ′ = {(y, t) | t < φ(y) & y ∈ W }.
Set x := (xy , xt ) and v := (vy , vt ) (where xy and vy belong to RN −1 and xt and vt
belong to R). There is some α > 0 such that
c.
(xy , xt )+]0, α] ((vy , vt ) + αBN ) ⊂ M
Thus, for any u := (uy , ut ) ∈ BN and for any θ ∈]0, α],
φ(xy + θ (vy + αuy )) ≥ xt + θ(vt + αut ).
In particular, for any u := (uy , ut ) ∈ BN and for any θ ∈]0, α/2],
α α
α > xt + θ vt + ut ,
≥ xt + θ vt + √
φ xy + θ vy + uy
2
2
2
because ( 21 uy ,
√
2
2 )
∈ BN . This actually means that
i αi α
(vy , vt ) + BN ⊂ RN \M,
(xy , xt ) + 0,
2
2
so that v belongs to DRN \M (x).
1645
SEMIPERMEABLE SURFACES I
2. Regularity of barrier solutions. We show here that, with any barrier solution, we can associate a Lipschitzian function p(·) : [0, T ] → RN such that kp(t)k = 1
and
0 = hx′ (t), p(t)i = H(x(t), p(t)) for almost every t ∈ [0, T ].
Moreover, p(t) is an exterior normal to M at x(t):
∀t ∈]0, T [,
−
TM (x(t)) = (p(t)) .
The function p(·) is called the adjoint of x(·). In the case when the hamiltonian
H is derivable, this adjoint coincides with the usual adjoint up to a multiplicative
coefficient, and (x(·), p(·)) satisfies the Pontryagin principle.
2.1. Two preliminary lemmas. We first estimate the variations of the contingent cone to closed sets with semipermeable boundaries along the barrier solutions.
LEMMA 2.1. Let M be as in Proposition 1.1 and let F satisfy (4). Assume that
x belongs to ∂M ∩ O and that x(·) ∈ SF (x) is a barrier solution (i.e., it remains in
∂M ∩ O′ on [0, T ]).
There is a constant C, which only depends on T and on the Lipschitz constant ℓ
of F , such that
(5)
∀0 ≤ t ≤ T, ∀v ∈ TM (x(t)), dTM (x) (v) ≤ Ctkvk.
Moreover,
(6)
(v) ≤ C(t − s)kvk.
∀0 ≤ s ≤ t ≤ T, ∀v ∈ TM
c (x(s)), dTM
c (x(t))
Proof of Lemma 2.1. We prove only (5), since the proof of (6) is essentially the
same. If v belongs to TM (x(t)), there exist hn → 0+ , vn → v such that x(t) + hn vn
belongs to M for any n. The Filippov theorem [12] provides the existence of solutions
yn (·) ∈ S−F (x(t) + hn vn ) such that
(7)
kx′ (t − s) + yn′ (s)k ≤ ℓeℓs hn kvn k for almost every s ∈ [0, t].
The solutions yn (·) remain in M on [0, T ] because M is (locally) invariant for −F
from Proposition 1.1.
. We now prove that the sequence {wn } converges, up to a
Set wn := yn (t)−x
hn
subsequence, to some w ∈ TM (x) such that kv − wk ≤ (eℓt − 1)kvk. Indeed,
yn (t) − x = (yn (t) − (x(t) + hn vn )) + (x(t) − x) + hn vn
Rt
= 0 (yn′ (s) + x′ (t − s))ds + hn vn .
Combining this latter equality with (7) yields
Z t
1
kwn − vn k ≤
kyn′ (s) + x′ (t − s)kds ≤ kvn k(eℓt − 1).
hn 0
Thus {wn } is bounded and converges, up to a subsequence, to some w which belongs
to TM (x) and satisfies
kw − vk ≤ kvk(eℓt − 1).
So Lemma 2.1 is proved by setting C := supt∈[0,T ]
eℓt −1
t .
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PIERRE CARDALIAGUET
We now compute DM (x) for some particular points x.
LEMMA 2.2. Let x(·) be a barrier solution on [0, T ] and assume that condition (4)
and the following condition are satisfied:
(8)
∀x ∈ O′ , ∀v ∈ ∂F (x), TF (x) (v) is a half-space.
For each t ∈]0, T [ where the derivative x′ (t) exists,
(9)
−Int[TF (x(t)) (x′ (t))] = DM (x(t))
and
(10)
Int[TF (x(t)) (x′ (t))] = DM
c (x(t)).
In particular, DM (x(t)) and DM
c (x(t)) are both equal to open half-spaces.
Assumption (8) plays a major role below. It is equivalent to
(i) ∂F (x) is a C 1 manifold for any x ∈ O′ .
(ii) F (x) is convex with a nonempty interior for any x ∈ O′ .
Proof of Lemma 2.2. Let t ∈]0, T [ be such that the derivative v := x′ (t) exists at
time t. Recall that v belongs to F (x(t)) ∩ TM (x(t)). Let us first prove that
(11)
−Int[TF (x(t)) (v)] ⊂ DM (x(t)).
Let w belong to the interior of −TF (x(t)) (v). Since F (x) is convex, there are some
λ > 0 and a > 0 such that w + aB is contained in λ(v − F (x(t))), i.e.,
v − τ w + τ aB ⊂ F (x(t)),
with τ := 1/λ. Since F is ℓ-Lipschitz, Lemma 1.3 implies that
aτ
a
B,
v − τw +
B ⊂ F (y).
∀y ∈ x(t) +
2τ ℓ
2
For h > 0 sufficiently small (say, h ∈ [0, ǫ] with ǫ > 0) the solutions of the differential
inclusion for −F starting from x(t + h) remain in x(t) + 2τa ℓ B on [0, h]. In particular,
s → x(t + h) − s(v − τ w + u) is a solution of the differential inclusion for −F on [0, h]
if kuk ≤ aτ
2 . Since x(t + h) belongs to M , the solutions of the differential inclusion
for −F starting from x(t + h) remain in M on [0, T ]. Thus
aτ (12)
B ⊂ M.
∀s ∈ [0, h], x(t + h) − s v − τ w +
2
Since v = x′ (t), there is some ǫ′ > 0 such that, for h ∈ [0, ǫ′ ],
haτ
.
4
Combining (12) with s = h with (13) yields, for any h ∈ [0, inf{ǫ, ǫ′ }],
(13)
kx(t + h) − x(t) − hvk ≤
x(t) + hτ w +
haτ
B ⊂ M.
4
Thus w belongs to DM (x(t)).
We can prove in the same way that
Int[TF (x(t)) (v)] ⊂ DM
c (x(t))
c is (locally) invariant by F . Since DM (x(t)) ∩ D c (x(t)) = ∅, and since
because M
M
both sets contain an open half-space and are open, DM (x(t)) and DM
c (x(t))
are, respectively, equal to the interiors of the half-spaces −TF (x(t)) (x′ (t)) and
TF (x(t)) (x′ (t)).
SEMIPERMEABLE SURFACES I
1647
2.2. The adjoint of a barrier solution.
THEOREM 2.1 (definition of the adjoint). Let M be a closed set with a semipermeable boundary and let x belong to ∂M ∩ O (cf. Proposition 1.1). Let x(·) ∈ SF (x)
be a barrier solution on [0, T ] and C be the constant defined by Lemma 2.1. There is
a 2C-Lipschitzian function p(·) : [0, T ] → RN such that kp(t)k = 1 for any t ∈ [0, T ]
and
−
∀t ∈]0, T [, TM (x(t)) = (p(t)) ,
−
where (p(t)) = {v ∈ RN | hp(t), vi ≤ 0}. The function p(·) is called the adjoint of
x(·) on [0, T ]. Moreover, if p(·) is the adjoint of some barrier solution x(·), then
(p(0))− ⊂ TM (x(0)).
(14)
The adjoint p(·) is uniquely defined. Theorem 2.1 states that the contingent cone
TM (x(t)) is a half-space for t > 0 and p(t) is the unique outward normal at x(t). This
means that, at x(t), the closed set M is “smooth.”
Proof of Theorem 2.1. Existence. If x′ (t) exists, Corollary 1.2 states that
DM (x(t)) ⊂ TM (x(t)) = RN \DRN \M (x(t)) = RN \DM
c (x(t)).
From Lemma 2.2, the left- and right-hand sides of the inclusions are half-spaces. Thus
TM (x(t)) is a half-space and there is some p(t) satisfying kp(t)k = 1 and TM (x(t)) =
(p(t))− .
Now fix any t ∈]0, T [. Since the solution x(·) is absolutely continuous, there are
tn → t+ and sn → t− such that the derivatives x′ (tn ) and x′ (sn ) exist. The sequences
(p(tn ))n∈N and (p(sn ))n∈N converge, respectively, to p1 and p2 (up to a subsequence).
Lemma 2.1 yields
(p1 )− ⊂ Limsup (p(tn ))− ⊂ TM (x(t))
(where Limsup denotes the Kuratowski upper limit [2]) and, for any v ∈ TM (x(t)),
0 = lim inf d(v, TM (x(sn ))) = lim inf hv, p(sn )i+ = hv, p2 i+
n
n
(where s+ := max{s, 0}), so that hv, p2 i ≤ 0.
Thus (p1 )− ⊂ TM (x(t)) ⊂ (p2 )− and TM (x(t)) is equal to a half-space. Let us
denote by p(t) the common value p1 = p2 . Then TM (x(t)) = (p(t))− . The function
p(·) is defined on ]0, T [.
Note that, for t = 0, the same proof shows that any upper limit p1 of the functions
p(tn ) satisfies (p1 )− ⊂ TM (x(0)). Let us now prove that p(·) is Lipschitz continuous
and so can be defined (uniquely) on [0, T ].
The adjoint is Lipschitzian. Let C be the constant of Lemma 2.1 and let 0 < s <
t < T . There are two cases.
(1) Either hp(t), p(s)i ≥ 0. Then we denote by v the projection of p(s) onto
TM (x(t)), and by w the projection of v onto TM (x(s)). From Lemma 2.1, the distance
between v and w is smaller than or equal to C(t − s)kvk. Since
v = p(s) − hp(t), p(s)ip(t) and w = v − hp(s), vip(s),
we have kv − wk = 1 − hp(t), p(s)i2 .
1648
PIERRE CARDALIAGUET
Moreover, kvk2 = 1 − hp(t), p(s)i2 = kv − wk. Combining this equation with
kv − wk ≤ C(t − s)kvk yields
kv − wk ≤ C 2 (t − s)2 .
Note that
kp(t) − p(s)k2 = 2 − 2hp(t), p(s)i.
We conclude that
kp(t) − p(s)k2 ≤ 2 − 2hp(t), p(s)i2 ≤ 2C 2 (t − s)2 .
Thus
(15)
kp(t) − p(s)k ≤ 2C|t − s|.
(2) Or hp(t), p(s)i ≤ 0. Then p(s) belongs to (p(t))− = TM (x(t)). Thus Lemma
2.1 states that
d(p(s), (p(s))− ) ≤ C(t − s).
Since the left-hand side is equal to 1, hp(t), p(s)i ≤ 0 cannot occur unless t−s ≥ 1/C.
In that case, equation (15) is fulfilled. Thus we have proved that p(·) is a 2C-Lipschitz
function.
LEMMA 2.3 (characterization of the adjoint). The adjoint p(·) of a barrier solution
x(·) on [0, T ] is the unique continuous function satisfying
(a) ∀t ∈ [0, T ], kp(t)k = 1,
(16)
(b) HF (x(t), p(t)) = hx′ (t), p(t)i = 0 for almost every t ∈ [0, T ].
COROLLARY 2.1. Under the notations and the assumptions of Lemma 2.3,
TM (x(t)) = −TF (x(t)) (x′ (t)) = (p(t))−
for almost every t ∈]0, T [.
COROLLARY 2.2. Assume, moreover, that F (x) is strictly convex for any x ∈ O′ .
Then any barrier solution is C 1 .
Proof. Indeed, the set-valued map t
Arg minv∈F (x(t)) hv, p(t)i is upper semicontinuous and, in fact, single-valued because F (x) is strictly convex. So it is continuous. From Lemma 2.3, x′ (t) is almost everywhere equal to the continuous function
t → Arg minv∈F (x(t)) hv, p(t)i and so is continuous.
Proof of Lemma 2.3 Assume that p(·) is the adjoint of x(·) on [0, T ]. We have
to prove (16). Let t ∈]0, T [. The set F (x(t)) is convex and has a nonempty interior
from assumption (8). Let w belong to the interior of F (x(t)). From Corollary 1.2,
w∈
/ TM (x(t)). Thus hw, p(t)i > 0 from Theorem 2.1. Since F (x(t)) = Int(F (x(t))),
we have proved HF (x(t), p(t)) ≥ 0. If x′ (t) exists, then x′ (t) belongs to F (x(t))
−
and so hx′ (t), p(t)i ≥ 0. Moreover, x′ (t) belongs to TM (x(t)) = (p(t)) , and thus
′
hx (t), p(t)i ≤ 0. So (16(b)) holds true because x(·) is almost everywhere derivable.
Conversely, let us now assume that some continuous function p(·) : [0, T ] → RN
satisfies (16). We have to prove that p(·) is the adjoint of x(·) on [0, T ]. Fix any
t ∈ (0, T ] where the derivative x′ (t) exists and where (16(b)) is fulfilled. Combining
SEMIPERMEABLE SURFACES I
1649
Lemma 2.2 and Theorem 2.1 yields that TM (x(t)) is the closure of the half-space
DM (x(t)). Thus
(17)
TM (x(t)) = −TF (x(t)) (x′ (t)) =
[
λ>0
λ(x′ (t) − F (x(t))).
Thanks to (16), for any λ ≥ 0 and any w ∈ F (x(t)), one has
hλ(x′ (t) − w), p(t)i ≤ 0.
(18)
Combining (17) and (18) yields
∀v ∈ TM (x(t)), hv, p(t)i ≤ 0.
So (p(t))− = TM (x(t)) for almost every t ∈ (0, T ]. In particular, p(t) coincide almost
everywhere with the adjoint of x(·), which is continuous. So p(·) is equal to the adjoint
of x(·).
We study here the regularity properties of the function which associates its adjoint
to a solution.
PROPOSITION 2.1. Assume that xn (·) are barrier solutions starting from xn ∈
∂M ∩ O and converging to some x(·) barrier solution starting from x ∈ ∂M ∩ O. If
pn (·) are the adjoint of xn (·) on [0, T ], then the pn (·) converge uniformly to the adjoint
of x(·) on [0, T ].
Proof of Proposition 2.1. Since the (pn (·)) are uniformly continuous, Ascoli’s
theorem states that pn (·) converge uniformly to some continuous function p(·) (up to
a subsequence). To prove that p(·) is the adjoint of x(·), it is sufficient to show that
p(·) satisfies (16). For any t ∈ [0, T ], kp(t)k = 1. From Lemma 2.3, for almost every
t ∈ (0, T ], hx′n (t), pn (t)i = 0. Thus
∀t ∈ [0, T ],
Z
t
hx′n (s), pn (s)ids = 0.
0
The sequence of functions pn (·) converges uniformly to p(·), and the sequence x′n (·)
converges weakly to x(·). Thus
∀t ∈ [0, T ],
Z
0
t
hx′ (s), p(s)ids = 0,
which implies that hx′ (t), p(t)i = 0 for almost every t ∈]0, T [.
Let t ∈ (0, T ] and v belong to F (x(t)). We are going to prove that hv, p(t)i ≥ 0.
From Michael’s theorem [22], a continuous function ṽ(·) : RN → V exists, such that
∀x ∈ RN ,
ṽ(x) ∈ F (x)
and
ṽ(x(t)) = v.
Note that ṽ(xn (t)) converge to ṽ(x(t)), and that hṽ(xn (t)), pn (t)i ≥ 0. Letting n →
+∞ yields hv, p(t)i ≥ 0 for any v ∈ F (x(t)) and any t ∈ (0, T ].
Thus p(·) satisfies (16) and is indeed the adjoint of x(·). Since any converging
subsequence of the uniformly continuous sequence (pn (·)) converges to the adjoint of
x(·), we have proved that the pn (·) converge to the adjoint of x(·).
1650
PIERRE CARDALIAGUET
2.3. Barrier solutions and the Pontryagin principle. We now prove that,
under some assumptions of differentiability of the hamiltonian H, barrier solutions
satisfy the Pontryagin principle.
In Isaacs’ pioneering work on differential games [18], semipermeable hypersurfaces
are constructed by using the method of characteristics, which is very close to the
Pontryagin principle. We show here that this method of construction is a priori
justified since the barrier solutions indeed satisfy the Pontryagin principle.
THEOREM 2.2. Assume that the set-valued map F satisfies (4), (8) and, moreover,
that its associated hamiltonian H is C 2 on RN × [RN \{0}].
Let M be a closed set with a semipermeable boundary. If x(·) is a barrier solution
and p(·) is its adjoint, and if q(·) : [0, T ] → RN is defined by
Z t
∂H
(x(s), p(s)), p(s) ds ,
∀t ∈ [0, T ], q(t) := p(t) exp −
∂x
0
then (x(·), q(·)) is a solution to the hamiltonian system
 ′
x (t) = ∂H

∂p (x(t), q(t)),


(19)
q ′ (t) = − ∂H
∂x (x(t), q(t)),



q(0) := p(0).
COROLLARY 2.3. Suppose that the assumptions of the previous theorem are satisfied. If x(·) is a barrier solution, then x(·) is C 1 on [0, T ]. Moreover, for any q 6= 0
such that
hq, x′ (0)i = H(x(0), q) = 0,
the solution (x(·), q(·)) of (19) with initial condition (x(0), q) satisfies
∀t ∈]0, T [, TM (x(t)) = (q(t))− .
Proof of Theorem 2.2. Since, for any t ∈ [0, T ], H(x(t), p(t)) = 0, since hx′ (t), p(t)i =
0 for almost every t ∈ [0, T ], and since H is differentiable, one has
x′ (t) = Arg
min hv, p(t)i =
v∈F (x(t))
∂H
(x(t), p(t))
∂p
for almost every t ∈ [0, T ]. In fact, these equalities hold true everywhere because
the right-hand side is continuous. Thus x′ (·) is defined everywhere on [0, T ] and is
continuous.
Let t ∈]0, T [ be such that p′ (t) exists. Let us prove that
∗
2
∂ H
′
(20)
(x(t), p(t)) p(t) + p (t) ≥ 0.
∀v ⊥ p(t), v,
∂x∂p
Since v ⊥ p(t), v belongs to the boundary of TM (x(t)) from Theorem 2.1, and so to
T∂M (x(t)). Thus there are hn → 0+ and vn → v with x(t) + hn vn ∈ ∂M . For any n,
let us consider the solutions xn (·) (with final conditions) to
( ′
xn (s) = ∂H
∂p (xn (s), p(s)) for s ∈ [0, t],
xn (t) := x(t) + hn vn .
SEMIPERMEABLE SURFACES I
1651
Note that the xn (·) converge to x(·) and that, moreover,
(21)
xn (s) − x(s)
→ z(s),
hn
∀s ∈ [t, T ],
where z(·) is the solution to
(
z ′ (s) =
∂2H
∂x∂p (x(s), p(s))z(s),
z(t) = v.
Since xn (t) ∈ ∂M , the functions s → xn (t−s) are solutions of the differential inclusion
for −F starting from M , and thus remain in M from the semipermeability of M . From
−
Theorem 2.1, TM (x(s)) = (p(s)) for every s. From (21), z(s) belongs to TM (x(s))
so that hz(s), p(s)i ≤ 0. In particular,
∀s ∈ [0, t], hz(s), p(s)i − hz(t), p(t)i ≤ 0,
because hz(t), p(t)i = hv, p(t)i = 0. Dividing by s − t and letting s → t− gives
hz ′ (t), p(t)i + hz(t), p′ (t)i ≥ 0,
and thus inequality (20) holds true.
Since H is positively homogeneous, Euler’s rule states that
∗
2
∂H
∂ H
(x(t), p(t)) p(t) =
(x(t), p(t)).
∂x∂p
∂x
From (20), for every t where p′ (t) exists, there is some λ(t) ∈ R such that
p′ (t) = −
∂H
+ λ(t)p(t).
∂x
(Note that one can compute λ(t) explicitly because kp(t)k = 1 implies that hp(t), p′ (t)i =
0, and thus λ(t) = h ∂H
∂x , p(t)i.)
Rt
Now let µ(t) := exp(− 0 λ(s)ds) and q(t) := µ(t)p(t). Using the fact that H is
positively homogeneous, it is easy shown that (x(·), q(·)) is a solution to the hamiltonian system.
Acknowledgment. We would like to warmly thank Pr. T. Rzeżuchowski for his
helpful suggestions and friendly advice.
REFERENCES
[1] L. AMBROSIO, P. CANNARSA, AND H.M. SONER, On the propagation of singularities of semiconvex functions, Ann. Scuola Sup. Pisa Ser. IV, XX (1993), pp. 597–616.
[2] J.-P. AUBIN AND H. FRANKOWSKA, Set-Valued Analysis, Birkhaüser, Boston, 1990.
[3] J.-P. AUBIN, Viability Theory, Birkhaüser, Boston, 1991.
[4] J.-P. AUBIN AND H. FRANKOWSKA, Partial differential inclusions governing feedback controls,
J. Convex Analysis, 2 (1995), pp. 19–40.
[5] J.-P. AUBIN AND H. FRANKOWSKA, Viability kernels of control systems, in Nonlinear Synthesis,
C. Byrnes and A. Kurzhanski, eds., Birkäuser, Boston, pp. 12–33.
[6] M. BARDI, A boundary value problem for the minimum time function, SIAM J. Control Optim.,
28 (1989), pp. 950–965.
[7] G. BARLES AND B. PERTHAME, Exit time problems in optimal control and vanishing method,
SIAM J. Control Optim., 26 (1988), pp. 1123–1134.
1652
PIERRE CARDALIAGUET
[8] P. CANNARSA AND H. FRANKOWSKA, Some characterizations of the value function in control
theory, SIAM J. Control Optim., 29 (1991), pp. 1322–1347.
[9] P. CANNARSA AND C. SINESTRARI, Convexity properties of the minimum time function, Calc.
Var., 3 (1995), pp. 273–298.
[10] P. CARDALIAGUET, M. QUINCAMPOIX, AND P. SAINT-PIERRE, Optimal times for constrained
non-linear control problems without local controllability, Appl. Math. Optim., to appear.
[11] N. CAROFF AND H. FRANKOWSKA, Conjugate points and shocks in nonlinear optimal control
Trans. Amer. Math. Soc., 348 (1996), pp. 3133–3153.
[12] A.F. FILIPPOV, Classical solutions of differential equations with multivalued right hand side,
SIAM J. Control, 5 (1967), pp. 609–621.
[13] H. FRANKOWSKA, Local controllability and infinitesimal generators of semi-groups of setvalued maps, SIAM J. Control Optim., 25 (1987), pp. 412–432.
[14] H. FRANKOWSKA, Control of Nonlinear Systems and Differential Inclusions, manuscript.
[15] H. FRANKOWSKA AND M. QUINCAMPOIX, Viability kernels of differential inclusions with
constraints: Algorithm and applications, Math. Systems Estimation Control, 1 (1992),
pp. 371–388.
[16] P. GRISVARD, Elliptic Problems in Nonsmooth Domains, Pitman, Boston, 1985.
[17] H.G. GUSEINOV, A.I. SUBBOTIN, AND V.N. USHAKOV, Derivatives for multivalued mappings
with applications to the game-theorical problems of control, Problems Control Inform. Theory, 14 (1985), pp. 295–298.
[18] R. ISAACS, Differential Games, John Wiley, New York, 1965.
[19] H. ISHII, Uniqueness of unbounded viscosity solutions of Hamilton-Jacobi equations, Indiana
Univ. Math. J., 33 (1984), pp. 721–748.
[20] S.N. KRUZKOV, Generalized solutions of Hamilton-Jacobi equations of eikonal type I, Math.
USSR-Sb, 27 (1975), pp. 406–446.
[21] P.L. LIONS, Generalized Solutions of Hamilton-Jacobi equations, Pitman, Boston, 1982.
[22] E. MICHAEL, Continuous selections, Ann. Math., 63 (1956), pp. 361–381.
[23] N.N. PETROV, Controllability of autonomous systems, Differential Equations, 4 (1968),
pp. 311–317.
[24] N.N. PETROV, On the Bellman function for time-optimal process problem, J. Appl. Math.
Mech., 34 (1970), pp. 785–791.
[25] M. QUINCAMPOIX, Differential inclusions and target problems, SIAM J. Control Optim.,
30 (1992), pp. 324–335.
[26] M. QUINCAMPOIX, Enveloppes d’invariance pour des inclusions différentielles Lipschitziennes
et applications aux problèmes de cibles, C. R. Acad. Sci. Paris Sér. I Math., 314 (1992),
pp. 343–347.