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320

But :

J ^^{r^ A*' )

{\\

--'

1{0<y<B)

^(n)

+ b(;-^

)Mo<y<E}\\

Ll(

n)) > °>

because the function to be integrated is positive. Finally, we obtain :

||((^

+ C?)fl)(.

)

.,A)||

L1(nx(0iOo))

<||

.,A)||

Ll(n)

, VAe(0,oo)

which proves the desired estimate. •

Proposition 2.13 Under the previous assumptions, the map T has a unique fixed point

T ED.

Proof.

We apply the same technique as the one used by Kelley [6]. Let (U

)

be the

sequence defined by : U

= T (£/„_i), with {To = T_. This sequence is increasing and

bounded by T

. Since the map T is compact, the sequence (U

)

lies in a compact subset

of C([0,E]). Then, there exists a subsequence (U^

)) € C([0,E]), which converges to

U e C([Q,Ej) and using the continuity of T we get : U = T{U). Similarly, the sequence

= F(V

-x), with V

= T

is decreasing and converges to V = T{V) e C([0,E]).

Moreover, if T = T(T) then U < T < V. Let U = ip{U) and V = ip(V) be the solutions

of (14). The difference W = V - U > 0 satisfies :

W" = 2nM{(I-(K + Q))(I-K)-

(L°(Tp-

(V))-L°(ip-

(U)))},

W(0) = 0,

W(E) = 0.

Since Sr e L°°(0, E) C L

(0, E) then W" G L

(0, E). Hence :

W"'(£;) - W'(Q)

l-E

W"{x)dx

2*J

M{(I-(K

Q))(I-

K)"

(

fjf^l^

)

}

(^)^-

In addition :

(

, A) (L° (V"

(7) , A) - L° (r

(0-) , A)) > 0, V(/x, A) e

[-1,1]

x (0, oo).

The operator (/

—

K)~ is positive because K is positive by definition. Then :

g=(I- K)-

(

(L° (V"

(F)

- L° (^"

((t7)

A)))

is positive. Thanks to Lemma 2.12, we have W'(E) - W'(0) > 0. Since W > 0 and

W^(0) = W{E) = 0, we get : W'{0) > 0 > iy'(E). Then we have W'{0) =

W'{E).

This

implies :

\\{K + Q)g\\

According to Lemma 2.12, g must be equal to zero and U = V. This concludes the

proof of the uniqueness. •

Then Propositions 2.4 and 2.13 allow us to conclude that there exists one and only

one solution (T, L) of the coupled system (l)-(3) in V x E. Remark 2.5 and Proposition

2.13 conclude the proof of the main Theorem 2.1. •

321

References

[1] V. Agoshkov, Boundary Value Problems for Transport Equations. Modelling and

simulation in Science, Engineering and Technology. Birkhauser, Basel (1998).

[2] F. Asllanaj, G. Jeandel, J.R. Roche and D. Schmitt, Existence and Uniqueness of

a steady state solution of a coupled radiative-conductive heat transfer problem for

a non grey and anisotropically participating medium. Preprint of the Institut Elie

Cartan, No 41, Nancy, France, (2001).

[3] F. Asllanaj, Etude et analyse numerique des transferts de chaleur couples par rayon-

nement et conduction dans les milieux semi-transparents: Application aux milieux

fibreux. Ph. D. Thesis, Universite Henri Poincare, Nancy, France, December 2001.

[4] H. Brezis and W. A. Strauss, Semi-linear second order elliptic equation in L

. J.

Math. Soc. Japan. 25(4) (1973).

[5] G. Da Prato, Somme d'applications non lineaires. Symposia Mathematica VII, 1st.

Naz. Alta Mat. Academic Press, p. 233-268 (1971).

[6] C.T. Kelley, Existence and uniqueness of solutions of nonlinear systems of conductive-

radiative heat transfer equations. Transport Theory Statist. Phys., 25 (1996), 249-

260.

[7] S.C. Lee, Radiation heat-transfer model for fibers oriented parallel to diffuse bound-

aries.

J. Thermophysics, 2 (1988), 303-308.

[8] R.H. Martin, Jr., Nonlinear operators and differential equations in Banach spaces.

Wiley-Interscience, New York (1976).

[9] M.N. Ozisik, Radiative Transfer and Interactions with Conduction and Convection.

Wiley-Interscience, New York (1973).

[10] M. Zlamal, A finite element solution of the nonlinear heat equation. RAIRO Numer-

ical Analysis, 17 (1980), 203-216.

Viscosity Lyapunov functions for almost sure

stability of degenerate diffusions*

Martino Bardi and Annalisa Cesaroni

Dipartimento di Matematica P. e A.

Universita di Padova

via Belzoni 7, 35131 Padova, Italy

Email : bardi@math.unipd.it ; acesar@math.tmipd.it

Abstract

The direct Lyapunov method for stability of dynamical systems is extended to

the almost sure stability of Ito stochastic differential equations by means of semi-

continuous Lyapunov functions satisfying a suitable system of partial differential

inequalities in viscosity sense. The a.s. stabilizability of controlled degenerate

dif-

fusion processes is also treated. The key tools are some geometric characterizations

of the invariance and viability properties of closed sets for controlled diffusions ob-

tained by PDE-viscosity methods.

1 Introduction

Beginning with the work of P.-L. Lions [21], the theory of viscosity solutions to second

order degenerate elliptic PDEs has given many contributions to the theory of optimal

control of degenerate diffusion processes. The link between the two fields is the Hamilton-

Jacobi-Bellman equation associated to the value function of an optimal control problem

via the Dynamic Programming Principle. The books by Fleming and Soner [14] and Yong

and Zhou [26] give excellent surveys of the subject, see also [6] for the connections of first

order Hamilton-Jacobi equations with deterministic control.

In this paper we use viscosity methods to study a problem where no optimization nor

value functions are involved. The issue is the stability of the origin, in a suitable sense,

for the iV-dimensional stochastic differential equation (SDE)

= f(X

)dt + a(X

)dB

*This research was partially supported by M.U.R.S.T., project "Analysis and control of deterministic

and stochastic evolution equations", and by the European Community, TMR Network "Viscosity solutions

and their applications".

322

323

where B

is an M-dimensional Brownian motion. There are many different notions of

stochastic stability, see, e.g., [20, 17, 15] and the references therein. Here we consider

the almost sure (Lyapunov) stability, a strong property that can be satisfied only by

truly degenerate diffusion processes. To explain our result let us first recall the classical

direct Lyapunov method for deterministic systems. Basically, a Lyapunov function V

for the ODE X[ = f(X

) is a positive definite and proper function decreasing along the

trajectories of the ODE. If V is differentiable the last condition is equivalent to the partial

differential inequality

DV(x)-f{x)<0

inR^UO}.

The existence of a smooth Lyapunov function implies the stability of the origin for the

ODE, but it is not a necessary condition (see, e.g., [16, 27]). Necessary and sufficient con-

ditions can be obtained by considering merely continuous, or even lower semicontinuous,

Lyapunov functions satisfying the differential inequality in a suitable weak sense. Similar

results were obtained also for the stabilizability of controlled deterministic systems (or,

more generally, differential inclusions), i.e., the problem of finding at least one trajectory

that remains in a neighborhood of the origin for any given small initial position. If a

the control function in the system X'

= f(X

, a

), the appropriate differential inequality

involves now a Hamilton-Jacobi-Bellman operator:

mmDV{x)-f(x,a)<0

inR

\{0}.

There is a large literature on this, making use of different tools of nonsmooth analysis.

Let us mention, for instance, [27, 1, 23, 5] using generalized Dini directional derivatives

(also known as contingent derivatives), [24, 19, 25] employing viscosity solutions, and

[12,

22] using proximal derivatives, and see also the references therein.

The main result of Section 1 is that the origin is almost surely stable for the SDE

if there is a lower semicontinuous Lyapunov function satisfying, again, a differential in-

equality, which now is

-DV(x)

•

f(x) - ]-Tr [a{x)a{x)

V(x)] > 0 in R

\ {0}, (1)

and, in addition, the system of equations

{x)DV{x) = 0

inl

\{0}.

(2)

Both (1) and (2) are interpreted in the viscosity sense and no ellipticity assumption is

made on the matrix aa

. Note that the last condition means that there is diffusion only

in the directions tangential to the level sets of V. The proof boils down to showing that

the sublevel sets of the Lyapunov function V are (a.s.) invariant for the SDE. This is

obtained by the stability properties of viscosity supersolutions and by a characterization of

invariant sets for SDE's due to the first author and Goatin

[7].

The necessary and sufficient

condition for the invariance of an arbitrary closed set K in [7] has a simple geometric

formulation in terms of the second order normal cone A/J(x) to K at a; this set has a

purely deterministic definition, in contrast to the stochastic contingent set introduced by

324

Aubin and Da Prato [2, 1] for earlier characterizations of invariance. Also the proof of

the invariance theorem in [7] makes use of PDE methods only (see also [8] for another

proof using just the most basic facts of the viscosity theory).

A sufficient condition for a.s. stability was obtained a few years ago by Aubin and

Da Prato [4] by means of Lyapunov functions satisfying an inequality involving a suitable

second order stochastic derivative of V, instead of (1) and (2). Presumably our condition

is equivalent to theirs, but we believe it is easier to check and more readily readable, at

least for people in the PDE community. Moreover, our methods extend to the problem of

a.s. stabilizability of controlled diffusion processes dX

= f(X

, a)dt + u(X

, a)dB

. The

result for this case is presented in Section 2. The inequality (1) must be replaced by the

second order Hamilton-Jacobi-Bellman inequality

max l-DV(x)

•

f(x, a) - ]-Tr [aa

(x, a)D

V(x)] 1 > 0 in R

\ {0},

whereas the system (2) takes a slightly more technical form, see Definition 3.2. Now one

needs a viability theorem for arbitrary closed sets K, namely, a condition ensuring that at

least one trajectory of the controlled SDE remains forever in K a.s. (this property is also

called controlled invariance). This problem was studied by Aubin and Da Prato [3] by

stochastic methods and by Buckdahn et al. [11] using viscosity solutions. We use instead

a characterization of viable sets obtained very recently by the first author and Jensen [8]

in terms of the second order normal cone NK(X) by means of PDE-viscosity methods.

Although we do not include in the present paper the full proof of the stabilizability

theorem, we report the results of [8] that are involved in it.

2 Lyapunov method for almost sure stability

Consider the Ito stochastic differential equation in M

(SDE) |

dXt = /(Xt)

+ a

(

dBt

< * > °'

| XQ = x.

where B

is an M-dimensional Brownian motion. Assume that / and a are continuous

functions defined in R

, taking values, respectively, in R-^ and in the space of N x M

matrices, and satisfying

\cr{x)-a{y)\<C\x-y\, Vz.j/eR", (3)

\m-f(y)\<C\x-y\, Vx,yeR

, (4)

The diffusion process described by (SDE) can be degenerate, because the matrix

a(x) := -a(x)a(x)

is merely positive semidefinite.

325

Definition 2.1 (a.s. stability). The null state X = 0 is almost surely stable for (SDE)

if for every e > 0 there exists S > 0 such that if\x\<Sthe corresponding trajectory X. of

(SDE) starting at x verifies \X

\ < e for allt > 0 almost surely.

Remark 2.2. This notion of stability for a stochastic system is considerably stronger

than the usual concepts of stochastic stability (see the monographs [20, 17]). Here we

require that, if the initial value is close enough to 0, the sample paths of the process

issuing from a; remain forever within a certain neighborhood of the origin with probability

not just in the mean, or with probability increasing to 1 as |a;| decreases to 0 (stability

in probability).

Definition 2.3 (Lyapunov function). A function V : R

—•

is a Lyapunov func-

tion for (SDE) if it satisfies the following conditions:

(i) it is lower semicontinuous;

(ii) it is positive definite, i.e., V(0) = 0 and V(x) > 0 for all x ^ 0;

(Hi) it is proper, i.e., lim|.

>00

V(x) = oo or, equivalently, the sets {x\V(x) < fi} are

bounded

for every /x

;

(iv) it is a viscosity supersolution of the equation

-DV(x)

•

f(x) - Tr [a(x)D

V(x)] =0 inR

{0};

(5)

(v) it is a bilateral viscosity supersolution of the equations

<7i(x)-DV(x) =

inM.

\{Q), i =

l,...,M,

(6)

where Oi(x) is the i-th column of the matrix a(x).

We recall that a bilateral viscosity supersolution, or lower semicontinuous viscosity

solutions as defined by Barron and Jensen (see [10, 6]), is a l.s.c. function such that a-

(x)

•

p = 0 for all p in the subdifferential

D~V(x).

The definition of viscosity supersolution of

second order degenerate elliptic equations such as (5) can be found, for instance, in [13].

Remark 2.4. To motivate this definition, let us assume that V is twice continuously

differentiable and look for a sufficient condition on V to be nonincreasing along the tra-

jectories of the stochastic system (SDE), i.e.,

V(X

) < V(X

) for t > s almost surely.

By Ito's formula the inequality dV(X

)/dt < 0 becomes

[DV(X

)

•

f(X

) + Tr (a(X

)

•

V(X

))} dt + [a

)DV(X

)} dW

< 0,

and by the properties of the Brownian motion one is led to the conditions

DV(X

)-f(X

)+Tr(a(X

V(X

)) < 0,

)DV(X

) = 0.

326

Remark 2.5. From conditions (i), (ii), and (Hi) in Definition 2.3 it is standard to infer

the following property of Lyapunov functions : for every e > 0 there exists 9 > 0 such

that, ifV(x) < 6, then \x\ < e.

If, in addition, V is continuous at 0, then the sublevel sets of V form a basis of

neighborhoods ofO. This property allows us to deduce the a.s. stability of the Ito stochastic

equation (SDE) from the invariance of the sublevel sets of

Lyapunov function, whenever

it exists and is continuous at 0.

In view of the last remark we can prove the Lyapunov stability in 0 of the system by

showing that the sublevel sets of a suitable l.s.c. function are invariant for (SDE). In [7]

it is proved the following Invariance Theorem, which states the equivalence between the

invariance of a closed set and a Nagumo-type geometric condition involving the second

order normal cone.

Definition 2.6. The 2nd order normal cone is defined as:

jvl(x) := {(p, Y) e R

x S(N) : for y -* x y e K

p-(y-x)

\(y-x)-

Y(y - x) > o(\y -

)},

where S(N) is the set of symmetric N x N matrices.

Theorem 2.7 (Invariance Theorem [7]). The trajectories X. of (SDE) starting in a

closed set K satisfy X

e K for allt > 0 almost surely if and only if

f(x)

•

p + trace [a(x)Y] > 0 Vz e dK,

V(p,

Y) € A/£(x). (7)

Now we can prove the following stability theorem generalizing the direct Lyapunov

method to nonsmooth Lyapunov functions and to stochastic dynamical systems.

Theorem 2.8 (a.s. stability). Let V be a Lyapunov function for (SDE) (Definition

2.3) continuous at 0. Then the null state X = 0 is almost surely stable for (SDE).

Proof.

Consider for \i > 0 the sublevel set of the function V:

K = {x\V(x) </i}.

We want to show that it is invariant, so we must check condition (7). We consider the

function U = —V: U is an upper semicontinuous function which satisfies

-p

•

f(x) - trace \a(x)

•

Y] < 0, (8)

for all (p, Y) contained in the second-order superjet of U at x

'+U(x) = {(jp,Y) eR

x S(N) : fory^x

U(y) < U{x) +p-{y-x) +

\{y-x)-

Y(y - x) + o(\y - x\

)}

and

o-i(x)-p = 0 i =

l,...,M,

(9)

for all p contained in the superdifferential D

U(x) of U at x. The set K can be written

as {x | U(x) >

—[J,}.

To prove the invariance of K we need the following lemma on the

change of unknown for second order partial differential equations:

327

Lemma 2.9. Let u be a viscosity subsolution of equation (8) and a bilateral subsolution

of equations (9). Let

<j>

e C

(2R)

strictly increasing with

cf>'

> 0. Then the map w =

4>ou

is a viscosity subsolution of equation (8) too.

Proof of the lemma. It is easy to check that, if (p,Y) e

2,+

w(x),

then

(il)'(w(x))p,il)'(w(x))Y + $"{w(x))p®p) e

u(x),

where ip is the inverse of

<f>

and p®p is

N x N matrix whose (i,j) entry is piPj. Then

—ip'(w(x))p

•

f(x)

—

trace [a(x)

•

ip'(w(x))Y]

—

trace [a(x)

•

tp"{w{x))p ®p\ < 0.

Moreover, if (p,Y) e J

w(x) then p

w{x) and ip\w(x))p e

u(x),

so by (9)

and '4)'(w(x)) > 0 we get

trace \a{x)

•

ip"{w{x))p ®p]= .;,/ ; '.' V[«r

(x)

•

^'{w{x))pf = 0.

Therefore

i>'(w{x)) [-p

•

f(x) - trace (a(x)

•

Y)] < 0,

which concludes the

proof.

•

We define now for every

> 0 a nondecreasing continuous real function

i>x(t)

t>-n,

X(t +

ju)

-/j, - - < t < -fi,

-1 t<-

We claim that the function i/j

o U is a viscosity subsolution of equation (8) for every

A. It is actually sufficient to choose a sequence ip

of smooth real maps with

ip'

> 0 and

converging uniformly on compact sets to ip

. Then for every n the map ip

°U is a viscosity

subsolution of equation (8) by Lemma 2.9. By the stability of viscosity subsolutions with

respect to uniform convergence (see, e.g., [13, 6]) we get the claim.

The sequence 4>\°U decreases as

A —*

+oo to the indicator function

«"-{

-.:t£

Since the sequence is decreasing and all the maps are upper semicontinuous, the point-

wise limit I coincides with the upper weak limit (or half relaxed semilimit) of the sequence

defined as

limsup*i/'A ° U(x) := inf

supj^

o U(y)\\x

—

y\ < S, 0 <

—

< <5},

A-.00

>° A

328

see [9, 13, 6]. It is well known that viscosity subsolutions are stable with respect to the

upper weak limit [9, 13, 6]. Therefore the indicator function of K is a viscosity subsolution

of equation (8).

It is easy to check by the definitions that for x € K : J

I(x) =

I(x),

and that

for x 6 dK : J^

I(x) = Nx(x). Therefore we have proved that

Vz G dK,

V(p,

Y) eNl{x): f(x)

•

p + trace \a(x)

•

Y] > 0,

and by the Invariance Theorem the closed set K is invariant. •

3 Viability and a.s. stabilizability of controlled sys-

tems

In this section we consider a controlled Ito stochastic differential equation:

= a(X

, a

)dB

+ f{X

)dt,

t > 0,

^ 1X

= *.

We assume that a

takes values in a given compact metric space A, f, a are continuous

functions defined in R

x A, taking values, respectively, in R^ and in the space of N x M

matrices, and satisfying

||cr(a;,Q;)

—

a(y,a)\\ < C\x

—

y\, for all x,y e

and all a e A, (10)

\f{x,a) - f{y,a)\ <

C\x-y\,

for all x,y € fi and alia £ A. (11)

For the precise definition of the set of admissible control functions A, or admissible

systems, we refer to [21, 14, 18].

Definition 3.1 (almost sure stabilizability). The null state X = 0 is almost surely

stabilizable for (CS) if for every e > 0 there exists S > 0 such that for every x with \x\ < S

there exists an admissible control function a. e A with the property that the corresponding

trajectory X. starting at x verifies \X

\ < s for allt>0 almost surely.

We define a(x,a) := ^a(x,a)a(x,a)

. The appropriate definition of a Lyapunov

function for almost sure stabilizability is the following.

Definition 3.2 (control Lyapunov function). A function V : R

—•

is a control

Lyapunov function for (CS) if it satisfies the conditions (i), (ii) and (Hi) of Definition

2.3

and

(iv ') it is a viscosity supersolution of the equation

max

{-DV(x)

•

f(x, a) - trace [a(x, a)D

V(x)] } = 0 in R

\ {0},

(v ') for every

(p,

Y) £ J

'~V(x) and i = 1,...

ai(x,a)

•

p = 0 Va e

argmax{—

•

f(x,a)

—

trace[a(x,a)Y]} .

329

If we want to follow the same approach as in the previous section, now we must prove

that the sublevel sets of the Lyapunov function are viable, or controlled invariant, for

(CS),

in the following sense.

Definition 3.3. A closed set K C R

is viable for the stochastic system (CS) if for all

initial points x £ K there exists an admissible control a, € A such that the corresponding

trajectory X, satisfies X

6 K for allt > 0 almost surely.

This property was studied by Aubin and Da Prato [3], Buckdahn et al. [11], and, more

recently, by Bardi and Jensen [8]. The last authors introduced a weaker notion named

e-viability:

Definition 3.4. The closed set K is e-viable for (CS) if for all bounded uniformly con-

tinuous functions I > 0, I = 0 in K, and all T > 0,

inf E 7 l(X

)dt = 0,

MxeK,

CC.&A J

where X, = X

is the solution of (CS) starting at x.

The main result of [8] is the following equivalence between the e-viability of a closed

set K and a Nagumo-type geometric condition.

Theorem 3.5 (e-viability theorem [8]). Given a closed set K C M

the following

properties are equivalent:

(i) K is e-viable for (CS);

(ii) for all

> 0 and all bounded uniformly continuous functions I > 0, I = 0 in K :

f+OO

inf E I l(X

)e-

dt = 0

Vxeif;

a.€A J

(Hi) the geometric condition

\/x E dK,

V(p,

y)eA/J(i), Ba<=A:f(x,a)-p + trace [a(x, a)Y] > 0 (12)

holds.

This theorem combined with an existence result for optimal controls in [18] has the

following consequence.

Corollary 3.6 (Viability theorem [8]). Assume in addition

{(a(x,a),

f(x,a)) : a e A} is convex for all x e K'. (13)

Then K is viable for (CS) if and only if (12) holds.

By means of these results we can prove a Lyapunov theorem on the a.s. stabilizability

of controlled degenerate diffusion. Its complete proof will appear in a forthcoming paper.