Rend. Sem. Mat. Univ. Poi. Torino Voi. 56 ,4 (1998) Control Theory and its Appi.

M. Bardi* » S. Bottacin

ON THE DIRICHLET PROBLEM FOR NONLINEAR DEGENERATE ELLIPTIC EQUATIONS

AND APPLICATIONS TO OPTIMAL CONTROL

Abstract. We construct a generalized viscosity solution of the Dirichlet problcm for fully

nonlinear degenerate elliptic equations in general domains by the Perron-Wiener-Brelot method. The result is designed for the Hamilton-Jacobi-Bellman-Isaacs equations of time-optimal stochastic control and differential gàmes with discon-tinuous value function. We study several properties of the generalized solution, in particular its approximation via vanishing viscosity and regularization of the do­main. The connection with optimal control is proved for a deterministi e minimum-time problem and for the problem of maximizing the expected escape time of a degenerate diffusion process from an open set.

Introduction

The theory of viscosity solutions provides a general framework for studying the partial differ­ential equations arising in the Dynamic Programming approach to deterministic and stochastic optimal control problems and differential games. This theory is designed for scalar fully nonlin­ear PDEs

(1 ) F(x, u(x), Du(x), D2u(x)) = 0 in fi,

where fi is a general open subset of R / v , with the monotonicity property

,2) F(x, r, p, X) < F(x, s, /;, Y) \ir <s and X — Y is positive semidefinite,

so it includes \st order Hamilton-Jacóbi equations and 2nd order PDEs that are degenerate elliptic or parabolic in a very general sense [18, 5],

The Hamilton-Jacobi-Bellman (briefly, HJB) equations in the theory of optimal control of diffusion processes are of the form

(3) supCau = 0, aeA

*PattiaIIy supported by M.U.R.S.T., projeets "Problemi nonlineari nell'analisi e nelle applicazioni fisiche, chimiche e biologiche" and "Analisi e controllo di equazioni di evoluzione deterministiche e sto­castiche", and by the European Community, TMR Network "Viscosity solutions and their applications".

13

14 M. Bardi - S. Bottacin

where a is the control variable and, for each oc; Ca is a linear nondivergence form operator

where / and e are the running cost and the discount' rate in the cost functional, b is the drift of the system, a = \oaT and a is the variance of the noise affecting the system (see Section 3.2). These equations satisfy (2) if and only if

(5) af, (x)ljj-j > 0 and ca(x) > 0, for ali x e Q, a e A, % eRN . ij J

and these conditions are automatically satisfied by operators coming from control theory. In the case of determi ni stic systems we have af- — 0 and the PDE is of ]st order. In the theory of two-person zero-sum deterministic and stochastic differential games the Isaacs' equation has the form

(6) sup inf Ca^u = 0,

where /3 is the control of the second player and Ca^ are linear operators of the form (4) and satisfying assumptions such as (5).

For many different problems it was proved that the value function is the unique continuous viscosity solution satisfying appropriate boundary conditions, see the books [22, 8,4, 5] and the references therein. This has a number of useful consequences, because we have PDE methods available to tackle séveral problems, such as the numerical càlculation of the value function, the synthesis of approximate optimal feedback controls, asymptotic problems (vanishing noise, penalization, risk-sensitive control, ergodic problems, singular perturbations .. . ). However, the theory is considerably less general for problems with discontinuoiis value function, because it is restricted to deterministic systems with a single controller, where the HJB equation is of first order with convex Hamiltonian in the p variables. The pioneering papers on this issue are due to Barles and Perthame [10] and Barron and Jensen[ll], who use different definitions of non-continuous viscosity solutions, see also [27, 28, 7, 39, 14], the surveys and comparisons of the different approaches in the books [8, 4, 5], and the references therein.

For cost functionals involving the exit Urne of the state from the set Q, the value function is discontinuous if the noise vanishes near some part of the boundary and there is not enough controllability of the drift; other possible sources of discontinuities are the lack of smoothness of d£2, even for nondegenerate noise, and the discontinuity or incompatibility of the boundary data, even if the drift is controllable (see [8, 4, 5] for examples). For these functionals the value should be the solution of the Dirichlet problem

r_. | F(x, u, Du, D~u) = 0 in Q , l u — g on 3Q,

where g(x) is the cost of exiting Q at x and we assume g e C(BQ). For 2nd order equations, or \st order equations with nonconvex Hamiltonian, there are no locai definitions of.weak solution and weak boundary conditions that ensure existence and uniqueness of a possibly discontinuoiis solution. However a global definition of generalized solution of (7) can be given by the following variant of the classica! Perron-Wiener-Brelot method in potential theory. We define

S : - [w e BUSC(ti) subsolution of (1), w <g on 8Q}

Z := {W e'ÀL5C(fi)supersolutionof(l), W > g on 3Q),

On the Dirichlet probìem 15

where BUSC(Q) (respectively, BLSC(S2)) denote the sets of bounded upper (respectively, lower) semicontinuous functions on £2, and we say that u : £2 -» R is a generalized solution of (7)if

(8) u(x) = sup w(x) = inf W(x).

With respect to the classical Wiener's definition of generalized solution of the Dirichlet probiem for the Lapalce equation in general nonsmooth domains [45] (see also [16,' 26]), we only replace sub- and superharmonic functions with viscosity sub- and supersolutions. In the classical theory the inequality s u p ^ s w < i n% e z W comes from the maximum principle, here it comes from the Comparìson Principle for viscosity sub- and supersolutions; this important result holds under some additional assumptions that are very reasonable for the HJB equations of control theory, see Section 1.1; for this topic we refer to Jensen [29] and Crandall, Ishii and Lions [18], The main difference with the classical theory is that the PWB solution for the Laplace equation is harmonic in Q and can be discontinuous only at boundary points where dù is very irregular, whereas here u can be discontinuous also in the interior and even if the boundary is smooth: this is because the very degenerate ellipticity (2) neither implies regularizing eflects, nor it guarantees that the boundary data are attained continuously. Note that, if a continuous viscosity solution of (7) exists, then it coincides with u and both the sup and the inf in (8) are attained.

Perron's method was extended to viscosity solutions by Ishii [27] (see Theorem 1), who used it to prove general existence results of continuous solutions. The PWB generalized solution of (7) of the form (8) was studied independently by the authors and Capuzzo-Dolcetta [4, I] and by M. Ramaswamy and S. Ramaswamy [38] for some special.cases of equations of the form (1), (2). In [4] this notion is called envelope solution and several properties are studied, in particular the equivalence with the generalized minimax solution of Subbotin [41, 42] and the connection with determinis.tic optimal control. The connection with pursuit-evasion ganies can be found in [41,42] within the Krasovskii-Subbotin theory, and in our paper with Falcone [3] for the Fleming value; in [3] we also study the convergence of a numerica] scheme.

The purposes of this paper are to extend the existence and basic properties of the PWB solution in [4, 1, 38] to more general operators, to prove some new continuity properties with respect to the data, in particular for the vanishing viscosity method and for approximations of the domain, and finally to show a connection with stochastic optimal control. For the sake of completeness we give ali the proofs even if some of them follow the sanie argument as in the quoted references.

Let us now describe the contents of the paper in some detail. In Subsection 1.1 we recali some known definitions and results. In Subsection 1.2 we prove the existence theorem under an assumption on the boundary data g that is reminiscent of the compatibility conditions in the theory of \st order Hamilton-Jacobi equations [34, 4]; this condition implies that the PWB solution is either the minimal supersolution or the maximal subsolution (i.e., either the inf or the sup in (8) is attained); and it is verified in time-optimal control problems. We recali that the classical Wiener Theorem asserts that for the Laplace equation any continuous boundary function g is resolutive (i.e., the PWB solution of the corresponding Dirichlet problem exists), and this was extended to some quasilinear nonuniformly elliptic equations, see the book of Heinonen, Kilpelàinen and Martio [25]. We do not know at the moment if this result can be extended to some class of fully nonlinear degenerate equations; however we prove in Subsection 2.1 that the set of resolutive boundary functions in our context is closed under uniform convergence as in the classical case (cfr. [26, 38]).

In Subsection 1.3 we show that the PWB solution is consistent with the notions of general­ized solution by Subbotin [41,42] and Ishii [27], and it satisfies the Dirichlet boundary condition

16 M. Bardi - S. Bottacin

in the weak. viscosity sense [10, 28, 18, 8, 4]. Subsection 2.1 is devoted to the stability of the PWB solution with respect to the uniform convergence of the boundary data and the operato»- F. Jn Subsection 2.2 we consider merely locai unilbrm perturbations of F, such as the vanishing viscosity, and prove a kind of stability provided the set Q is simultaneously approximated from the interior.

In Subsection 2.3 we prove that for a nested sequence of open subsets Qn of £2 such that UH &n = ^» 'f un is the PWB solution of the Dirichlet problem in Q„, the solution u of (7) satisfies

(9) u(x) = limw„(jt), xeQ. n

This allows to approximate u with more regular solutions u„ when dQ is not smooth and Q„ are chosen with smooth boundary. This approximation procedure goes back to Wiener [44] again, and it is standard in elliptic theory for nonsmooth domains where (9) is often used to define a generalized solution of (7), see e.g. [30, 23, 12, 33]. In Subsection 2.4 we characterize the boundary points where the data are attained continuously in terni sof the existence of suitable locai barriers.

The last section is devoted to two applications of the previous theory to optimal control. The first (Subsection 3.1.) is the classical minimum time problem for deterministic nonlinear systems with a closed target. In this case the lower semicontinuous envelope of the value function is the

• PWB solution of the homogeneous Dirichlet problem for the Bellman equation. The proof we give heré is different from the one in [7,4] and simpler. The second application (Subsection 3.2) is about the problem of maximizing the expected discounted time that a controlied degenerate diffusion process spends in Q. Here we prove that the value function itself is the PWB solution of the. appropriate problem. In both cases g == 0 is a subsolution of the Dirichlet problem, which implies that the PWB solution is also the minimal supersolution.

It is worth to mention some recent papers using related methods. The thesis of Bettini [13] studies upper and lower semicontinuous solutions of the Cauchy problem l'or degenerate parabolic and \st order equations with applications to finite horizon differential games. Our paper [2] extends some results of the present one to boundary value problems where the data are prescribed only on a suitable part of d&. The first author, Goatin and Ishii [6] study the boundary value problem for (1) with Dirichlet conditions in the viscosity sense; they construct a PWB-type generalized solution that is also the limit of approximations of Q from the outside, instead of the inside. This solution is in general different from ours and it is related to control problems involving the exit time from S2, instead of £2.

1. Generalized solutions of the Dirichlet problem

1.1. Preliminaries

Let F be a continuous function

F :QxRxRN.x S(N) -*R,

where £2 is an open subset of R^, S(N) is the set of symmetric N x N matrices equipped with its usuai order, and assume that F satisfies (2). Consider the partial differential equation

(10) F(x,u(x), Du(x), D2u(x)) = 0 inft,

On the Dirichlet probkm 17

where u : Q ->• R, DM denotes the gradient of u and D2// denotes the Hessian matrix of second derivatives of u. From now on subsolutions, supersolutions and solutions of this equation will be understood in the viscosity sénse; we refer to [18, 5] for the definitions. For a general subset E of RN we indicate with USC(E), respectively LSC(E), the set of ali functions E -^ R upper, respectively lower, semicontinuous, and .with BUSC(E), BLSC(E) the subsets of functions that are also bounded.

DEFINITION 1. We will say that equation (10) satisfies the Comparison Principle iffor alt. subsolutions w e BUSC(Q) and supersolutions W e BLSC(Q) of(JO) sudi that w < W on 0 Q, the inequality w < W holds in Q.

We refer to [29, 18] for the strategy of proof of some comparison principles, examples and references. Many results of this type for first order equations can be found in [8, 4].

The main examples we are interested in are the Isaacs equations:

(11) supinf£a^H(jc)=0 a fi

and

(12) infsup£aA<(jt) = 0 , fi a

where

C^uix) = _< ,« • " W " J J - + bf(x)^- + c"-0(x)u - f-hx). '•' dXjOXj ' dXj

Here F is

F(x,r, p, X) = supinf{-trace (aa^(x)X) + ba^(x) • p + ca^(x)r - fa^(x)}. a fi

If, for ali* e Q,aa^(x) = ^cra^(x)(acl^(x))T, where att'P(x) is a matrix of order N x M, r

denotes the transpose matrix, aa'P, ba'P, ca-P, fa'P are bounded and uniformly continuous in S2, uniformly with respect to a, fi, then F is continuous, and it is proper if in addition ca'P > 0 for ali a, fi.

Isaacs equations satisfy the Comparison Principle if Q is bounded and there are positive constants K\, K^, and C such that

(13) F(x, t, p, X) - F(x, s, q, Y) < max{K\trace (K - X), K\ (t - s)} + K2\p - q\,

for ali Y < X and t < s,

(14) \\o-a<P(x)-o-a<P(y)\\ < C\x-yl forali*, ye Mandali a, fi

(15) \ba-p{x)-ba'P{y)\ < C\x-y\, for ali x, ye Mandali a, fi,

see Corollary 5.11 in [29]. In particular condition (13) is satisfied if and only if

max{Xa'P(x), ca<P(x)} > K > 0 for ali x e ?2, a e A, fi e B ,

where Xa,P(x) is the smallest eigenvalue of aa-^(x). Note that this class of equations contains as special cases the Hamilton-Jacobi-Bellman equations of óptimal stochastic control (3) and linear degenerate elliptic equations with Lipschitz coefficients.

18 M. Bardi - S. Boltàcin

Given a function u : fi -* [—oo, +00], we indicate with u* and u*, respectively, the upper and the lower semicontinuous envelope of u, that is,

M*(JC) := lim sup{w(y) : y e fi, \y — x\ < r}, r\0

u*(x) := lim inf{w(y) : y e fi, | y - * | < r } . r \0 " ' '

PROPOSITION 1. L<?J 51 (respectively Z) be a set offimctìons such that for ali w e S (re­spectively W e Z) w* is a subsolution (respectively W* is a supersolution) of(10). Defme the function

u(x):~ sup w(x), JC e fi, (respectively u(x) := inf W(x)) . weS ' WeZ

lfu is locally bounded, then u* is a subsolution (respectively u* is a supersolution) of(10).

The proof of Proposition 1 is an easy variant o.f Lemma 4.2 in [18].

PROPOSITION 2. Let wn e BUSC(Q) be a sequence ofsubsolutions (respectively W„ e BLSC(fi) a sequence of supersolulions) of(10), such that w„(x) \ u(x) far ali x e fi (respec­tively Wn (x) y u(x)) and u is a locally bounded function. Then u is a subsolution (respectively supersolution) of(IO).

For the proof see, for instance, [4]. We recali that, for a general subset E ofRN and x e E, »econd order supei

pairs (p, X) such that the second order superdifferential of u at JC is the subset J~E M(JC) of R^ X S(N) given by the

u(x) < u(x) •+- p • (x — x) + -X(x — JC) -,(x — x) •+ oQx — JC|~)

for E B x -» x. The opposite inequality defines the second order subdifferential of u at x,

Jg~u(x).

LEMMA 1. Let u* be a subsolution of(10). Ifu* fails to be a supersolution at some point 2 — x e fi, Le. the re exist (/?, X) e J^ M*(JC) such that

F(£, «*(*), p,X)<0,

then for ali le > 0 small enough, the re exists U^ : fi -> R such that Uj£ is subsolution of(10) and

f Uk(x)>u(x), supa(£/jfc-M) > 0, [ Ufrix) = u(x)for ali x è fi such that \x — x\ > k .

The proof is an easy variant of Lemma 4.4 in [18]. The Iastresultof this subsecti-on is Tshii's extension of Perron's method to viscosity solutions [27].

THEOREM 1. Assume there exists a subsolution u \ and a supersolution un of(10) such that «I < i*2> and consider the functions

V(x) := sup{w(x) : u\ < w < un, w* subsolution o/'(10)},

W(x) := \nf{w(x) : u\ < w < un_, u>* supersolution 0/(10)}.

Then U*, W* are subsolutions of(10) and (/*, W* are supersolutions of(IO).

On the Dirtchlet problem 19

1.2. Existence of solutions by the PWB method

In this section we present a notion of weak solution for the boundary value problem

F(x,u, Du, D~u) =0 in fi, ( 1 6 ) ' u = g o n d a ,

where F satisfies the assumptions of Subsection 1.1 and g : Ofi -> R is continuous. We recali that<S, Z are the setsof ali subsolutions and ali supersolutions of (16) defined in the Introduction.

DEFINITION2. The function defined by

i¥(,(x) := sup w(x), w<=S

is the lower envelope viscosity solution, or Perron-Wiener-Brelot lower solution, of(16). We will refer to it as the lower e-solution. The function defined by

~HR(x):= inf W(x),

is the upper envelope viscosity solution, or PWB upper solution, of(J6), briefiy upper e-solution. If H_„ — Hg, then

Hg •= Kg = Hg

is the envelope viscosity solution or PWB solution of (16), briefiy e-solution. In this case the data g are called risolutive.

Observe that Kg < Hs by the Comparison Principle, so the e-solùtion exists if the inequal-ity > holds as well. Next we prove the existence theorem for e-solutions, which is the main result of this section. We will need the following notión of global barrier, that is much weaker than the classical one.

DEFINITION 3. We say that w is a lower (respectively, upperj barrier al a point x € dQ if w e S (respectively, w e Z) and

lim w(y) = g(x). y-*x

THEOREM 2. Assume that the Comparison Principle holds, and that S, Z are nonempty.

i) Ifthere exists a lower barrier at ali points x e dQ, then Hg = m i n ^ e ^ W " ' t n e e-solution of(Ì6).

ii) Ifthere exists an upper barrier at ali points x e SQ, then Hg = max^g^ w is the.e-solution of(16).

Proof Let w be the lower barrier at x € <9fi, then by definition w < / / „ . Thus

(1L>)*(*) = lim inf Ke(y) > lim inf w(y) - g(x). iS y-*x * _v—>JC

By Theorem 1 (Kg)* is a supersolution of (10), so we can.conclude that (//,,)* e Z. Then

(K8)*> ~Hg > Kg, so H_g = 77g and Kg € Z.

D

20 M. Bardi - S. Bottacin

EXAMPLE 1. Considerane linear problem with Lipschitz coefficients

f -cijj (x)uXìXj (x) + bj(x)uXj (x) + c(x)u(x) =0 in Q , { ' \ u(x) = g{x) onOft,

with the matrix cijj (x) nonnegative semidefìnite and such that a\ \ (x) > \x > 0 for ali x e Q. In this case we can show that ali continuous functions on 3Q are resolutive. Thè proof follows the classical one for the Laplace equation, the only hard point is checking the superposition principle for viscosity sub- and supersolutions. This can be done by the same methods and under the sanie assumptions as the Comparison Principle.

1.3. Consistency properties and examples

The next results give a characterization of the e-solution as pointwise limit of sequences of sub and supersolutions of (16). If the equation (10) is of first order,. this. property is essentially Subbotin's definition of (generalized) minimax solution of (16) [4.1,42].

THEOREM 3. Assume that the Comparison Principle holds, and that S, Z are nonempty.

i) Ifthere exists a e S continuous at each point ofdQ and such that u_= g on dQ, then tfiere exists a sequence wn e S such that wn /* H^.

ii) Ifthere exists u e Z continuous at each point of(ìQ and such that 77 = g on d£2, then the re exists a sequence Wn e Z such that Wn \ Hg. .

Proof. We give the proof only for /), the same proof works for ii). By Theorem.2 Hg = mmWe.Z W. Given e > 0 the function

(18) u€(x) := sup{it)(x) : w e S, w(x) = ii(x) ifdist(.r, dQ) < e],

is bounded, and u$ < uè for e < 8. We detìne

V(x) := lim («i/„)*(jf),

and note that, by definition, Hg > ue > (wc),*, and then Hg > V. We claim that (we)* is supersolution of (10) in the set

S2€ := [x e Q : dist (x, dQ) > e}.

To prove this claim we assume by contradiction that (z^)* fails to be a supersolution at ;y e Q6. Note that, by Proposition 1, (u€)* is a subsolution of (10). Then by Lemma 1, for ali k > 0 small enough, there exists U^ such that Uj£ is subsolution of (10) and

(19) sup(£4 - u€) > 0, Uk(x) = u€(x) if\x-y\>k.

We fix k < dist (y, 3Q) — e, so that U^ix) = ue(x) = M(JC) for ali x such that dist(x, <ìQ) < €. Then U£(x) = u(x), soU£ e S and by the definition of ue -we obtain U% < we. This gives a contradiction with (19) and proves the claim.

By Proposition 2 V is a supersolution of (10) in Ù. Moreover if x e <ìQ, for ali e > 0, (ue)*(x) — g(x), because u€(x) = i±(x) if dist (x, dQ) < e by definition, i± is continuous and u = g on 3fì. Then V > g on 3Q, and so V e Z.'

On the Dirichlet probìem 21

To complete the proof we define tu,, := (MI/,,)*, and observe that this is a nondecreasing sequence in <S whose pointwise limit is > V by definition of V. On the other hand w„ < Hg by definition of Hg, and we nave shown that Hg = V, so w„ f Hg,

D

COROLLARY 1. Assume the hypotheses of Theo rem 3. Then Hg is the e-solution of(l6) ìf andonly ifthere exisl two sequences offimctions wn e S, W„ e Z, sudi that wn = VV„ = g on dQ and far ali x e Q

Wn(x) -> Hg(x), Wn(x) -> Hg(x) as n -> oo .

REMARK 1. It is easy to see from the proof of Theorem 3, that in case /), the e-solution Hg satisfies

Hg (x) — sup Ut (x) x 6 Q , €

where

(20) ue(x) := sup{w;(jc) : w e S, w(x) = u_(x) for x e £2 \ S€},

and S€, € e]0, 1], is any family of open sets such that (M)e e £>, (H)6 D (»)§ l'or e < 8 and U ©e = V.

EXAMPLE 2. Consider the Isaacs equation (11) and assume the sufficient conditions for the Comparison Principle.

• rf

g = 0 and fa'P(x) > 0 for ali x e Q, a e A, )3 e D ,

then w = 0 is subsolution of the PDE, so the assumption /) of Theorem 3 is satisrìed.

• If the domain Q is bounded with smooth boundary and there exist a e A and \x > 0 such that

rfjP(x)b$j > /x|£|2 for ali peB, xe~Q, t-eRN ,

then there exists a classical solution u of

f inf£"'0w=O inS2,

\ u_— g on 0Q ,

see e.g. Chapt. 17 of [24]. Then «. is a supersolution of (11), so the hypothesis ii) of Theorem 3 is satisfied.

Next we compare e-solutions with Ishii'sdefinitions of non-continuous viscosity solution and of boundary conditions in viscosity sense. We recali that a function u e BU SC(Q) (respec­tively ù e BLSC{Q)) is a viscosity subsolution (respectively a viscosity supersolution) of the boundary condition

(21) u = g or F(x, u, Du, D"u) = 0 on DS2,

22 M. Bardi - S. Bottacin

if forali x e d£2 and <f) e C2(£!) such that u—(f> attains a locai maximum (respectively minimum) at x, we have

(u - g)(x) < 0 (resp. > 0) or F(x, u(x), D<j)(x), D2(()(x)) < 0 (resp. > 0).

An equivalentdefinition can be given by means of the semijets /—' u(x), J^ u(x) instead of the test functions, see [18].

PROPOSITION 3. IfH_„ : £2 —» R is the lower e-solution (respectively, Hs is the upper

e-solution) of(16), then Ht is a subsolution (respectively, Hg^ is a supersohition) of (10) and ofthe boundary condition (21).

Proof. If H_„ is the lower e-solution, then by Proposition 1, Ht is a subsolution of (10). It remains to check the boundary condition.

Fix an y e d& such that Kg(y) > g(y)> and 4> e C2(Q) such that H_* - (f) attains a locai maximum at y. We can assume, without loss of generality, that

K*8(y) = <P(y), (Kg - </>)(•*) < H * - .vi3 for ali x e Q n B(y, r).

By definition of H_* there exists a sequence of points x„ -* y such that

(Ke ~ (f>)(x„) > - - for ali « . 6 n

Moreover, since H_g is the lower e-solution, there exists a sequence of functions wn e S such that

KP(XII) < Wn(.Xn) for a'l n . ò n

Since the function wn — 0 is upper semicontinuous, it attains a maximum at y„ e £J fi B(y, /-), such that, for n big enough,

2 -\ < (u)„ - <f>)(y„) < -\)>n ~ .Vi •

So as n —> oo

y„ -» y, w;i(y/i) - • <f>(y) = K*s(y) > g(y) •

Note that )'/( £ 3£2, because y,j e 8Q would imply u;„(y„) < g(yn), which gives a contradiction to the continuity of g at y. Therefore, since wn is a subsolution of (10), we have

F(y„, w„(y„), Z><M)'„), D20(y,,)) < 0,

and letting n -^ oo we get

/r(.y,H*O'),^(.v),/)20(.v))<o,

by the continuity of F.

•

On the Dirichlet problem 23

REMARK 2. By. Proposition 3, if the e-solution Ho of (16) exists, it is a non-continuous viscosity solution of (10) (21) in the sense of Ishii [27]. These solutions, however, are npt unique in general. An e-solution satisfies also the Dirichlet problem in the sense that it is a non-continuous solution of (10) in Ishii's sense and Hg(x) = g(x) for ali x e B£2, but neither this property characterizes it. We refer to [4] for explicit examples and more details.

REMARK 3. Note that, by Proposition 3, if the e-solution Hg is continuous at ali points of d£2] with £2\ e £2, we can apply the Comparison Principle to the upper and lower semicontinu-ous envelopes of Hg and obtain that it is continuous in'ftp if the equation is uniformly elliptic in fij we can also apply in £2f the locai regularity theory for continuous viscosity solutions developed by Caffarelli [17] and Trudinger [43].

2. Properties of the generalized solutions

2.1. Continuous dependence under uniform convergence of the data

We begin this section by proving a result about continuous dependence of the e-solution on the boundary data of the Dirichlet Problem. It states that the set of resolutive data js closed with respect to uniform convergence. Throughout the paper we denote with =* the uniform convergence.

THEOREM 4. Let F : Q x E x RN x S(N) -> R be continuous and proper, and let gn : 3Q —> M. be continuous. Assume that {gn}n is a sequence of resolutive data sudi that gn^g on 3Q. Then g is resolutive and H^^Hg on Q.

The proof of this theorem is very similar to the classical one for the Laplace equation [26]. We need the following result:

LEMMA 2. For ali c> 0, H_(f>+C) ^ Kg + e and ~H(g+c) < ~Hg + e.

Proof. Let

Sc := {w e BUSC(Q) : w is subsolution of (10), w < g + e on <ÌQ].

Fix u e Sc, and consider the function v(x) = u(x) — e. Since F is proper it is easy to see that v eS. Then

K(s+c):= S U P u - S U P v + c := Kg + c • U&SC U€<S

n

Proof of Theorem 4. Fix € > 0, the uniform convergence implies 3/n : V/? > in: g„ — e < g < gn + e. Since gn is resolutive by Lemma 2, we get

• H8n - € < HL{gn-e) < Ks < K{gn+e) < Hg„ + < •

Therefore HSn=ìH_o- The proof that Hgn=$Hg, is similar.

D

24 M. Bardi - S. Bottacin

Next result proves the continuous dependence of e-solutions with respect to the data of the Dirichlet Problem, assuming that the equations Fn are strictly decreasing in /•', uniformly in n.

THEOREM 5. Let Fn : £2 x R x RN x S(N) ->• R is continuous and proper, g : 8Q -» R is continuous. Suppose that V/z, V5 > 0 3e > 0 such that

Fn(x,r-8,p,X) + € < Fn(x, r, p, X)

for ali (x,r, p,X) e~QxRxRN x S(N), and Fn=$ F on Q x R x RN x S(N). Suppose g is resolutive for the problems

(22) Fn (x, u, Du, D u) = 0 in Q ,

u = g ondQ,.

Suppose gn : OQ -> R is continuous, gn^g on dQ and gn is resolutive for the problem

(23) Fu(x, u, Du, D2w) = 0 in Q , u = gn on 3Q

Then g is resolutive for (16) and / / " =£ Hg, where / / " is the e-solution of(23).

Proof Step 1. For fixed 8 > 0 we want to show that there exists ni such that for ali n > m: .\K'g-&g\<8, where //£ is the e-solution of (22).

We claim that there exists m such that / /" — 8 < H_ and Hg < H1, + 8 for ali n > ni. Then

Kg-s<Kg <Hg<H',g+8 = H!g + 8.

This proves in particular Hgl=$.H_g and H"=$Hg, and then Hg = / / ? , so g is resolutive for (16).

It remains to prove the claim. Let

Sg := [v subsolution of Fn = 0 in Q, u < g o n d£2).

Fix v e S1', and consider the function u = v - 8. By hypothesis there exists an e > 0 such that 0 .

convergence of Fn to F we get, for n large enough,

F(x, u(x), p, X) < Fn(x,u(x), p,X) + € < Fn(x,v(x), p, X) < 0,

9 4. F„(^:, u(x), p, X) + 6 < F„(*, D(X), /;, X), for ali (p, X) e JQ V(X). Then using the uniform

2 4- ^ 4-so w is a subsolution of the equation Fn = 0 because 7f ì I»(JC) = J£ u(x).

We have shown that for ali v e SL' there exists u e S such that v — u + 8, and this proves the first claim. The proof of the second claim is similar.

Step 2. Using the argument of proof of Theorem 4 with the problem

/•24\ ( Fm(x,u, Du, D2u l H - gn

2") = 0 in fi, on 3fi,

we see that fixing 8 > 0, there exists /; such that for ali n > p: \W* — //",' | < 8 for ali m.

Step 3. Using again arguments of proof of Theorem 4, we see that fixing 8 > 0 there exists a such that for ali n,m> a: \H™ - H'f \ < 8. 1 — 1 1—gn — g m 1 —

On the Dirìchlet problem 25

Step 4. Now take 8 > 0, then there exists p such that for ali n, ni > p:

Similarly \H™m -~H8\< 3<5. But H%m = 7/™n, and this complete the proof.

D

2.2. Continuous dependence under locai unitomi conyergence of the operator

In this subsection we study the continuous dependence of e-solutions with respect to perturba-tions of the operator, depending on a parameter h, that are not uniform over ali Jp x R x R^ x S(N) as they were in Theorem 5, but only on compact subsets of £2 x R x R^. x S(N). A typical example we have in mind is the vanishing viscosity approximation, but similar arguments work for discrete approximation schemes, see [3]. We are able to pass to the limit under merely locai perturbations of the operator by approximating Q with a nested family of open sets 0 6 , solving the problem in each ©6, and then letting e, fi go to 0 "with li linked to e" in the following sense;

DEFINITION 4. Let v%, u : Y -» R, for e > 0, li > 0, Y £RN. We say that v€h converges

to u as (e, h) \ (0, 0) with h linked to e at the poìnt x, and write

(25) lim vUx) = u(x) ( e , / ; ) \ ( 0 , 0 ) n • • •

h<h(e) •

iffar ali y > 0, there exist afunction h :]0, +oo[—>]0, +oo[ and'i > 0 such that

\v€h(y) ~ u(x)\< Y, forali y e Y : \x - y\ < h(e)

forali e <"£, h < h(€).

Tojustify this definition we note that:

0 it implies that for any x and e„ \ 0 there is a sequence hn \ 0 such that vf" (x„) -> u(x) for any sequence xn such that \x — xn | < hn, e.g. xn = x for ali n, and the same holds for any sequence h'n > hn\.

ii) iflim/j\^Q v^(x) exists for ali small e and its limit as € \ 0 exists, then it coincides with the limit of Definition 4, that is,

lim vf(x) = lim lim u f ( » . (e,/i)\(0,0) n €\0h\0 n

REMARK 4. If the convergence of Definition 4 occurs on a compact set K where the limit u is continuous, then by a standard compactness argument we obtain the uniform convergence in the following sense:

DEFINITION 5. Let K be a subset ofRN and vjr, u : K -> R for ali e, h > 0. We say that v€

f% converge uniformly on K to u as (e, li) \ (0, 0) with h linked to e if for any y > 0 there are

? > 0 and h :]0, +oo[-»]0, +oo[ such that

sup|i;| -u\<y K

far alle < €,h < h(e).

26 M. Bardi - S. Bottacin

The main result of this subsection is the following. Recali that a family of functions u;6; :

fi -» R is locally uniformly bounded if for each compact set K e fi there exists a Constant C^ such that sup/f | i^ | < C^ for ali h, € > 0. In the proof we use the weak limits in the viscosity sense and the stability of viscosity solutions and of the Dirichlet boundary condition in viscosity sense (21) with respect to such limits.

THEOREM 6. Assume the Comparison Principle holds, Z ^ 0 and let i±be a continuous subsolution of( 16) such that u_' = g on dfi. For any € e]0, 1], let S€ be an open set sudi that 0 e C fi, andfar h e]0, 1) let vi be a non-continuous viscosity solution (le., vj* isa subsolution andv^ is a super solution) of the problem

r ,. . Fit(x, u, Du, D2u) = 0 in 0 e , u(x) = u(x) or F/i(x, u, Du, D~u) = 0 • on (KM)e ,

where Fj-, : 0 e x R x R^ x S(N) -> R is continuous and proper. Suppose {v^} is locally

uniformly bounded, vj7 > w in fi, and extend vj} := w in fi \ 0 e . Finally assume that F/?

converges uniformly to F on any compact subset of fi x R x R^ x S(N) as li \ 0, and

0 6 2 0a i /e<«,Uo<€<l®€ = «- . Then v^ converges to the e-solution Hg of(16) with li linked to e, that is, (25) holds for ali

x e fi; moreover the convergence is uniform (as in Def. 5) on any compact subset of fi where Hg is continuous.

Proof. Note that the hypotheses of Theorem 2 are satisfied, so the e-solution Hg exists. Consider the weak limits

v^(x) • := ìim'mf Vfr(x) := supinf{i>| (y) : \x — y\ < 8, 0 < h < 8},

v€(x) := limsup*uj(jt) := inf sup{^(y) : \x - y\ < 8, 0 < h.< 8}. h\0 s>0

By a standard result in the theory of viscosity solutions, see [10, 18, 8, 4], v€ and v€ are respec-tively supersolution and subsolution of

(27) I F(x, u, Du, D~u) = 0 in 0 e , . I u(x) = u(x) or F(x, u, Du, D^u) = 0 on (KH)e .

We claim that ùe is also a subsolution of (16). Indeed v*h = u in fi \ 0 e , so v€ = u in the interior of fi \ 0 e and then in this set it is a subsolution. In 0 e we have already seen that v€ = (v€)* is a subsolution. It remains to check what happens on 3 0 é . Given x e <KH)e, we must prove that

2 4-forall(/?,X) eJnV€(x) we have

(28) . Fh(x,v€(x),p,X)<0.

l s t Case: ve(x) > u(x). Sincere satisfies the boundary condition on O0e of problem (27), 2 4-_ *

then for ali (/?, X) e J^ve(x) (28) holds. Then the same inequality holds for ali (/?, X) e

J^+ve(x) as well, because J^+v€(x) e y , +i7e(i). 2 n d Case: ve(x) = u(x). Fix (p, X) e J^+ve(x), by definition

. . 1 . , . 9 ue(x) < ve(x) + p • (x — x) + -X(x — x) • (x - x)*+ o(\x — jt|*")

On the Dihchlet problem 27

for ali x -» x. Since v€ > u and v€(x) = u(x), we get

u_(x) < u_(x) +•/; • (x — x) + -X(x — x) • (x - x) + o(\x — x\~),

2 4-that is (/?, X) e J^ u_(x). Now, since M is a subsolution, we conclude

F(x,v€(x),p,X) = F(x,u(x)tp,X)<0.

We now claim that

(29) «e < i^ < ùe < //g in fì,

where u€ is defined by (20). Indeed, since v^ is a supersolution in 0C and v_€ > w, by the Comparison Principle y^ > w in &€ for any w e S such that «j = «on (KH)6. Moreover v_€ = i± on fi\®e, so we get v^ > we in fi. To prove the last inequality we note that Hg is a supersolution of (16) by Theorem 2, which implies v€ < Hg by Comparison Principle.

Now fix x € fi, € > 0, y > 0 and note that, by definition of lower weak limit, there exists h = h(x, €, y) > 0 such that

ve(x)-y < vi(y)

for ali h < h and y e fi D /?(*, /i). Similarly there exists H = &(.*:, e, y) > 0 such that

v%(y) <v€(x) + y

for ali h < k and ;y e fin B(x, k). From"Remarle 1, we know that Hg = sup(r u€, so there exists ? such that

Hg(x) — y < MeOO, forali e < ?.

Then, using (29), we get

. Hg(x)-2y <vf2(y) < Hg(x) + y

for ali e <I,h <h := min{/z, k} and y e fi fi Z?(;t, li), and this completes the proof.

•

REMARK 5. Theorem 6 applies in particular if vf are the solutions of the following vanish-ing viscosity approximation of (10)

( m j -hAv + F(x, v, Dv, D2v) = 0 in 0 € , ( \ \ v = u on 306 .

Since F is degenerate elliptic, the PDE in (30) is uniformly elliptic for ali li > 0. Therefore we can choose a family of nested 0C with smooth boundary and obtain that the approximating vj are much smoother than the e-solution of (16). Indeed (30) has a classical solution if, for instance, either F is' smooth and F(x, -, -, •) is convex, or the PDE (10) is a Hamilton-Jacobi-Bellman equation (3) where the linear operators Ca have smooth coefficients, see [21, 24, 31]. In the nonconvex case, under somestructural assumptions, the continuity of the solution of (30) follows from a barrier argument (see, e.g., [5]), and then it is twice differentiable almost every where by a result in [43], see al so [17].

28 M. Bardi - S. Bottacin

2.3. Continuous dependence under increasing approximation of the domain

In this subsection we prove the continuity of thee-solution of (16) with respect to approximations of the domain fi from the interior. Note that, if vj = v€ for ali h in Theorem 6, then v€(x) -> Hg (x) for ali x e fi as e \ 0. This is the case, for instance, if v€ is the unique e-solution of

F(x,u, Du, D2u) = 0 in 0 6 , u =.u_ on 0(M)6 ,

by Proposition 3. The main result of this subsection extends this remark to more general ap­proximations of fi from the interior, where the condition G€ e fi is dropped. We need first a monotonicity property of e-solutions with respect to the increasing of the domain.

tN LEMMA 3. Assume the Comparison Principle holds and lei fi] e fio e R/v, f/' respec-

y H2 he the e-solution in fij, respectively fi2, ofthe problem

F(x, u, Du, D2u) = 0 in fi/, (31)

[ u = g on Ofi,- ,

with g : fi2 -» M continuous and subsolution of(3J) with i = 2. Ifwe defme

Hkx) = \ H^ ***** _ ^ l g(x) ifx e Q2\Qi,

then H2 > H[g in Qo-

Proof. By definition of e-solution //- > g in fi2, so H2 is also supersolution of (31) in fi|.

Therefore HÌ > HÌ in fij because / / ' is the smallest supersolution in fi|, and this compleles the proof.

• THEOREM 7. Assume that the hypotheses of Theorem 3 i) hojd with i± continuous and fi

bounded, Let {fi,,} be a sequence ofopen subsets o/fi, sudi that fi,, e fi,/+j and (J;/ fi,, = fi. Let uu be the e-solution of the problem

n _ , ( F(x, u, Du, D2u \ u = u_

2-)=0 in fi,, , on Ofi„

Ifwe extend un := u in fi \ fi», then Unix) /• Hs(x) for ali x e fi, where H^ is the e-solution of(16).

Proof. Note that for ali n there exists an e„ > 0 such that fif/) = [x e fi : dist (x, Ofi) > e,,} e fi,,. Consider the e-solution u(ll of problem

( F(x,u, Du,D2u)=0 infie„, u = u on Ofi

If we set uÉII = u in fi \ fie„, by Theorem 6 we get ue/ì —» //# in fi, as remarked at the beginning of this subsection. Finally by Lemma 3 we have Hs > u„ > u€n in fi, and so u„ ->• Hs in fi.

D

On the Dirìchlet problem 29

REMARK 6. If DQ is not smooth and F is uniformly elliptic Theorem 7 can be used as an approximation result by choosing Qn with smooth boundary. In faci, under some structural assumptions, the solution un of (32) turns out to be continuous by a barrier argument (see, e.g., [5]), and then it is twice differentiable almost everywhere by a result in [43], see also [17]. If, in addition, F is smooth and F(x, -, -, •) is convex, or the PDE (10) is a HJB equation (3) where the linear operators Ca have smooth coefficients, then un is of class C~, see [21, 24, 31, 17] and the references therein. The Lipschitz continuity of u„ holds also if F is not uniformly elliptic bui it is coercive in the p variables.

2.4. Continuity at the boundary

In this section we study the behavior of the e-solution at boundary points and characterize the points where the boundary data are attained continuously by means of barriers.

PROPOSITION 4. Assume that hypothesis i) (respectively il)) of Theorem 2 holds. Then the e-solution Hg of(J6) takes up the boundary data g continuously atXQ e BQ, Le., Iimx_^X() Hg (x) = g(xo), Ifand only Ifthere is an upper (respectively lower) barrier at XQ (see Defmitlon 3).

Proof. The necessity is obvious because Theorem 2 /) implies that Hg e Z, so Ho is an upper barrier at x if it attains continuously the data at x.

Now we assume W is an upper barrier at x. Then W > Hg, because W e Z and Hg is the minimal element of Z. Therefore

g(x) < Hs(x) < liminf Hs(y) < limsup Hs(y) < lim W(y) = s(x),

so lim_v_>jc Hg (y) = g(x) = Hg (x).

•

In the classical theory of linear elliptic equations, locai barriers suffice to characterize boundary continuity of weak solutions. Similar results can be proved in our fully nonlinear context. Here we limit ourselves to a simple result on the Dirichlet problem with homogeneous boundary data for the Isaacs equation

f supinf{-a": V / J C , . + b^uXi + ca^u - /«•*} = 0 in « , (33) a P J J

u = 0 on 3Q.

DEFINITION 6. We saythat W e BLSC(B(x$,r) fi Q) wlthr. > 0 is an upper locai barrier for problem (33) at XQ 6 BQ if

l) W > 0 Is a supersolutlon ofth'e PDE in (33) in B(XQ, r) 0 Q,

li) W(xo) = 0, W(x) > p > 0 for ali \x - XQ\ = r,

ili) W is continuous at XQ.

PROPOSITION 5. Assume the Comparison Princlple holds for (33), fa^ > 0 for ali a, fi and let Hg be the e-solutlon ofproblem (33). Then Hg takes up the boundary data continuously at XQ e dQ, ifand only ifthere exists an upper locai barrier W at XQ.

30 M. Bardi - S. Bottacin

Proof. We recali that Hg exists'because the function u = 0 is a lower barrier for ali points JC e HQ by the fact that fa,P > 0, and so we can apply Theorem 2. Consider a supersolution w of (33). We claim that the function V defined by

I pW(x)Aw(x) ifx e B(xQLr)nn, {x)~ \ W(x) ìfx eQ\B(x0,r),

is an upper barrier at XQ for p > 0 large enough. It is easy to check that pW is a supersolution of (33) in B(XQ, r) fi £2, so V is a supersolution in B(XQ, r) n Q (by Proposition "!)• and in Q \ B(xo, r). Since w is bounded, by property ii) in Definition 6, we can fìx p and e > 0 such that V(x) = w(x) for ali x e £2 satisfying r — e < \x - XQ\ < r. Then V is supersolution even on 8B(XQ, r) fi Q. Moreover ìt is obvious that V > 0 on 8Q and V(XQ) == 0. We have proved that V is supersolution of (33) in 2.

It remains to prove that limA_>X() V(x) = 0. Since the Constant 0 is a subsolution of (33) and V is a supersolution, we have V > 0. Then we reach the conclusion by Hi) of Definition 6.

" ' .. D

EXAMPLE 3. We construct an upper locai barrier for (33) under the assumptions of Propo­sition 5 and supposing in addition

8Q is C" in a neighbourhood of XQ e BQ ,

there exists an a* such that for ali fi either

(34) a?j ^(xo)niixo)nj(xo) > c> 0

or

(35) -a"j ^(X0)CIXÌXJ(XQ) + b" ^(XQ)IIJ(XQ) > c> 0'

where n denotes the exterior normal to Q and ci is the signed clistance from OQ

óisl(x,dQ) ifjcefì,--dist(jc,9Q) \fxeRN\Q

à(x) = Ì j :_w_. ^ :r_. _^N

Assumptions (34) and (35) are the naturai counterpart for the Isaacs equation in (33) of the conditions for boundary regularity of solutions to linear equations in Chapt. 1 of [37]. We claim that

W(x) = 1 - ^-s(d(x)+k\x-xo\2)

is an upper.locai barrier at XQ for a suitable choice of 8, X > 0. Indeed it is easy to compute

-c^f (JCO) WXìXj (XQ) + bf* (XQ) WXÌ (XQ) + e01'? (x0) W(XQ) - f'P (XQ) =

-5tf ?• (xo)dXjXj (XQ) + 82ac*jP (x0)dXi (xQ)dXj (x0) + 8b"^ (x0)dXi (x0)

-28\Tr[aa'P(x0)]-fa<fi(xo).

Next we choose ex* as above and assume first (34). In this case, since the coefficients are bounded and continuous and dìsC2, we can make W a supersolution of the PDE in (33) in a neighborhood of XQ by taking 8 large enough. If, instead, (35) holds, we choose first X small and then 8 large to get the same conclusion.

On the Dirìchlet problem 31

3. Applications to optimal control

3.1. A deterministic minimum-time problem

Our first example of application of the previous theory is the time-optimal control of nonlinear deterministic systems with a closed and nonempty target T C R^. For this minimum-time problem we prove that the lower semicontinuous envelope of the value function is the e-solution of the associated Dirich.let problem for the Bellman equation. This result can be also found in [7] and [4], but we give here a different and simpler proof. Consider the system

( 3 6 ) ( * « » = , .

where a e A := {a : [0, oo) ->-A measurable) is the set of admissible controls, with

A a compact space, / : R^ x A -» R^ continuous ( . 31 > 0 such that (/(*, a) - f(y, a)) • (x - y) < L\x - y\2 ,

for ali x, y e R^, a e A. Under these assumptions, for any a e A there exists a unique trajectory of the system (36) defined for ali t, that we denote yx(t, a) or .y.r(0- We also defìne the minimum lime for the system to reach the target using the control a e A:

t ; = f inf{r > 0 : yx(t, a)eT}, if {/ > 0 : yx(t, a) e V) ? 0, x ' [ +oo otherwise.

The value function fòr this problem, named minimum Urne function, is

T(x)= inf tx(a), xeRN . a&A

Consider now the Kruzkov transformatiorì of the minimum time

) - r r w , ifT(x)<oo, otherwise.

The new unknown v is itself the value function of a time-optimal control problem with a discount factor, and from its knowledge one recovers immediately the minimum time function T. We remark that in general v has no continuity properties without further assumptions; however, it is lower semicontinuous if f(x, A) is a convex set for ali x, so in such a case D = u* (see, e.g., [7,4]).

The Dirichlet problem associated to v by the Dynamic Programming method is

j v + H(x,Dv) [ v = Q,

™ , . , » V - , ^ J ) = 0 , i n R / v \ r , (38) \v = o, i n a r , where

H(x, p) := max{—f(x, a) • p) — 1 .

A Comparison Principle for this problem can be found, for instance, in [4].

THEOREM 8. Assume (37). Then u* is the e-solution and the minimal supersolution of(38).

32 M. Bardi - S. Bottacin

Proof. Note that by (37) and the fact that w = 0 is a subsolution of (38), the hypotheses of Theorem 2 i) are satisfied, so the e-solution exists and it is a supersolution. It is well known that u* is a supersolution of v + H(x, Dv) = 0 in R^ \ T, see, e.g., [28, 8, 4]; moreover t>* > 0 on Or, so u* is a supersolution of (38). In order to prove that u* is the lower e-solution we construct a sequence of subsolutions of (38) converging to v*.

Fix € > 0, and consider the set

V€ :={x (=RN :dist(jt,3r) <€} ,

let T€ be the minimum time function for the problem with target r6, and ve its Kruzkov trans-formation. By standard results [28, 8, 4]"i>e is a non-continuous viscosity solution of

j v = 0or v + H(x,Dv) = 0, in dre .

With the same argument we used in Theorem 6, we can see that v* is a subsolution of (38). We define

u(x) := supine*)

and will prove that u — v*. By the Comparison Principle u | < v* for ali e > 0, then u(x) < v*(x). To prove the

opposite inequality we observe it is obvious in r and assume by contradiction there exists a point x <£ T such that

(39) supv€(x) < supu*(jt) < v*(jc).

Consider first the case v*(x) < 1, that is, T*(x) < +oo. Then there exists S > 0 such that

(40) T*(x) < T*(x) - 8 < +oo, for ali e > 0.

By defìnition of minimum time,. for ali € there is a control a€ such that

e

(41) tl(fl€)<T€(x)+-<+oo.

Let zf è Te be the point reached at time t\{a€) by the trajectory starting from x, using control a€. By standard estimates on the trajectories, we have for ali €

|z€| = lyia|(a€))l-<(|Jc| + >/2Af7'(Jc))^7 '( i ).

where M := L + sup{|/(0, a)| : a e A}. So, for some R > 0, z€ e Z?(0, /?) for ali e. Then we can find subsequences such that

(42) z€„ z e ar, t„ := r|" (a€„) -> 7, as n -* oo.

From this, (40) and (41) we get

(43) j<T^x)-~.

On the Diiichletproblem 33

Let y€ be the solution of the system

f yf = f(y^a€n) 1 < t„ , 1 y(t£(a€n)) = z,

that is, the trajectory moving backward from z using control a(jl ; and set x„ := y€n (0). In order to prove that xn -> x we consider the solution y€/l of

{ y - f(y,a€„) t <t„, 1 y(tn) = z€„,

that is, the trajectory moving backward from z€n and using control a€n. Note that yè„(0) = x. By differentiating \y€n — J€n |

2, using (37) and then integrating we get, for ali / < t„,

1.^,(0-ye„(0l2 < \z€n-z\2+.f2L\y€n(s)-y€tt(s)\2ds.

Then by Gronwall's lemma, for ali t < t„,

\y^)-y,n(t)\<\z€l,-z\eL(t"~'\

which gives, for t = 0,

\x-xn\<\Zen-z\eLtl' ,

By letting n -> oo, we get that x„ -> x. By definition of minimum time T(xn) < t„, so letting /? -^ oowe obtai-n T*(x) < 7, which

gives the desired contradiction with (43). The remaining case is i>*(i) =. 1. By (39) T*(x) < K < +oo for ali e. By using the

previous argumént we get (42) with J < +oo and T*(x) < 1. This is a contradiction with r*(i) = +co and completes the proof.

•

3.2. Maximizing the mean escape time of a degenerate diffusion process

In this subsection we study a stochastic control problem having as a special case the problem of maximizing the expected discounted time spent by a controlied diffusion process in a given open set £2 e R^. A number of engineering applications of this problem are listed in [19], where, however, a different cost criterion is proposed and a nondegeneracy assumption is made on the diffusion matrix. We consider a probability space (Q', T, P) with a right-continuous increasing filtration of complete sub-a fields {J/}, a Brownian motion Bt in RM ^"/-adapted, a compact set A, and cali A the set of progressively measurable processes ott taking values in A. We are given bounded and cohtinuous maps a from RN x A into the set of N x M matrices and b : RN x A ~» RN satisfying (14), (15) and consider the controlied stochastic differential equation

(SDE) , t=cra'(Xt)dBt -ba'{Xt)dt, / > 0. { dXt = & 1 X0 = x .

For any a, e A (SD E) has a pathwise unique solution Xt which is T\ -progressively measurable and has continuous sample paths. We are given also two bounded and uniformly continuous

34 M, Bardi - S. Bottacin

maps'/, e :,R x A -+R, ca(x) > co > 0 for ali x, a, and consider the payoff functional

/ (*, a.) := E (f'Aa' fa'(Xt)e~Ió <?*{X*)dsdt\ f

where E denotes the expectation and

tx(a.)\=\nf[t*0:Xt &Q},

where, as usuai, tx((x.) = +og if Xt e ft forali t > 0. We want to maximize this payoff, so we consider the value function

v(x) := sup /(*,«.) .

Note that.for / = e = 1 the problem becomes the maximization of the mea.n discounted time E{\ — e~tx(fil-ì) spent by the trajectories of (SDE) in Q.

The Hamilton-Jacobi-Bellman operator and the Dirichlet problem associated to v by the Dynamic Programming method are

F(x, u, Du, D2u) := mìn{-afj(x)uXjXj + ba(x) • Du +ca(x)u - fa(x)}, aeA •' J

.T where the matrix (a/j) is ^aa' , and

»2«0=0 in £2, on 9 £2,

^44s f F(x, u, Du, D2u) } M = 0

see, for instance, [40,35, 36, 22, 32] and the references therein. The proof that the value function satisfies the Hamilton-Jacobi-Bellman PDE is based on the Dynamic Programming Principle

(45) v{x) = sup E ( f6AtX fa> (X,)<T fi <*™chdt + v(X9Atx )e~ C'x c"x <*'>d") , a.e A VO /

where tx = tx(a), for ali x e Q and ali J^-measurable stopping times 0. Although the DPP (45) is generally believed to be true under the current assumptions (see, e.g., [35]), we were able to find its proof in the literature only under some additionalconditions, sudi as the convexity of the set .

{(aa(x), ba{x), fa(x), ca(x)) : a e A}

for ali x e £2, see [20] (this is true, in particular, whe.n relaxed controls are used), or the inde-pendence of the variance'of the noise from the control [15], i.e., aa(x) = a(x) for ali x, or the continuity of v [35]. As recalled in Subsection 1.1 a Comparison Principle for (44) can be found in [29], see also [18] and the references therein.

In order to prove that v is the e-solution of (44), we approximate Q with a nested family of open sets with the properties

(46) e€ e Q, e e]0, 1],' 0e 3 ®s for e < 8, ( J 6 e = Q . e

For each e > 0 we cali v€ the value function of the same control problem with tx-replaced with

tUot.):=mf[t>Q\Xt<t®€}

I

On the Dirìchlet problem 35

in the definition of the payoff J. In the next theorem we assume that each ue satislìes the DPP (45) with tx replaced with f|.

Finally, we make the additional assumption

(47) fa(x) > 0 for al] x e Q, a e A .

which ensures that w = 0 is a subsolution of (44). The main result of this subsection is the following.

THEOREM 9. Under the previous assUmptions the vaine fimction v is the e-solution and the minimal supersolution of (44), and

v = sup v€ = lim v€ . 0<€<l e \ °

Proof. Note that v€ is nondecreasing as e \ 0, so l im^o v€ exists and equals the sup. By Theorem 3 with g = 0, u = 0, there exists the e-solution HQ of (44). We consider the lunctions u€ defìned by (20) and claim that

Then

(48) H0= sup ve, 0<€<I

because HQ = sup€ U2€ by Remark 1. We prove the claim in three steps. Step 1. By standard methods [35, 9], the Dynamic Programming Principle for v€ implies

that u6 is a non-continuous viscosity solution of the Hamilton-Jacobi-Bellman equation F — 0 in ©e and v* is a viscosity subsolution of the boundary condition

(49) u = 0or F(x,u, Du, D2u) = 0 on 30 e ,

as defìned in Subsection 1.3. Step 2. Since (i%)* is a supersolution of the PDE F — 0 in S€ and (u6)* > 0 on ()©é, the

Comparison Principle implies (i>€)* > u> for any subsolution w of (44) such that w — 0 on <KH)€.

Since <)©«? e Q \ ©2e by (46), we obtain U2€ < Ve* by the definition (20) of U2€- • Step 3. We claim that v* is a subsolution of (44). In fact we noted before that it is a

subsolution of the PDE in ©€, and this is true also in Q \ ©e where v* = 0 by (47), whereas the boundary condition-is trivial. It remains to check the PDE at ali points of (KM)e... Given x <= (KM)e, we must prove that for ali 4> e C2(£2) such that v^ — (p attains a locai maximum at x, we nave

(50) F(x,vl(x),D<f)(x),D2<f>(x)) < 0 .

\st Case: v*(x) > 0. Since vj satislìes (49), for ali <p e C2(©f) such that u* - 0 attains a locai maximum at x (50) holds. Then the same inequality holds for ali (p e C~(Q) as well.

2nd Case: U*(JC) = 0. Since v* — 0 attains a locai maximum at i , for ali x near i we nave

v* (x) — v* (x) < (f>(x) — <fr (x).

By Taylor's formula for 0 at i and the fact that vj (x) > 0, we gel

D(j)(x) • {x — x) > o{\x - jc|),

36 M. Bardi - S. Bottacin

and this implies D<f>(x) = 0. Then Taylor's formula for (f> gives also

(x - x) • D2(j){x)(x - x) > o{\x - x\2) ,'

and this implies D24>(x) > 0, as it is easy to check. Then

F(JC, v*(x), D0(Jc), P2<l>(x)) = F(Jc,0,0, £>20(i)) < 0

because a" > 0 and fa > 0 for ali * and a. This completes the proof that v* is a subsolution of (44). Now the Comparison Principle yields v£ < //Q, since A/o is a supersolution of (44).

It remains to prove that v = supo<e<i v€. To this purpose we take a sequence €n \ 0 and define

Mx\a) := E (jtX ^ f«>(Xt)e-tìcax(X«)dsdl) .

We claim that

Iim Jn(x, a ) = sup 7/i(x, a ) = J(x, a ) for ali a. and'x . « " n '

The monotonicity of (£" follows from (46) and it implies the monotonicity of ./„ by (47). Let

r := supt€x"(a.) <tx(a.),

n

and note that tx((x.) = +oo if x == +oo. In the case r < +oo, Xfen e BG€ll implies Xx e 3Q, so T = tx(a.) again.. This and (47) yield the claim by the Lebesgue monotone cónvergence theorem. Then

v(x) = sup sup Jn(x, oe.) = sup sup •/„(.*:, a,) = supi^,, = supi^ , a. n n a, n . e

so (48) gives v = HQ and completes the proof. D

REMARK 7. From Theorem 9 it is easy to get a Verification theorem by taking the su-persolutions of (44) as verification functions. We consider a presynthesis ot^x\ that is, a map (*(') : Q -> A, and say it is optimal at x() if J(xa, a(*<>)) = v(xtì). Then Theorem 9 gives im-mediately the following sufficient condition of optimality: ifthere exists a verification funaioli W sudi that W(x0) < J(x0, a^x"'), then a'"' is optimal at x(,; moreover, a characterization of global optimality is the following: a^ is optimal in Q if and only if ./(•, a^) is a. verification function.

REMARK 8. We can combine Theorem 9 with the results of Subsection 2.2 to approximate the value function v with smooth value functions. Consider a Brownian motion Bt in R^ Fr adapted and replace the stochastic differential equation in (SDE) with

dXt ^o-at(Xt)dBt-ba'(X()dt + V2hdBt, / > 0 ,

for h > 0. For a family of nested open sets with the properties (46) consider the value function vè

h of the problem of maximizing the payoff functional ./ with tx replaced with t€x. Assume for

simplicity that aa,ba,ca, fa are smooth (otherwise we can approximate them by mollificatipn). Then u is the classical solution of (30), where F is the HJB operator of this subsection and u = 0, by the results in [21, 24, 36, 31], and it is possible to synthesize an optimal Markov control policy for the problem with e, h > 0 by standard methods (see, e.g., [22]). By Theorem 6 vf converges to v as e, h \ 0 with h linked to e.

On the Dirìchlet problem 37

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AMS Subject Classification: 49L25,35J65, 35J70,93E20.

Martino BARDI, Sandra BOTTACIN Dipartimento di Matematica P. e A. Università di Padova via Belzoni 7,1-35131 Padova, Italy