数理解析研究所講究録
911 巻 1995 年 20-28

20

On the state constraint boundary value problem
for Isaacs equations
埼玉大学理学部

小池茂昭(Shigeaki Koike)\S 1. Introduction
We consider the following Isaacs equations of first-order:.

$\min_{b\in B}\max_{a\in A}\mathrm{f}\lambda u(x)-\langle g(x, a, b), Du(X)\rangle-f(x, a, b)\}=0$

for$x\in\Omega$sets in$\mathrm{R}^{n}$, where$\mathrm{R}^{N}$$\Omega\subset \mathrm{R}^{n}$for some

is an open bounded set,

$N\in \mathrm{N},$$f$and$g$$A$(1)

and $B$ are compact

are given continuous real-valued and-valued functions, respectively, on$\overline{\Omega}\cross \mathrm{A}\cross B$and$\lambda>0$

is aconstant.
Soner [5] first characterized the value function associated with thestate constraint (SC in short) problem arising in deterministic optimalcontrol (i.e. $\neq B=1$ ) as the unique viscosity solution of (1) (i.e. Bell-man equation) among continuous viscosity solutions under a boundary
condition, which he proposed via dynamic programming principle.
Recently, for first-order Bellman equations, Ishii and myself in [3] haveproposed a new boundary condition, which is naturally derived from the21SC requirement. In [3], it was shown that the value function is the uniqueviscosity solution among possibly discontinuous viscosity solutions

under the new boundary condition.

Our question here is what is the natural boundary condition for theIsaacs equations (1) under the SC requirement.

Our aim here is to present the definition of value for the SC problem
of differential games in a reasonable way\S 2. Value function
We shall define the value for the SC problem associated with (1).

In what follows, we suppose the continuity assumptions:$\exists\omega_{0}$: a modulus of continuity$\mathrm{s}.\mathrm{t}$.
,$(i)|g(X, a, b)-g(x, \text{\^{a}}, \hat{b})|\leq\omega_{0}(|a-\hat{a}|+|b-\hat{b}|)$$(A1)$$(ii)|f(x, a, b)-f(\hat{X}, \text{\^{a}}, \hat{b})|\leq\omega_{0}(|x-\hat{X}|+|a-\hat{a}|+|b-\hat{b}|)$
$\{$

for

$\forall x,\hat{x}\in\overline{\Omega},\forall a,$(iii)$\text{\^{a}}\in A,$$\forall b,\hat{b}\in B$, and$(a,b) \in A\cross\sup B\{||f(\cdot, a, b)||_{c_{(\overline{\Omega}}})+||g(\cdot, a, b)||c^{0},1(\overline{\Omega})\}<\infty$.

Notations[Controls by the player$I$][Controls by the player $II$ ][States for$(\alpha,\beta,$$A=.${ :$[0,$ $\infty)arrow A|\alpha$: measurable}$B=${ :$[0,$ $\infty)arrow B|\beta$: measurable}]

X$x)\in A\cross B\cross\overline{\Omega}$$\alpha$$\beta$$(\cdot;x, \alpha, \beta)$

is the unique solution of22

for $t>0$$\frac{dX}{dt}(t)=g(X(t), \alpha(t),\beta(t))$$\{$$X(0)=x$ .

Roughly speaking, our SC problem is as follows: For eachplayers$I$$J(x, \alpha,\beta)$

and II, respectively,over$\alpha\in A$

and($‘ \mathrm{m}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{m}\mathrm{i}\mathrm{z}\mathrm{e}$$\beta\in B$$x\in\overline{\Omega}$, theand maximize” a functional

for which $X(t;x, \alpha,\beta)$ stays in$\overline{\Omega}$

for any

time $t\geq 0$ . Then, we will derive a function depending on the , which$x$

we will call the value function.
We shall characterize the boundary value problem which the valuefunction satisfies in the sense of viscosity solutions.
Notations[Admissible pairs of controls for $(x,$$AD_{s}(x)=$$s)\in\overline{\Omega}\cross(0,${ $(\alpha,\beta)\in A\cross B|X(t;x,$We will suppose that
[Admissible controls by$AD_{\infty}(x)\neq\emptyset$$I$

for $(x,$]$]$$\alpha,\beta)\in\overline{\Omega}$

for all$x\in\overline{\Omega}$$s)\in\overline{\Omega}\cross(0,$$A_{s}(x)=\{\alpha\in A|\exists\beta\in B\mathrm{s}.\mathrm{t}.

[Admissible controls by II for $(x,$$\infty$$\infty$$s]$}.

.

$]$(\alpha,\beta)\in AD_{s}(X)\}$$s)\in\overline{\Omega}\cross(0,$$B_{s}(x)=\{\beta\in B|\exists\alpha\in A\mathrm{s}.\mathrm{t}.

]for $t\in[0,$$\infty$]$]$(\alpha,\beta)\in AD_{s}(x)\}$[Strategies]$\Delta_{s}=\{\delta:Barrow A|\mathrm{F}\mathrm{o}\mathrm{r}\forall\beta,\hat{\beta}\in\beta,\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{n}\delta[\beta]=\delta t\in(0,S],\mathrm{i}\mathrm{f}\beta=\hat{\beta}\mathrm{a}.\mathrm{e}.\mathrm{i}\mathrm{n}_{\mathrm{i}\mathrm{n}()}(\mathrm{o},t)[\hat{\beta}]\mathrm{a}.\mathrm{e}.0,\mathrm{f}\mathrm{o}\mathrm{r}t$.

$\}$

23

[Admissible strategies]$\Delta_{s}(x)=$Note that, for$B${$\delta\in\Delta_{s}|(\delta[\beta],\beta)\in AD_{s}(x)$$\forall x\in\Omega$, by $(A1),$$\exists s>0\mathrm{s}.\mathrm{t}$for$\forall\beta\in B_{s}(x)$}. $A_{s}(x)=A$ and $B_{s}(x)=$.
Now we define the value

$V$

on

$\overline{\Omega}$

byinf$V(x)\overline{=}$ $\mathit{5}\in\Delta_{\infty}(x)\rho B_{\infty}\sup_{\in(x)}\int_{0}^{\infty}e^{-x_{t}}f(x(t;x, \alpha,\beta), \alpha(t),\beta(t))dt$

Note that$V$.

coincides with that for the control problem in [3] if $\neq B=1$ .

\S 3. SC problem for (1)For each$(x, b)\in\overline{\Omega}\cross B$, we define the following subsets of$A(x, b)=\{a\in A|\mathrm{f}\mathrm{o}\mathrm{r}\exists r>\mathrm{o}_{\mathrm{s}}t\in[\mathrm{o},\cdot r]\mathrm{i}\mathrm{f}y.\in\overline{\Omega}\cap B_{r}(x)\mathrm{t}.X(t,y,a,b)\in\overline{\Omega}$$A$:$\}$.
We shall use the following assumptions:$\exists r,$$(A2)$

$\{$

for

$s>0\mathrm{s}.\mathrm{t}.$$\mathrm{a}.\mathrm{a}$and, if $b\in B$ and. $t\in[0, s]$ , and$X(t;x, a,\beta)\in\overline{\Omega}$$\beta\in B$$x\in\partial\Omega$satisfy $|\beta(t)-b|<r$, then $A(x, b)\neq\emptyset$ ,for $t\in 1^{0,s}$ ] and $a\in A(x, b)$ .

Notations$H(_{X,r},p;a, b)=\lambda r-\langle \mathit{9}(x, a, b),p\rangle-f(_{X,a}, b)$for$(x, r,p, a, b)\in\overline{\Omega}\cross \mathrm{R}\cross \mathrm{R}^{n}\cross A\cross B$.

$H(x, r,p)= \min_{\in bB}\max_{a\in A}H(X, r,p;a, b)$$\prime b(xb)r,p;a,$$H_{in}(_{X,r,p})= \inf_{\in Ba\in A}\sup,H(X,$$b)$

24

for

$(x, r,p)\in\overline{\Omega}\cross \mathrm{R}\cross \mathrm{R}^{n}$.

We shall adapt the following definition of viscosity solutions for theSC problems of (1).
Definition. We call$u$a subsolution (resp., supersolution and solution)

for the $SC$ problem of (1) if it is a viscosity subsolution (resp., supersolution and solution) of$fI_{\mathrm{r}}(x, u(x),$Remarks.

$Du(X))=0$ for$H_{in}(x, r,\mathrm{P})=H(x, r,p)$$H_{in}^{*}(X, r,p)=H(x, r,p)$

for

for

$x\in\overline{\Omega}$.

$(SC)$$(x, r,p)\in\Omega\cross \mathrm{R}\cross \mathrm{R}^{n}$$(x, r,p)\in\partial\Omega\cross \mathrm{R}\cross \mathrm{R}^{n}$$H_{*}(x, r,p)--Hin(x, r,p)$ for$x\in\partial\Omega\cross \mathrm{R}\cross \mathrm{R}^{n}$.

.

.

Theorem 1. ([4] cf. [2]) Assume $(A1)$ and $(A2)$ . Then, the valuefunction$V$

is a viscosity solution of $(SC)$ .

\S 4. Comparison and uniqueness results
Now, we present our comparison result for $(SC)$ , which implies thatthe value function constructed in section 2 is the unique viscosity solutionof $(SC)$ and that it is continuous.
We first introduce the following subsets: For $b\in B$ and$G(x, b)\equiv \mathrm{t}\mathrm{h}\mathrm{e}$$x\in\overline{\Omega}$convex hull of $\{g(x, a, b)|a\in A(x, b)\}$ .

, we let25

We use the following hypotheses concerning on the vector fields.
For$(A3)$(i)$\{$$z\in\partial\Omega,$$\exists\theta\in(0,1),$$r>0$ and$\bigcup_{0<\iota}<rBt\theta(x+t\xi)\subset\Omega$

(ii)

for

$\xi\in S^{n-1}\mathrm{s}.\mathrm{t}$.

$x\in\overline{\Omega}\cap B_{r}(z)$$G(z, b) \cap\bigcup_{t>0}B_{t\theta}(t\xi)\neq\emptyset$.

for $b\in B$ .

We remark that assumption $(A3)$ allows us to treat the case when thevector fields are tangential to$\partial\Omega$

if a convex combination of those dire-cts inside rather perpendicularly.
We shall also suppose the nondegeneracy of the convex combinationsof the vector fields appearing in our boundary condition;$(A4)$$\inf_{x\in\partial\Omega,b\in B}\{|\eta||\eta\in G(x, b)\}>0$.

We first present a key lemma:
Lemma. ([4] cf. [2],$[3_{\lrcorner}^{\mathrm{t}}$) Let $z\in\partial\Omega_{f}r>0_{f}\xi\in S^{n-1}$ and $\theta\in(0,1)$

satisfy that$\bigcup_{0}<t<rBt\theta(x+t\xi)\subset\Omega$

Then, there are constants$C^{1}(\overline{\Omega}\cross\overline{\Omega})$$C_{0},$

such that, for $x,$

$C_{1}\geq 1,$

for$x\in\overline{\Omega}\cap B_{r}(z)$$\sigma\in(0,1-\theta)$$y\in\overline{\Omega}\cap B_{r}(z)$and a function$\xi’,$$D_{x}\psi(x, y))\leq 0$

provided$|D_{x}\psi(x, y)|\leq C_{1}|x-y|$$\psi\in$,$c_{0}^{-1}|X-y|^{2}\leq\psi(x, y)\leq C_{0}|X-y|2$$\langle$.

$x\in\partial\Omega$

and,$\xi’\in B_{\theta+\sigma}(\xi)$

(3)

,(4)

and $D_{x}\psi(X, y)+D_{y}\psi(X, y)=0$ .

Remark. Note that (4) is stronger than the as sociated requirement in [3].

26

Theorem 2. ([5]) Assume $(A1)-(A4)$ . Let$u$and$v$

be a viscosity

sub- and supersolution of $(SC)$ , respectively. Then, we have$u^{*}\leq v_{*}$in$\overline{\Omega}$.

Ou tline of proof. For simplicity, we suppose that

$u$

and

$v$

are upper

and lower semicontinuous, respectively.
It is enough to get a contradiction when there exist $\Theta>0$ andsuch that $\ominus=u(z)-v(z)>u(x)-v(x)$ for any

Choose the$C^{1}$-function$\psi$.

$x\in\partial\Omega\backslash \{z\}$from Lemma for this . For$z$$\alpha,$$z\in\partial\Omega$$\mu>0$ ,weset$\Psi_{\alpha}(x, y)=u(x)-v(y)-\alpha\psi(x, y)+\mu\langle\xi, x-y\rangle$

where $\mu>0$ will be fixed later and $\alpha>0$ will be sent to$\infty$

,
. Let

$(x_{\alpha}, y_{\alpha})$satisfy$\Psi_{\alpha}(x_{\alpha}, y\alpha)=\mathrm{m}\mathrm{a}_{\frac{\mathrm{x}}{\Omega}}\Psi(\alpha Xx,y\in’ y)\geq\ominus$

A standard observation using$\Psi_{\alpha}(x_{\alpha}, y_{\alpha})\geq\Psi_{\alpha}(z, z)$.
together with (3)implies$\lim_{\alphaarrow\infty}x_{\alpha}=\lim_{\alphaarrow\infty}y_{\alpha}=z,\lim_{\alphaarrow\infty}u(x_{\alpha})=u(z),\lim_{\alphaarrow\infty}v(y\alpha)=v(z)$We shall write

$x$

and

$y$

in place ofA difficulty arises only when$x_{\alpha}$$x\in\partial\Omega$and$y_{\alpha}$, respectively.

;$H_{in}(x, u(X),$ $\alpha D_{x}\psi(x, y)-\mu\xi)\leq 0$

Thus, there is$\hat{b}\in B$, for some$l\in \mathrm{N}$.

, such that

$H(x, u(x),$ $\alpha Dx\psi(X, y)-\mu\xi;a,\hat{b})\leq 0$.

(5)

27

for all $a\in A(x,\hat{b})$ . In view of $(A3)-(ii)$ , we can choose$\{a_{k}\in A(z,\hat{b})\}_{k}l=1$

and$\eta(x)\equiv\Sigma_{k=1}^{l}t_{kg}(x, a_{k},\hat{b})$$A(z,\hat{b})(\subset A(x,\hat{b}))$andsuch that$\sum_{k=1}t_{k}\iota=1$

Set$\{t_{k}>0\}_{k}\iota_{=1}$$k1 \sum_{=}^{l}t_{k}g(Z, a_{k})\hat{b})\in\bigcup_{t>0^{B_{t\theta}}}(t\xi)$.

. Taking a convex combination over$a\in$

in the above inequality, we see that
(6)$-\langle\eta(_{X}), \alpha Dx\psi(X, y)-\mu\xi\rangle\leq C$

for a constant $C>0$ independent of$(A4)$$\mu$$\alpha,$. In view of $(A1)-(iii)$ and, there is a contant $k>0$ independent of$B_{\theta}(\xi)$. By (5), for large $\alpha>1$ , we see that$\alpha,$$\mu$such that $k\eta(x)\in$$k\eta(x)\in B_{\theta+\sigma}(\xi)$. Hence,by(6), we have$\mu k|\eta(X)|\sqrt{1-(\theta+\sigma)^{2}}\leq C$

for some $C>0$ . Therefore, we get a contradiction for a large $\mu>1$ .

$\blacksquare$Now, according to Theorem 2, it is easy to show the uniqueness andcontinuity of the lower and upper value functions.

Corollary. Assume $(A1)-(A4)$ .

solution of $(SC)$ . Moreover,$Then_{\mathrm{Z}}V$$V\in C(\overline{\Omega})$is the unique viscosity.

References[1] P. DUPUIS AND H. ISHII, On oblique derivative problems for fully

nonlinear second-order elliptic partial differential equations on nonsmooth28

domains, Nonlinear An al., 15 (1990), 1123-1138.
[2] L. C. EVANS AND P. E. SOUGANIDIS, Differential games and repre-sentation formulas for solutions ofequations,$\mathrm{H}\mathrm{a}\mathrm{m}\mathrm{i}\mathrm{l}\mathrm{t}\mathrm{o}\mathrm{n}- \mathrm{J}\mathrm{a}\mathrm{C}\mathrm{o}\mathrm{b}\mathrm{i}- \mathrm{I}_{\mathrm{S}}\mathrm{a}\mathrm{a}\mathrm{c}\mathrm{s}$Indiana Univ. Math. J., 33 (1984), 773-797.
[3] H. ISHII AND S. KOIKE, A new formulation of state constraints
problem of first-order PDEs, to appear.

[4] S. KOIKE, On the state constraint problem for differential games,preprint.
[5] M. H. SONER, Optimal control with state-space constraint , SIAM
$I$

J.

$Co\mathrm{n}$trol&Optim., 24 (1986), 552-562.