数理解析研究所講究録
第 1542 巻 2007 年 33-40

33

Asymptotic solutions of Hamilton-Jacobi equations
with state constraints
Hiroyoshi Mitake (三竹 大寿) *Graduate school of Science and Engineering, Waseda University(早稲田大学大学院理工学研究科)
This article reviews recent results on the asymptotic behavior of solutions of the
Cauchy problem for Hamilton-Jacobi equations and describe briefly some results obtained in [26].

1Introduction.

We are concemed with the Cauchy problem for Hamilton-Jacobi equations:$=0$$u_{t}+H(x, Du(x,t))$$\{$$u(\cdot, 0)$$=u_{0}$inin$\Omega\cross(0, \infty)$$\Omega$

,

(1.1)

,

, which is assumed$H=H(x,p)$ is a function:
where is a domain ofis the unknown function,)
to be coercive and convex in the variable
:
$Du=$
:
is a given initial data. The, andfunction $H$ will be called the Hamiltonian.
In recent years, many researchers have investigated in the large-time behavior of
the solution $u(x, t)$ of (1.1) asand established convergence results which claim
under appropriate hypotheses that there exist a constant and a solution $v\in C(\Omega)$ of$H(x, Du)=c$ in such that$\Omega$$\Omega\cross \mathbb{R}^{n}arrow \mathbb{R}$$\mathbb{R}^{n},$

$p,$

$u$

$(\partial u/\partial x_{1}, \ldots , \partial u/\partial x_{n})$$u_{t}=\partial u/\partial t,$$\Omega\cross[0, \infty$$u_{0}$$arrow \mathbb{R}$$\Omegaarrow \mathbb{R}$$tarrow\infty$$c$$\Omega$$u(x, t)+ct-v(x)arrow 0$

locally uniformly for$x\in\Omega$as$tarrow\infty$

.

(1.2)

In this paper, we consider (1.1) with state constraints and establish a convergence result.
Associated with the Cauchy problem (C) is the additive eigenvalue problem for $H$ :$H(x, Du(x))=a$

Here one seeks for a pair $(u, a)$ ofof (1.3). If $(u, a)$ is such a pair, we calleigenvatue.

$\Omega$

.

(1.3)

and
such that is a solution
an additive eigenfun ction and an additive$u\in C(\overline{\Omega})$

$u$

in

’e-mail address: take-taketshiro2@akane.waseda.jp$a\in \mathbb{R}$

$u$

$a$

34

, the function
A simple observation related to this is that, for any$v(x)-ct$ is a solution of (1.1) if and only if $(v, c)$ is a solution of the additive eigenvalue
problem for $H$ . We call such a function $v(x)-ct$ an asymptotic solution of (1.1) provided$(v, c)$ is a solution of the additive eigenvalue problem for $H$ .
The following classical theorem solves the additive eigenvalue problem for $H$ whichis periodic in .
$(v, c)\in C(\Omega\cross \mathbb{R})$$x$

Theorem 1.1 (LIons-Papanicolaoll-Varadhan[24]). Let
be coercive and periodic in . Then for any$x$$H(x,p+Du)=a$
has a solution$\Omega=\mathbb{R}^{n}$and$H\in C(\Omega\cross \mathbb{R}^{n})$$p\in \mathbb{R}^{n_{J}}$$(v, c)\in C(\Omega)\cross \mathbb{R}$and the constantin$c$$\Omega$is unique.

Next we look back on a short history of asymptotic problems. This study goes backwhere
to the works of Kru\v{z}kov [22], Lions [23] and Barles [1], who studied thedependonvariable.
not$H=H(p)$does
andIn the case where $H=H(x,p)$ depends both on and , the first general results
were obtained by Namah-Roquejoffre [28] and Fathi $[11, 12]$ :$ca_{\iota}se$$x$$\Omega=\mathbb{R}^{n}$

$x$

$p$

Theorem 1.2. Let $M$ be a compact manifold without boundary. Let $H$ :be smooth, superlinear and stnctly convex. Then for any $u_{0}\in C(M)$ and a solutionof (1.1), there emsts a solution $(v, c)$ of (1.3) such that the convergence (1.2) holds.

$M\cross \mathbb{R}^{n}arrow \mathbb{R}$$u$

Afterwaxds $Ro$ quejoffre [30] and Davini-Siconolfi [9] Improved the above $approa\iota h$ .
visof pMial differentlal eqllations
By tother approaA based on thecosity solutions, this type of results have been obtained by Namah-Roquejoffre [28] and$\bm{t}d$$th\infty ry$$Barles- Sollganidis[4]$ .

More recently the large-time asymptotic problem of the same kind has been studied
by Fujita-Ishii-Loreti [14], Barles-Roquejoffie [3], Ishii [18],in the case where
and Ichihara-Ishii [16].
on the $large- t\ddagger mea_{\iota}symptotic$ problem
On the other htd, there are not mry$b_{011}ndary$conditions. Hamilton-Jacobi
which treat Hamilton-Jacobi $eq_{11}ations$ with
with the$eq_{l1}ations$ on
to be set onbealso
consideredcan-dimensional torus
covered by theperiodIc boundary. The periodic boundary conditIon Is$b_{011}ndary$$q_{l1}oted$ above. As far
condition andperiodic
the author knows, only thecondition are treated for the large-time asymptotic problem.
the Dirichlet$a_{\iota}symptotic$
problem for Hamilton-Jacobi equations with state
We here study theconstraints or, in other words, with the state constraint $bo\tau idary$ condition:$\Omega=\mathbb{R}^{n}$$res\iota 1lts$$\mathbb{R}^{n}$$n$$th\tau lS$$res\tau 1lts$$a_{\iota}s$$bound_{\mathfrak{N}}y$(C)$\{\begin{array}{ll}u_{t}+H(x, Du(x, t))\leq 0 in \Omega\cross(0, \ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}), (1.4)u_{t}+H(x, Du(x, t))\geq 0 in \overline{\Omega}\cross(0, \infty), (1.5)u(\cdot, 0)=u_{0} on \overline{\Omega}.

(1.6)\end{array}$

State constraint problems arise naturally in optimal control, and their dynamic
programming equations have the form $(1.4)-(1.5)$ , where the boundary condition is35

implicit in the fact that inequality (1.5) is required on the closure S2. This formulation,
been introduced by Soner [31]. Pairs
In terms of PDE, of state constraint problems
of inequalities such as $(1.4)-(1.5)$ are referred as Hamilton-Jacobi equations with state
constraints or the state constraint problem for Hamilton-Jacobi equations.
The additive eigenvalue problem with state constraints is formulated as follows:$ha_{\iota}s$$(E)_{a}$$\{\begin{array}{ll}H(x, Du(x))\leq a in \Omega, (1.7)H(x, Du(x))\geq a on \overline{\Omega}.

(1.8)\end{array}$

The additive eigenvalue problem for $H$ gives the “stationary states” for solutions of(C) as our main result shows. We call a function $v(x)-ct$ an asymptotic solution of(C) provided $(v, c)$ is a solution of (1.7) and (1.8).
Our main purpose of this paper is to show that under appropriate hypotheses on $H$
and any solution $u(x, t)$ of (C) converges to an asymptotic solution $v(x)-ct$ inas $tarrow\infty$ . That is, as $tarrow\infty$ ,$C(\Omega\gamma$$\Omega$$u(x, t)+ct-v(x)arrow 0$

uniformly for.

$x\in\overline{\Omega}$

Assumptions.

2

for some $k,$ $l\in N$ and $r>0$ . Write $U(x, r)=\{y\in \mathbb{R}^{n}||x-y|<r\}$ .
We denote by $C(A, B),$ $C^{0,1}(A, B)$ and $LSC(A, B)$ the sets of continuous, Lipschitz
continuous and lower semicontinuous functions on $A$ with values in $B$ , respectively. We
denote by $W^{1,\infty}(A, B)$ the set of functions on $A$ with values in $B$ which is differentiable
and the distributional first derivatives are bounded almost everywhere on $A$ . Whenthe set $B$ is clear by the context, we may omit writing $B$ in the above notation: for
instance, we may write $C(A)$ for $C(A, B)$ . We also use the symbol $AC([a, b], B)$ to
denot the set of absolutely continuous fimctions on $[a, b]$ with values in $B$ .
) $arrow[0, \infty$ ) a modulus if it is continuous an nondeWe call a function $m$ :Let$A\subset \mathbb{R}^{k},$$B\subset \mathbb{R}^{l}$

$e$

$d$

$[0, \infty$) and if $m(O)=0$ .
creasing on
We make the following assumptions on the Hamiltonian $H$ , the initial data
the domain :$[0, \infty$$\Omega$(H1)$H\in C(\overline{\Omega}\cross \mathbb{R}^{n})$.

(H2) The function $parrow H(x,p)$ is strict convex for each$x\in\overline{\Omega}$.

(H3) The function $H$ is coercive, i.e.
$\lim_{farrow\infty}\inf\{H(x,p)|x\in\overline{\Omega},p\in \mathbb{R}^{n}\backslash U(0,r)\}=\infty$(u1)$u_{0}\in C(\overline{\Omega})\cap W^{1,\infty}(\Omega)$(B1) The domain$\Omega$.

.

is bounded and a H\"older domain with exponent $\alpha>2/3$ .

$\tau\ovalbox{\tt\small REJECT}$

and

36

Remark 2.1. In fact, we can remove the restriction that$u_{0}\in W^{1,\infty}(\Omega)$in [26].

. See Section 7

Remark 2.2. An equivalent formulation of (B1) is that there exists a constant $2/3<$
such that for any
and $b_{z}>0$ ,and for some$z\in\partial\Omega$$\alpha\leq 1$$\eta_{z}\in \mathbb{R}^{n}$$\pi_{\cap U(z,b_{*})}\bigcup_{x\in}\bigcup_{0<\ell<b_{*}}x+s^{\alpha_{7}}lz\cdot$

3

Solutions of (C).

Now we give a comparison result for (C).

Theorem 3.1 (Theorem 2.1 in [26]). Let $T>0$ , and let
and $v\in$$v_{\ell}+H(x,Du)\leq0$$v_{t}+H(x,Dv)\geq
0$ onin
satisfyand$u\leq v$ onin the vzscosity sense, respectively. Then, if $u\leq v$ on.
$u\in C(\overline{\Omega}\cross[o, \eta)$$LSC(\overline{\Omega}\cross[0, T])$$\Omega\cross(0, T)$$\overline{\Omega}\cross(0, T)$$\overline{\Omega}\cross\{0\},$$\overline{\Omega}\cross[0, T)$

For a proof, we refer to the reader [26, Section 4]. Uniqueness of solutIons of (C)follows from the above theorem. It Is worth pointing out that in the literature on(C) or its stationary version, it is usually assumed for uniqueness of solutions thatis a Lipschitz domain. Here we take advantage of assumption (H3) to $re$ duce thestandard Lipschitz regularity of to the H\"older regularity (B1), which seems to bea new observation. This new generality of domains
is obtained with help of thecoercivity assumption (H3) on $H$ .
) $arrow R$ defined by
We consider the flmction : S72$\Omega$$\Omega$$\Omega$$u$$\cross[0, \infty$$u(x, t)$ $:= \inf\{\int_{0}^{t}L(\gamma(s),\dot{\gamma}(s))ds+u_{0}(\gamma(0))|\gamma\in C(x;t)\}$ ,(3.1)

where is the Lagrangian of $H,$ $i.e,$ $L(x,p);= \sup_{\xi\in R^{n}}\{p\cdot\xi-H(x,\xi)\}$ and $C(x;t)$denotes the set of all trajectories
such that $\gamma(t)=x$ .
The regularity and continuity of is obtained by our next theorem.
$L$$\gamma\in AC([0, t],\Omega\gamma$$u$Theorem 3.2. Let
(a)

$u$

be the junction defined by (3.1). Then$u\in C^{0,1}(\Omega\cross[0, \infty))\cap C(\overline{\Omega}\cross[0, \infty))$;(b) There is a constant $C>0such$ that $|Du(x, t)|+|u_{t}(x, t)|\leq Ca.e$ .
$\Omega\cross(0, \infty)$

(c)

$u$

is

$a$

;

(unique) solution of (C).

$(x, t)\in$37

4Additive eigenvalue problems.

We define the constantby$c_{H}$$:= \inf$$c_{H}${$a\in \mathbb{R}|(1.7)$

has a solution},(4.1)

and consider the following inequalities:$H(x, Du(x))\leq c_{H}$

in

,

(4.2)

$H(x, Du(x))\geq c_{H}$

on .

(4.3)

$\Omega$$\overline{\Omega}$

The following theorem ensures the existence of the additive eigenvalue problem andthe uniqueness of the constant.

Theorem 4.1. Problem$(E)_{a}$has a solution$v\in C(\Omega\gamma$

if and only if $a=c_{H}$ .

Using Theorem 3.1 and Theorem 4.1, we see that the functim $u(x, t)+c_{H}t$ is bounded), where is the solution of (C).
on$\overline{\Omega}\cross[0, \infty$$u$Proposition 4.2. There vists a constant $C>0$ such that$e$$|u(x, t)+c_{H}t|\leq C$

on$5\cross[0, \infty$).

We assume that $c_{H}=0$ by replacing $H$ by $H-c_{H}$ .
The following lemma is important for our proof of Theorem 5.2.
be a viscosity solution
andLemma 4.3 (Theorem 8.1 in [26]). Let$\gamma\inC((-\infty,0$], ) such that $\gamma(0)=x$ and for any. Then there exists a curveof$x\in\overline{\Omega}$$\phi\in C(\overline{\Omega})$$\overline{\Omega}$$(E)_{0}$$[a, b]\subset(-\infty, 0]_{f}$$\gamma\in AC([a, b],\overline{\Omega})$andFollowing $[30, 18]$ , we call curves satisfying (4.4) extremal curves forto denote the set of all extremal curves for .
inafter we write(4.4)$\phi$and here-$\phi$$\mathcal{E}(\phi)$5.

$\int_{a}^{b}L(\gamma(s),\dot{\gamma}(s))+c_{H}ds=\phi(\gamma(b))-\phi(\gamma(a))$

Convergence.

Let $u(x, t)$ be the unique viscosity solution of (C).

Lemma 5.1 (Proposition 8.2 in [26]). There enist a constant $\delta\in(0,1)$ and a modulus$T>\tau\geq 0$,
is a solution of
for which ifthen$u_{0}\in C(\overline{\Omega}),$$\omega$-.

$\phi$$(E)_{0},$$\gamma\in \mathcal{E}(\phi),$$and\mapsto^{\tau-\tau}\leq\delta$$( \gamma(0), T)-u(\gamma(-T), \tau)\leq\phi(\gamma(0))-\phi(\gamma(-T))+\frac{\tau T}{T-\tau}\omega(\frac{\tau}{T-\tau})$.

38

Lemma 5.1 is a variant of [18, Proposition 7.1]. We remark that the “strict” convexity of $H$ is only needed in Lemma 5.1 in our approach to Theorem 5.2. We here notethat it is known that there are some examples of Hamilton-Jacobi equations to showthat the HanliltonIan is not strict convex but convex and the convergence (1.2) is nottrue. We refer the reader to [1, 15, 4, 5, 19].
We state our main theorem:

Theorem 5.2 (Theorem 2.2 in [26]). For any there exists a solutionof the additive eigenvalue problem for $H$ such that if
solution of (C), then, as$u_{0}$$(v, c)\in C(\overline{\Omega})\cross \mathbb{R}$$u\in C(\overline{\Omega}\cross[0, \infty))$is the viscosity$tarrow\infty_{f}$$u(x, t)+ct-v(x)arrow 0$ uniformly on St.

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