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Ergodic problem for the Hamilton-Jacobi-Bellman equation. II. (English) Zbl 0903.49018
Consider the O.D.E. $$\dot x_\alpha=b(x_\alpha(t), \alpha(t))$$, $$x_\alpha(0)=x\in \overline\Omega$$, $$x_\alpha(t)\in \overline\Omega$$ for all $$t\geq 0$$, with periodic boundary conditions if $$\overline\Omega$$ is a $$n$$-dimensional torus (or with Neumann boundary condition if $$\Omega$$ is a bounded open subset in $$R^n$$). In this setting the control variable $$\alpha$$ is a measurable function from $$[0,\infty)$$ to some metric space $$A$$. Let $$u_\lambda(x)$$ be the value function $$u_\lambda(x)=\text{ inf}_\alpha \int_0^\infty e^{-\lambda s} f(x_\alpha(s),\alpha(s)) ds$$, where $$f$$ is Lipschitz on $$\overline\Omega\times A$$. It is proved that if any point $$x\in \overline\Omega$$ is exactly uniformly controllable, then there exists a constant $$d_f$$ such that $$\text{ lim}_{\lambda \downarrow 0}\lambda u_\lambda(x)=d_f$$ uniformly in $$x\in\overline\Omega$$. Similar results are given when the exact controllability assumption is replaced by an approximate controllability condition. The finite horizon problem is also studied. The same kind of result is proved when $$x$$ is controllable to any point $$y\in Z_1$$, where $$Z_1$$ is a nonempty closed invariant subset strictly contained in $$\overline\Omega$$. This paper is the continuation of [M. Arisawa, Ann. Inst. Henri Poincaré, Anal. Non Linéaire 14, No. 4, 415-438 (1997; Zbl 0892.49015)].

##### MSC:
 49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games 49L99 Hamilton-Jacobi theories 49L20 Dynamic programming in optimal control and differential games
##### Keywords:
Hamilton-Jacobi equations; ergodic attractor
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##### References:
 [1] Arisawa, M., Ergodic problem for the Hamilton-Jacobi-Bellman equation - existence of the ergodic attractor -, Ann. I.H.P., anal. non lin., Vol. 14, 415-438, (1987) · Zbl 0892.49015 [2] Arisawa, M.; Lions, P.L., A continuity result in the state constraints system, (), No. 3 · Zbl 0953.49004 [3] Capuzzo-Dolcetta, I.; Lions, P.L., Hamilton-Jacobi equations with state constraints, Trans. amer. math. soc., Vol. 318, 643-683, (1990) · Zbl 0702.49019 [4] Capuzzo-Dolcetta, I.; Menaldi, J.L., Asymptotic behavior of the first order obstacle problem, Journal of differential equations, Vol. 75, No. 2, (1988) · Zbl 0674.35011 [5] Crandall, M.G.; Lions, P.L., Viscosity solutions of Hamilton-Jacobi equations, Trans. amer. math. soc., Vol. 277, 1-42, (1983) · Zbl 0599.35024 [6] Lions, P.L., Generalized solutions of Hamilton-Jacobi equations, () · Zbl 0659.35015 [7] Lions, P.L., Neumann type boundary conditions for hamilmton-Jacobi equations, Duke J. math., Vol. 52, 793-820, (1985) · Zbl 0599.35025 [8] Simon, B., Functional integration and quantum physics, (1979), Academic Press · Zbl 0434.28013 [9] Soner, H.M.; Soner, H.M., Optimal control with state-space constraint II, SIAM J. control optim., SIAM J. control optim., Vol. 24, 1110-1122, (1986) · Zbl 0619.49013
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