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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:
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