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On Bellman equations of ergodic control in \(\mathbb{R}{}^ n\). (English) Zbl 0779.35038

\(Lu+H(x,\nabla u)+\lambda=q(x)\), \(x\in\mathbb{R}^ n\), \(H\) the Hamiltonian, \(\lambda\) a constant, \(L\) a second order differential operator (locally uniform elliptic) is called Bellman equation of ergodic control in \(\mathbb{R}^ n\). At first (chapter 1) the authors give an interesting discussion of problems and results, which are connected with this equation, and come to the conclusion, that a satisfactory theory for equations of this type has not been obtained so far.
Then (chapters 2 and 3) the solvability of \(Lu+H+\alpha u=q\) in \(\mathbb{R}^ n\) with \(\alpha>0\) is proved and the passage to the limit \(\alpha\to 0\) and asymptotic estimates for \(| x|\to\infty\) follow. Chapter 4 deals with uniqueness: it exists, assuming some conditions (especially \(H(x,p)\geq c\sum a_{ij} p_ i p_ j\), \(c>0\), and \(H(x,\beta p)\geq \beta^ 2 H(x,p)\) \(\forall x,p\) and for \(\beta\geq 1\)), at most one unique pair \(u\) (\(u\) up to a constant), \(\lambda\), satisfying the equation, where \(\lambda\) is a constant and \(u\in W^{1,\infty}_{\text{loc}}\) with \(u(\infty)=\infty\). Additionally uniqueness is shown with other conditions for \(H\).
The authors close their paper with some remarks concerning the quadratic growth of the Hamiltonian and the unboundedness of the solutions, which in the present paper arises from the fact, that the domain is the whole space (the solution is locally bounded).

MSC:

35J60 Nonlinear elliptic equations
49J20 Existence theories for optimal control problems involving partial differential equations
49L20 Dynamic programming in optimal control and differential games
35B37 PDE in connection with control problems (MSC2000)
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