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On the solutions of the equation arising from the singular limit of some eigen problems. (English) Zbl 0920.49015
McEneaney, William M. (ed.) et al., Stochastic analysis, control, optimization and applications. A volume in honor of Wendell H. Fleming, on the occasion of his 70th birthday. Boston: Birkhäuser. 135-150 (1999).
The principal eigenvalue $$\mu^\varepsilon$$ and the corresponding normalized eigenvector $$\varphi^\varepsilon$$ are considered for matrices $$A^\varepsilon$$ with nonnegative elements $$A^\varepsilon(i,j)$$ and $$A^\varepsilon(i,j)\approx \exp( \frac{1}{\varepsilon}V(i,j))$$ for $$\varepsilon\rightarrow 0.$$ Limiting equations are discussed for $$\varepsilon\log \mu^\varepsilon$$ and $$\varepsilon\log \varphi^\varepsilon(i)$$ as $$\varepsilon\rightarrow 0.$$ There are relations to the Bellman equation of stochastic control theory.
For the entire collection see [Zbl 0905.00042].
Reviewer: W.Grecksch (Halle)

##### MSC:
 49L20 Dynamic programming in optimal control and differential games 15A18 Eigenvalues, singular values, and eigenvectors 93E20 Optimal stochastic control