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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)

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