×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite