Risk-Sensitive Control and an Abstract Collatz–Wielandt Formula

Ari Arapostathis (), Vivek S. Borkar () and K. Suresh Kumar ()
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Ari Arapostathis: The University of Texas at Austin
Vivek S. Borkar: Indian Institute of Technology
K. Suresh Kumar: Indian Institute of Technology

Journal of Theoretical Probability, 2016, vol. 29, issue 4, 1458-1484

Abstract: Abstract The ‘value’ of infinite horizon risk-sensitive control is the principal eigenvalue of a certain positive operator. For the case of compact domain, Chang has built upon a nonlinear version of the Krein–Rutman theorem to give a ‘min–max’ characterization of this eigenvalue which may be viewed as a generalization of the classical Collatz–Wielandt formula for the Perron–Frobenius eigenvalue of a nonnegative irreducible matrix. We apply this formula to the Nisio semigroup associated with risk-sensitive control and derive a variational characterization of the optimal risk-sensitive cost. For the linear, i.e., uncontrolled case, this is seen to reduce to the celebrated Donsker–Varadhan formula for principal eigenvalue of a second-order elliptic operator.

Keywords: Risk-sensitive control; Collatz–Wielandt formula; Nisio semigroup; Variational formulation; Principal eigenvalue; Donsker–Varadhan functional; Primary 60J60; Secondary 60F10; 93E20 (search for similar items in EconPapers)
Date: 2016
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DOI: 10.1007/s10959-015-0616-x

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