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Risk-Sensitive Control and an Abstract Collatz–Wielandt Formula

Author

Listed:
  • Ari Arapostathis

    (The University of Texas at Austin)

  • Vivek S. Borkar

    (Indian Institute of Technology)

  • K. Suresh Kumar

    (Indian Institute of Technology)

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.

Suggested Citation

  • Ari Arapostathis & Vivek S. Borkar & K. Suresh Kumar, 2016. "Risk-Sensitive Control and an Abstract Collatz–Wielandt Formula," Journal of Theoretical Probability, Springer, vol. 29(4), pages 1458-1484, December.
    • Handle: RePEc:spr:jotpro:v:29:y:2016:i:4:d:10.1007_s10959-015-0616-x
    DOI: 10.1007/s10959-015-0616-x
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