Risk-Sensitive Control and an Abstract Collatz-Wielandt Formula

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.