Cumulants in risk-sensitive control: the full-state-feedback cost variance case

Abstract

The risk-sensitive optimal stochastic control problem has an interpretation in terms of managing the value of a denumerable linear combination of the cumulants of a traditional performance index. This paper considers in detail the foundations for a full-state-feedback solution to the problem of controlling the second cumulant of a cost function, given modest constraints on the first cumulant. The formulation is carried out for a class of nonlinear stochastic differential equations, associated with an appropriate class of non-quadratic performance indices. A Hamilton-Jacobi framework is adopted, and the defining equations for solving the linear, quadratic case are determined. The method is then applied to a situation in which a building is to be protected from earthquakes.