A hamiltonian formulation of risk-sensitive Linear/quadratic/gaussian control

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Abstract
We consider the optimal-control problem with time-homogeneous linear (and in general non-Markov) dynamics and risk-sensitive criterion. Appeal to the extremal principle prescribed by the risk-sensitive certainty equivalence principle (RSCEP) yields a symmetric equation system, indicating that the extended hamiltonian formulation generalizes naturally to the risk-sensitive case. The conjugate variable of a hamiltonian formulation now has an interpretation in terms of forecasted process noise, and the RSCEP in fact provides a stochastic maximum principle for which all variables have a clear interpretation and the desired measurability properties. In the infinite-horizon case (meaningful under generalized controllability conditions) optimal control is determined explicitly in terms of a canonical factorization. For the case of imperfect process observation, the RSCEP leads to coupled-equation systems that can again be solved in terms of canonical factorizations in the time-invariant (stationary) case.