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Path Integral Control methods were developed for stochastic optimal controlcovering a wide class of finite horizon formulations with control affinenonlinear dynamics. Characteristic for this class is that the HJB equation islinear and consequently the value function can be expressed as a conditionalexpectation of the exponentially weighted cost-to-go evaluated overtrajectories with uncontrolled system dynamics, hence the name. Subsequently itwas shown that under the same assumptions Path Integral Control generalises tofinite horizon risk sensitive stochastic optimal control problems. Here westudy whether the HJB of infinite horizon formulations can be made linear aswell. Our interest in infinite horizon formulations is motivated by thestationarity of the associated value function and their inherent dynamicstability seeking nature. Technically a stationary value function may ease thesolution of the associated linear HJB. Second we argue this may offer aninteresting starting point for off-linear Reinforcement Learning applications.We show formally that the discounted and average cost formulations arerespectively intractable and tractable.

Risk Sensitive Path Integral Control for Infinite Horizon Problem Formulations