Stabilization and Optimal Control of a Multi Input-Delayed SDE System
Résumé
In this paper, we design a controller for a input-delayed Stochastic Differential Equation (SDE) with distinct input delays and a stochastic drift. Our objective is to steer the system to a desired final state on average while minimizing variance over time, thereby improving robustness to disturbances. We first establish a controllability result, highlighting lower bounds for the variance, demonstrating that the system cannot reduce variance beyond strict structural limits. Under standard controllability conditions, we then design a controller that drives the mean of the states while ensuring bounded variance. Finally, we analyze the optimal control problem for variance minimization over the entire trajectory. Under additional controllability assumptions, we show that the optimal control can achieve any variance level above the fundamental structural limit.
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