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Recent work on data-driven control and reinforcement learning has renewedinterest in a relative old field in control theory: model-free optimal controlapproaches which work directly with a cost function and do not rely uponperfect knowledge of a system model. Instead, an "oracle" returns an estimateof the cost associated to, for example, a proposed linear feedback law to solvea linear-quadratic regulator problem. This estimate, and an estimate of thegradient of the cost, might be obtained by performing experiments on thephysical system being controlled. This motivates in turn the analysis ofsteepest descent algorithms and their associated gradient differentialequations. This note studies the effect of errors in the estimation of thegradient, framed in the language of input to state stability, where the inputrepresents a perturbation from the true gradient. Since one needs to studysystems evolving on proper open subsets of Euclidean space, a self-containedreview of input to state stability definitions and theorems for systems thatevolve on such sets is included. The results are then applied to the study ofnoisy gradient systems, as well as the associated steepest descent algorithms.

Remarks on input to state stability of perturbed gradient flows, motivated by model-free feedback control learning