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Infinite-dimensional nonlinear systems Sampled-data control Gradient descent method Stochastic approximation methods work. First, discrete-time gradient descent method is considered and semi-global practical asymptotic stability with respect to an ultimate bound is shown. Next, under the more complicated setting where the sampled measurements of the plant’s output are corrupted by an additive noise, three basic stochas-tic approximation methods are analysed; namely finite-difference, random directions, and simultaneous perturbation. Semi-global convergence to an optimum with probability one is established. A tuning pa-rameter within the sampled-data framework is the period of the synchronised sampler and hold device, which is also the waiting time during which the system dynamics settle to within a controllable neigh-bourhood of the steady-state input–output behaviour. © 2015 Elsevier Ltd. All rights reserved. 1