On the convergence of random search algorithms in continuous time with applications to adaptive control

This paper considers the problem of random search in the case where a gradient is used to bring the solution toward a local minimum, and a white noise perturbation is added to drive the solution toward the global minimum. Such an algorithm has been suggested by several authors (see, for example, Khas\´minskii [1], Yudin [2], Gurin [3], and Vaysbord[4],[5]). The problem is considered in terms of the "differential generator" of the stochastic process. It is shown that the algorithm does not converge to a global minimum. However, in the case where the value of the function at the global minimum is known, but the point at which the global minimum occurs is not known, the results show that this search technique can be used to keep the system\´s state at this point.