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This paper introduces a new method to tackle the issue of the almost sure convergence of stochastic approximation algorithms defined from a differential inclusion. Under the assumption of slowly decaying step-sizes, we establish that the set of essential accumulation points of the iterates belongs to the Birkhoff centre associated with the differential inclusion. Unlike previous works, our results do not rely on the notion of asymptotic pseudotrajectories, predominant technique to address the convergence problem. They follow as a consequence of Young's superposition principle for closed measures. This perspective bridges the gap between Young's principle and the notion of invariant measure of set-valued dynamical systems introduced by Faure and Roth. Also, the proposed method allows for obtaining sufficient conditions under which the velocities locally compensate around any essential accumulation point. [ABSTRACT FROM AUTHOR]

This work addresses the occupation measure relaxation of calculus of variations problems, which is an infinite-dimensional linear programming reformulation amenable to numerical approximation by a hierarchy of semidefinite optimization problems. We address the problem of equivalence of this relaxation to the original problem. Our main result provides sufficient conditions for this equivalence. These conditions, revolving around the convexity of the data, are simple and apply in very general settings that may be of arbitrary dimensions and may include pointwise and integral constraints, thereby considerably strengthening the existing results. Our conditions are also extended to optimal control problems. In addition, we demonstrate how these results can be applied in nonconvex settings, showing that the occupation measure relaxation is at least as strong as the convexification using the convex envelope; in doing so, we prove that a certain weakening of the occupation measure relaxation is equivalent to the convex envelope. This opens the way to application of the occupation measure relaxation in situations where the convex envelope relaxation is known to be equivalent to the original problem, which includes problems in magnetism and elasticity. [ABSTRACT FROM AUTHOR]