A practical variational approach to stochastic optimal control via state moment equations

By using a very simple remark on the moment equations of stochastic Markov processes, one can use the Euler-Lagrange variational approach (with Hamiltonian) to solve some stochastic optimal control problems. As an example one considers a linear-quadratic-Gaussian problem with small state dependent noise, which is solved by a perturbation technique (expansions with respect the small parameter). Another application example is provided by the tracking control of mechanical systems with noisy sliding equations which works via the stochastic Hamiltonʹs principle of mechanical systems. Finally, an economic example shows that the approach also works with moments of fractional order.