Examples of nonclassical feedback control problems

We consider a control system with “nonclassical” dynamics: ? = f(t, x, u, D _x u), where the right hand side depends also on the first order partial derivatives of the feedback control function. Given a probability distribution on the initial data, we seek a feedback u = u(t, x) which minimizes the expected value of a cost functional. Various relaxed formulations of this problem are introduced. In particular, three specific examples are studied, showing the equivalence or non-equivalence of these approximations.