Non-classical problems of optimal feedback control

The paper is concerned with problems of optimal feedback control with "non-classical" dynamics ẋ=f(t,x,u,Du), where the evolution of the state x depends also on the Jacobian matrix Du=(∂u _i/∂x _j) of the feedback control function u=u(t, x). Given a probability measure μ on the set of initial states, we seek feedback controls u({dot operator}) which minimize the expected value of a cost functional. After introducing a basic framework for the study of these problems, this paper focuses on three main issues: (i) necessary conditions for optimality, (ii) equivalence with a relaxed feedback control problem in standard form, and (iii) dependence of the expected minimum cost on the probability measure μ.