Generalized synthesis for singular nonlinear optimal control problems

We consider the problem of minimizing a quadratic noncoercive functional along the trajectories of a control-affine system. Due to lack of coercivity, existence of “classical” minimizers cannot, in general, be guaranteed. Under appropriate commutativity assumptions the problem can be extended into the space of generalized controls of class $W_{-1,\infty}$ and reduced into a new problem which is generically coercive but nonconvex. We show how to extend further the problem in order to include generalized controls which are “generalized derivatives of one-parameter families of regular probability measures”, thus achieving convexification. Generalized trajectories for this type of controls exist only in a weak sense. We discuss a version of the maximum principle suitable to this class of problems and show how a generalized synthesis can be obtained.