Jacobson type necessary optimality conditions for general control systems

This paper is devoted to a second order maximum principle and sensitivity relations for the Mayer problem arising in optimal control theory. The control system under consideration involves arbitrary closed, time dependent control sets U(t) and arbitrary closed sets of initial conditions. Optimal controls are supposed to be merely measurable. We prove that to every optimal trajectory-control pair (x̅(·); u̅(·)) corresponds a solution p̅(·) of the adjoint system (as in the Pontryagin maximum principle) and a matrix solution W(·) of an adjoint matrix differential equation that satisfy some second order transversality and maximality conditions. We then show that in the case when the system dynamics are differentiable with respect to the input, this approach leads to pointwise Jacobson like necessary optimality conditions for general control systems and measurable optimal controls that may take values on the boundary of control constraints, drastically improving some results known up to now. Finally we provide second order sensitivity relations along x̅(·) involving both p̅(·) and W(·).