Jacobson type necessary optimality conditions for general control systems

Abstract

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 (x(·),ū(·)) corresponds a solutionp(·) 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 alongx(·) involving bothp(·) and W (·).

First and second order necessary conditions for stochastic optimal controls

“…Using this method, in [17,11], some second order integral type necessary conditions for deterministic optimal controls were established. It was shown in [10,12] that these necessary conditions imply pointwise ones.…”

Pointwise second-order necessary optimality conditions and second-order sensitivity relations in optimal control

“…To the best of our knowledge this definition of second-order normals never appeared in the literature before [11], where we used it to express the second-order transversality conditions. A second-order normal cone was defined in [3] (without using second-order tangents) by…”