M.M.A. Ferreira

An Euler–Lagrange Inclusion for Optimal Control Problems with State Constraints

New first-order necessary conditions for optimality for control problems with pathwise state constraints are given. These conditions are a variant of a nonsmooth maximum principle which includes a joint subdifferential of the Hamiltonian – a condition called Euler–Lagrange inclusion (ELI). The main novelty of the result provided here is the ability to address state constraints while using an ELI.The ELI conditions have a number of desirable properties. Namely, they are, in some cases, able to convey more information about minimizers, and for the normal convex problems they are sufficient conditions of optimality. It is shown that these strengths are retained in the presence of state constraints.

Unmaximized necessary conditions for constrained control problems

New first-order necessary conditions of optimality for control problems with mixed state-control and pure state constraints are derived. In contrast to known results these conditions hold when the Jacobian of the active state-control constraints with respect to the control has full rank. A crucial feature of these conditions is that they are stated in terms of a joint subdifferential and do not involve the maximization of the Hamiltonian. The main novelty of the result is precisely the ability to address state-control and pure state constraints, generalizing previously proved results. The conditions developed are, in some cases, stronger than the standard nonsmooth maximum principle, since they can reduce the set of candidates to minimizers.

Nondegeneracy and normality in necessary conditions involving Hamiltonian inclusions for state-constrained optimal control problems

Similarly to other standard versions of the maximum principle, recently derived necessary conditions of optimality involving Hamiltonian inclusions are satisfied by a degenerate set of multipliers when applied to problems to which the initial state is fixed and it is in the boundary of the state constraint set. In such case, the necessary conditions do not provide useful information to select minimizers. A constraint qualification under which nondegenerate necessary conditions based on a standard maximum principle was previously defined. In this paper we show that when the velocity set is convex the same constraint qualification permits nondegenerate necessary conditions involving Hamiltonian inclusions. This is of relevance since it covers problems in which the set of multipliers produced by Hamiltonian inclusion conditions is smaller than those generated by standard maximum principles. Furthermore, we show that the constraint qualification can be strengthened so that normality can be established.

On Optimality Conditions for Control Problems with Constraints

Abstract We report on optimality conditions for control problems with mixed state control constraints and pure state constraints. Our main goal is to unify previously developed work and to illustrate the different approaches used. We describe necessary conditions that include an Euler-Lagrange inclusion for a weak minimizer as well as necessary conditions in the form of a maximum principle for a strong minimizer. The sufficiency of the same conditions for a certain class of these problems is also analysed

Necessary conditions for some optimal control problems with state-dependent control constraints

Necessary conditions of optimality in the form of maximum principles are proved for some optimal control problems with state-dependent control constraints. The authors consider problems with constraints of the form x/spl dot/(t)=f/sup 1/(t,x(t))+f/sup 2/(t,u(t)) 0=b/sup 1/(t,x(t))+b/sup 2/(t,u(t)) together with end point constraints and pointwise set constraints on the control variable. An optimality condition in the form of a strong maximum principle is derived under a convexity hypothesis. The authors highlight through example the importance of convexity for the validity of their maximum principle. Moreover, the authors show that without such hypothesis no weak version of their maximum principle is valid.

Optimal control problems with state-dependent control constraints

We consider optimal control problems with state-dependent control constraints to which we associate a differential inclusion control problems. Necessary optimality conditions are obtained for the latter using proximal normal analysis. It is then shown that calmness guarantees the nontriviality of such conditions. The calmness assumption on the differential inclusion problem permits the derivation of optimality conditions in the form of a maximum principle for the original problem.<<ETX>>