Abstract
Standard versions of the maximum principle for optimal control problems with pathwise state inequality constraints are satisfied by a trivial set of multipliers in the case when the left endpoint is fixed and lies in the boundary of the state constraint set, and so give no useful information about optimal controls. Recent papers have addressed the problem of overcoming this degenerate feature of the necessary conditions. In these papers it is typically shown that, if a constraint qualification is imposed, requiring existence of inward pointing velocities, then sets of multipliers exist in addition to the trivial ones. A simple, new approach for deriving nondegenerate necessary conditions is presented, which permits relaxation of hypotheses previously imposed, concerning data regularity and convexity of the velocity set. (C) 1999 Academic Press.

Abstract

Abstract
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.

Abstract
We propose a Model Predictive Control (MPC) framework to generate feedback controls for time-varying nonlinear systems with input constraints. One of the main features of this framework is to allow the feedback laws to be discontinuous and thereby enlarge the class of nonlinear systems that can be stabilized by continuous-time MPC. We consider a continuous-time MPC framework and perform a continuous-time stability analysis while considering that the inter-sampling times are nonzero and that the open-loop optimal control problems are solved at every sampling instant. The feedback law generated by MPC is not a function of the state at every instant of time, rather it is a function of the state at the last sampling instant. The trajectories resulting from this "sampling-feedback" are well-defined even when the feedback is discontinuous. Important classes of nonlinear systems that could not be stabilized by a continuous feedback, such as the nonholonomic systems, can now be addressed in a continuous-time MPC framework.

Abstract

Abstract
In this paper, first order necessary conditions of optimality for control problems with pathwise state constraints are provided. These conditions axe able to avoid the degeneracy phenomenon which affects standard versions of the maximum principle. A common. feature of the endpoint degeneracy phenomena described in the literature is identified and it is shown that similar phenomena might occur other than at the endpoints of the trajectory. A strengthened form of the maximum principle is established under suitable constraint qualifications, in which the (measure) multiplier associated with the pathwise state constraint is not involved in the nontriviality condition. Copyright (C) 2001 IFAC.