Abstract

Abstract
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.

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. Copyright (C) 2003 IFAC.

Abstract
We propose an MPC framework to address impulsive dynamical systems, i.e. systems in which the state trajectories are permitted to have discontinuities (jumps) in response to impulsive controls. We describe how previous results on sampled-data trajectories, invariance of sets, optimal control, and stability in the context of impulsive systems can be used and adapted to a sampled-data MPC framework. We provide sufficient conditions on the design parameters of the optimal control problems that guarantee: (i) feasibility of the sequence of optimal control problems, and (ii) stability of the controlled system. © 2012 IFAC.

Abstract
In this work, we discuss normal forms of necessary conditions of optimality (NCO) for optimal control problems subject to pathwise state constraints and for problems in the calculus of variations with inequality constraints. It is known that standard forms of the NCO may fail to provide information that is useful to identify optimal solutions, namely when the multiplier associated with the objective function takes the value zero. The normal forms of the NCO guarantee that the conditions remain always informative, which is of importance in critical applications where decisions based on optimization are taken, such as autonomous systems. Based on a previous nondegenerate maximum principle for optimal control problems, we extend the strengthness of these conditions to normality while applying them to the particular case of calculus of variations problems. We compare our results with existent normal forms of NCO for dynamic optimization problems and conclude that, when applied to calculus of variations problems, we may say that, under similar conditions, we can apply such result to a wider class of problems, having less regularity on the data.

Abstract
We propose a new model predictive control (MPC) framework to generate feedback controls for time-varying nonlinear systems with input constraints. We provide a set of conditions on the design parameters that permits to verify a priori the stabilizing properties of the control strategies considered. The supplied sufficient conditions for stability can also be used to analyse the stability of most previous MPC schemes. The class of nonlinear systems addressed is significantly enlarged by removing the traditional assumptions on the continuity of the optimal controls and on the stabilizability of the linearized system. Some important classes of nonlinear systems, including some nonholonomic systems, can now be stabilized by MPC. In addition, we can exploit increased flexibility in the choice of design parameters to reduce the constraints of the optimal control problem, and thereby reduce the computational effort in the optimization algorithms used to implement MPC.