Abstract
This note proposes a model predictive control (MPC) algorithm for the solution of a robust control problem for continuous-time systems. Discontinuous feedback strategies are allowed in the solution of the min-max problems to be solved. The use of such strategies allows MPC to address a large class of nonlinear systems, including among others nonholonomic systems. Robust stability conditions to ensure steering to a certain set under bounded disturbances are established. The use of bang-bang feedbacks described by a small number of parameters is proposed, reducing considerably the computational burden associated with solving a differential game. The applicability of the proposed algorithm is tested to control a unicycle mobile robot.

Abstract
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.

Abstract
New first-order necessary conditions of optimality for control problems with state constraints are provided. These conditions are a variant of the nonsmooth maximum principle in which an Euler-Lagrange inclusion is involved. The main novelty of the result is precisely the ability to address state constraints, generalizing a known Euler-Lagrange inclusion for optimal control problems. The conditions developed are, in some cases, stronger than the standard maximum principle, since they can reduce the set of candidates to minimizers. © 2001 EUCA.

Abstract

Abstract
Necessary Conditions of Optimality (NCO) play an important role in the characterization of and search for solutions of Optimization Problems. They enable us to identify a small set of candidates to local minimizers among the overall set of admissible solutions. However, in constrained optimization problems it may happen that necessary conditions of optimality are merely state a relation between the constraints and do not use the objective function to select candidates to minimizers. To avoid this phenomenon, it is necessary to strengthened the NCO. Here, we overview and describe strengthened forms NCO for calculus of variations with inequality constraints.

Abstract