【正文】 can only be modelled appropriately as nonholonomic systems. Some examples are the wheeled vehicles, robot manipulators, and many other mechanical systems. A difficulty encountered in controlling this kind of systems is that any linearization around the origin is uncontrollable and therefore any linear control methods are useless to tackle them. But, perhaps the main challenging characteristic of the nonholonomic systems is that it is not possible to stabilize it if just timeinvariant continuous feedbacks are allowed [1]. However, if we allow discontinuous feedbacks, it might not be clear what is the solution of the dynamic differential equation. (See [4, 8] for a further discussion of this issue). A solution concept that has been proved successful in dealing with stabilization by disconti nuous feedbacks for a generalclass of controllablesystems is the concept of “samplingfeedback” solution proposed in [5]. It can be seenthat sampleddata MPC framework described can be bined naturally with a “samplingfeedback” law and thus define a trajectory in a way which is verysimilar to the concept introduced in [5]. Those trajectories are, under mild conditions, welldefined even when the feedback law is discontinuous. There are in the literature a few works allowing discontinuous feedback laws in the context of MPC. (See [8] for a survey of such works.) The essential feature of those frameworks to allow discontinuities is simply the sampleddata feature — appropriate use of a positive intersampling time, bined with an appropriate interpretation of a solution to a discontinuous differential equation. 4 Barbalat’s Lemma and Variants Barbalat’s lemma is a wellknown and powerful tool to deduce asymptotic stability of nonlinear systems, especially timevarying systems, using Lyapunovlike approaches (see . [17] for a discussion and applications). Simple variants of this lemma have been used successfully to prove stability results for Model Predictive Control (MPC) of nonlinear and timevarying systems [7, 15]. In fact, in all the sampled data MPC frameworks cited above, Barbalat’slemma, or a modification of it, is used as an important step to prove stabilityof the MPC schemes. It is shown that if certain design param eters (objectivefunction, terminal set, etc.) are conveniently selected, then the value function is monotone decreasing. Then, applying Barbalat’s lemma, attractiveness of the trajectory of the nominal model can be established (. x(t) → 0 as t → ∞ ). This stability property can be deduced for a very general class of nonlinear systems: including timevarying systems, nonholonomic systems, systems allowing discontinuous feedbacks, etc. A recent work on robust MPC of nonlinear systems [9] used a generalization of Barbalat’s lemma as an important step to prove stability of the algorithm. However, it is our believe that such generalization of the lemma might provide a useful tool to analyse stability in other robust continuoustime MPC approaches, such as the one described here for timevarying systems. A standard result in Calculus states that if a function is lower bounded and decreasing, then it converges to a limit. However, we cannot conclude whether its derivative will decrease or not unless we impose some smoothness property on f˙(t). We have in this way a wellknown form of the Barbalat’s lemma (see . [17]). 5 Nominal Stability A stability analysis can be carried out to show that if the design parameters are conveniently selected (. selected to satisfy a certain sufficient stability condition, see . [7]), then a certain MPC value function V is shown to be monotone decreasing. More precisely, for some δ 0small enough and for any 。see . [17] for a discussion and applications). To show that an MPC strategy is stabilizing (in the nominal case), it is shown that if certain design parameters (objective function, terminal set, etc.) are conveniently selected, then the value function is mono tone decreasing. Then, applying Barbalat’s lemma, attractiveness of the trajectory of the nominal model can be established