This paper deals with (global) finite-gain input/output stabilization of linear systems with saturated controls. For neutrally stable systems, it is shown that the linear feedback law suggested by the passivity approach indeed provides stability, with respect to every L p-norm. Explicit bounds on closed-loop gains are obtained, and they are related to the norms for the respective systems without saturation. These results do not extend to the class of systems for which the state matrix has eigenvalues on the imaginary axis with nonsimple (size > 1) Jordan blocks, contradicting what may be expected from the fact that such systems are globally asymptotically stabilizable in the state-space sense; this is shown in particular for the double integrator.

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Abstract. When applying methods of optimal control to motion planning or stabilization problems, some theoretical or numerical difficulties may arise, due to the presence of specific trajectories, namely, singular minimizing trajectories of the underlying optimal control problem. In this article, we provide characterizations for singular trajectories of control-affine systems. We prove that, under generic assumptions, such trajectories share nice properties, related to computational aspects; more precisely, we show that, for a generic system -with respect to the Whitney topology -, all nontrivial singular trajectories are of minimal order and of corank one. These results, established both for driftless and for control-affine systems, extend results of [13,14]. As a consequence, for generic systems having more than two vector fields, and for a fixed cost, there do not exist minimizing singular trajectories. We also prove that, given a control system satisfying the LARC, singular trajectories are strictly abnormal, generically with respect to the cost. We then show how these results can be used to derive regularity results for the value function and in the theory of Hamilton-Jacobi equations, which in turn have applications for stabilization and motion planning, both from the theoretical and implementation issues.1. Introduction. When addressing standard issues of control theory such as motion planning and stabilization, one may adopt an approach based on optimal control, e.g., Hamilton-Jacobi type methods and shooting algorithms. One is then immediately facing intrinsic difficulties due to the possible presence of singular trajectories. It is therefore important to characterize these trajectories, by studying in particular their existence, optimality status, and the related computational aspects. In this paper, we provide answers to the aforementioned questions for control-affine systems, under generic assumptions, and then investigate consequences in optimal control and its applications.Let M be a smooth (i.e. C ∞ ) manifold of dimension n. Consider the control-affine system

Consider the control system (Σ) given byẋ = x(f + ug), where x ∈ SO(3), |u| ≤ 1 and f, g ∈ so(3) define two perpendicular left-invariant vector fields normalized so that f = cos(α) and g = sin(α), α ∈]0, π/4[. In this paper, we provide an upper bound and a lower bound for N (α), the maximum number of switchings for time-optimal trajectories of (Σ). More precisely, we show that N S (α) ≤ N (α) ≤ N S (α) + 4, where N S (α) is a suitable integer function of α such thatThe result is obtained by studying the time optimal synthesis of a projected control problem on RP 2 , where the projection is defined by an appropriate Hopf fibration. Finally, we study the projected control problem on the unit sphere S 2 . It exhibits interesting features which will be partly rigorously derived and partially described by numerical simulations.

In this paper, we consider linear switched systemsẋ(t) = A u(t) x(t), x ∈ R n , u ∈ U , and the problem of asymptotic stability for arbitrary switching functions, uniform with respect to switching (UAS for short). We first prove that, given a UAS system, it is always possible to build a common polynomial Lyapunov function. Then our main result is that the degree of that common polynomial Lyapunov function is not uniformly bounded over all the UAS systems. This result answers a question raised by Dayawansa and Martin. A generalization to a class of piecewise-polynomial Lyapunov functions is given.