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

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.

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 the rolling problem (R) without spinning nor slipping of a smooth connected oriented complete Riemannian manifold (M, g) onto a space form ( M, ĝ) of the same dimension n 2. This amounts to study an n-dimensional distribution D R , that we call the rolling distribution, and which is defined in terms of the Levi-Civita connections ∇ g and ∇ ĝ . We then address the issue of the complete controllability of the control system associated to D R . The key remark is that the state space Q carries the structure of a principal bundle compatible with D R . It implies that the orbits obtained by rolling along loops of (M, g) become Lie subgroups of the structure group of π Q,M . Moreover, these orbits can be realized as holonomy groups of either certain vector bundle connections ∇ Rol , called the rolling connections, when the curvature of the space form is non-zero, or of an affine connection (in the sense of Kobayashi and Nomizu, 1996 [14]) in the zero curvature case. As a consequence, we prove that the rolling (R) onto an Euclidean space is completely controllable if and only if the holonomy group of (M, g) is equal to SO(n). Moreover, when ( M, ĝ) has positive (constant) curvature we prove that, if the action of the holonomy group of ∇ Rol is not transitive, then (M, g) admits ( M, ĝ) as its universal covering. In addition, we show that, for n even and n 16, the rolling problem (R) of (M, g) against the space form ( M, ĝ) of positive curvature c > 0, is completely controllable if and only if (M, g) is not of constant curvature c.

In this paper, we consider the control system Σ defined by the rolling of a strictly convex surface S of IR 3 on a plane without slipping or spinning. The purpose of this paper is to present the numerical implementation of a constructive planning algorithm for Σ, which is based on a continuation method. The performances of that algorithm, both in robustness and convergence speed, are illustrated through several examples.

In this paper, a variable gain super-twisting algorithm based on a barrier function is proposed for a class of first order disturbed systems with uncertain control coefficient and whose disturbances derivatives are bounded but they are unknown. The specific feature of this algorithm is that it can ensure the convergence of the output variable and maintain it in a predefined neighborhood of zero independent of the upper bound of the disturbances derivatives. Moreover, thanks to the structure of the barrier function, it forces the gain to decrease together with the output variable which yields the non-overestimation of the control gain.