The averaged control system of fast oscillating control systems

For control systems that either have a fast explicit periodic dependence on time and bounded controls or have periodic solutions and small controls, we define an average control system that takes into account all possible variations of the control, and prove that its solutions approximate all solutions of the oscillating system as oscillations go faster. The dimension of its velocity set is characterized geometrically. When it is maximum the average system defines a Finsler metric, not twice differentiable in general. For minimum time control, this average system allows one to give a rigorous proof that averaging the Hamiltonian given by the maximum principle is a valid approximation.Comment: (2012

Exponential Stabilization of Nonholonomic Systems by Means of Oscillating Controls

This paper is devoted to the stabilization problem for nonlinear driftless control systems by means of a time-varying feedback control. It is assumed that the vector fields of the system together with their first order Lie brackets span the whole tangent space at the equilibrium. A family of trigonometric open-loop controls is constructed to approximate the gradient flow associated with a Lyapunov function. These controls are applied for the derivation of a time-varying feedback law under the sampling strategy. By using Lyapunov's direct method, we prove that the controller proposed ensures exponential stability of the equilibrium. As an example, this control design procedure is applied to stabilize the Brockett integrator.Comment: 25 pages, 2 figure

Metric approximation of minimum time control systems

Slow-fast affine control systems with one fast angle are considered. An approximation based on standard averaging of the extremal is defined. When the drift of the original system is small enough, this approximation is metric, and minimum time trajectories of the original system converge towards geodesics of a Finsler metric. The asymmetry of the metric accounts for the presence of the drift on the slow part of the original dynamics. The example of the J2 effect in the two-body case in space mechanics is examined. A critical ratio between the J2 drift and the thrust level of the engine is defined in terms of the averaged metric. The qualitative behaviour of the minimum time for the real system is analyzed thanks to this ratio

Time Versus Energy in the Averaged Optimal Coplanar Kepler Transfer towards Circular Orbits

International audienceThe aim of this note is to compare the averaged optimal coplanar transfer towards circular orbits when the costs are the transfer time transfer and the energy consumption. While the energy case leads to analyze a 2D Riemannian metric using the standard tools of Riemannian geometry (curvature computations, geodesic convexity), the time minimal case is associated to a Finsler metric which is not smooth. Nevertheless a qualitative analysis of the geodesic flow is given in this article to describe the optimal transfers. In particular we prove geodesic convexity of the elliptic domain