Applied and theoretical aspects of the controllability of nonholonomic systems.

摘要

In this thesis, we are concerned with two issues regarding the controllability of non-holonomic control systems.; The first one is the motion planning problem for the above systems: given two points p and q in the state space, we propose a method called the Continuation Method (CM) for finding "practically and efficiently" a control that steers p to q. The CM is based on the analysis of {dollar}phi,{dollar} an end-point mapping the control space H onto the state space M. The CM consists of two main steps: the first one requires the determination of the singular set S of {dollar}phi{dollar} and its image {dollar}phi(S).{dollar} When the control system under scrutiny is driftless and affine in the control, S is exactly the set of controls giving rise to abnormal extremals. The second step consists of "lifting" {dollar}Csp1{dollar} paths {dollar}pi:lbrack0,1rbrackto M phi(S){dollar} to paths {dollar}Pi:lbrack 0,1 rbrackto H{dollar} such that {dollar}phi(Pi)=pi.{dollar} This is achieved if a special o.d.e. in H called the Path Lifting Equation (PLE) has a global solution. First we study in detail the problem of local existence and uniqueness of the PLE. We show that the regularity of the initial condition for the PLE is preserved as long as the PLE has a solution. We provide an alternate proof to a result of Sussmann stating that, for control-affine systems subject to the Strong Bracket Generating Condition, then if H is the space of square-integrable inputs over (0,1), we have that for every {dollar}Csp1{dollar} path {dollar}pi:lbrack 0,1rbrackto Mphi(S),{dollar} the corresponding PLE has a global solution. We extend this result to other Hilbert spaces and to the case of the Motion Planning Problem with obstacles. Finally we study two particular cases where there exist non-trivial abnormal extremals.; The second problem we address is the Dubins problem (D) i.e. the study of the structure of the time optimal trajectories for a model of a car which is moving forward on a two dimensional Riemannian manifold M and can turn with a prescribed bound {dollar}varepsilon>0{dollar} on the radius of curvature. We devote our attention to the controllability issue. For problem (D), the state space N is the unit tangent bundle of M. Through the detailed analysis of (D) on the Poincare half-plane, we show that completeness of the manifold is not enough to insure controllability. A result of Lobry says that if M is compact then (D) is completely controllable. In the non-compact case we introduce an asymptotic flatness condition (AF) expressed in terms of the Gaussian curvature of M. This condition turns out to be sufficient for controllability. Finally we consider the Poincare half-plane and describe completely the structure of the time optimal trajectories.