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this paper is as follows: in Section 2, we collect some mathematical preliminaries from the literature on controllability of nonlinear systems and on classification of free Lie algebras. These are drawn from classical references in control theory [4, 17, 18, 36, 40] and Lie algebras [15, 43]. In Section 3, using some outstanding results of Brockett on optimal steering of certain classes of systems as motivation [5], we discuss the use of sinusoidal inputs for steering systems of first order, i.e., systems where controllability is achieved after just one level of Lie brackets of the input vector fields. Section 4 attempts to expand the domain of applicability of these results to more complex systems, where several orders of Lie brackets are needed to obtain the full Lie algebra associated with the input distribution. The 4 MURRAY AND SASTRY

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Abstract. We extend Hamilton–Jacobi theory to Lagrange–Dirac (or implicit Lagrangian) systems, a generalized formulation of Lagrangian mechanics that can incorporate degenerate Lagrangians as well as holonomic and nonholonomic constraints. We refer to the generalized Hamilton–Jacobi equation as the Dirac–Hamilton–Jacobi equation. For non-degenerate Lagrangian systems with nonholonomic constraints, the theory specializes to the recently developed nonholonomic Hamilton–Jacobi theory. We are particularly interested in applications to a certain class of degenerate nonholonomic Lagrangian systems with symmetries, which we refer to as weakly degenerate Chaplygin systems, that arise as simplified models of nonholonomic mechanical systems; these systems are shown to reduce to non-degenerate almost Hamiltonian systems, i.e., generalized Hamiltonian systems defined with non-closed two-forms. Accordingly, the Dirac–Hamilton– Jacobi equation reduces to a variant of the nonholonomic Hamilton–Jacobi equation associated with the reduced system. We illustrate through a few examples how the Dirac–Hamilton–Jacobi equation can be used to exactly integrate the equations of motion.

The motion control problem is considered for the nonholonomic systems with unknown dynamic parameters. The proposed control algorithm is based on a method of recursive aim inequalities which allows us to solve the problem of the adaptive stabilization in the presence of uniformly bounded disturbances affecting the system. Unlike other works, the presented algorithm of the parameters estimation is given in a form of a differential equation instead of a discretetime algorithm. 1 Introduction This paper addresses adaptive control of mechanical systems with linear homogeneous constraints. Assume that the system under consideration is defined on the connected Riemannian n-dimensional real smooth configuration manifold M with local coordinates q = (q 1 ; . . . ; q n ). Linear homogeneous constraints are defined by a k-dimensional smooth regular codistribution \Delta ? = span f! 1 ; . . . ; ! k g ae T M and given in the local coordinates as ! j (q) q = 0; j = 1; . . . ; k; (1:1) where ...