Andrei A. Agrachev

Lie-Algebraic Stability Criteria for Switched Systems

It was recently shown that a family of exponentially stable linear systems whose matrices generate a solvable Lie algebra possesses a quadratic common Lyapunov function, which implies that the corresponding switched linear system is exponentially stable for arbitrary switching. In this paper we prove that the same properties hold under the weaker condition that the Lie algebra generated by given matrices can be decomposed into a sum of a solvable ideal and a subalgebra with a compact Lie group. The corresponding local stability result for nonlinear switched systems is also established. Moreover, we demonstrate that if a Lie algebra fails to satisfy the above condition, then it can be generated by a family of stable matrices such that the corresponding switched linear system is not stable. Relevant facts from the theory of Lie algebras are collected at the end of the paper for easy reference.

Abnormal sub-Riemannian geodesics: Morse index and rigidity

Abstract Considering a smooth manifold Μ provided with a sub-Riemannian structure, i.e. with Riemannian metric and nonintegrable vector distribution, we set a problem of finding for two given points q0, q1 ∈ Μ a length minimizer among Lipschitzian paths tangent to the vector distribution (admissible) and connecting these points. Extremals of this variational problem are called sub-Riemannian geodesics and we single out the abnormal sub-Riemannian geodesics, which correspond to the vanishing Lagrange multiplier for the length functional. These abnormal geodesics are not related to the Riemannian structure but only to the vector distribution and, in fact, are singular points in the set of admissible paths connecting q0 and q1. Developing the Legendre-Jacobi-Morse-type theory of 2nd variation for abnormal geodesics we investigate some of their specific properties such as weak minimality and rigidity-isolatedness in the space of admissible paths connecting the two given points.

Strong Optimality for a Bang-Bang Trajectory

In this paper we give sufficient conditions for a bang-bang regular extremal to be a strong local optimum for a control problem in the Mayer form; strong means that we consider the C0 topology in the state space. The controls appear linearly and take values in a polyhedron, and the state space and the end point constraints are finite-dimensional smooth manifolds. In the case of bang-bang extremals, the kernel of the first variation of the problem is trivial, and hence the usual second variation, which is defined on the kernel of the first one, does not give any information. We consider the finite-dimensional subproblem generated by perturbing the switching times, and we prove that the sufficient second order optimality conditions for this finite-dimensional subproblem yield local strong optimality. We give an explicit algorithm to check the positivity of the second variation which is based on the properties of the Hamiltonian fields.

Local Invariants of Smooth Control Systems

Methods are presented for locally studying smooth nonlinear control systems on the manifoldM. The technique of chronological calculus is intensively exploited. The concept of chronological connection is introduced and is used when obtaining the invariant expressions in the form of Lie bracket polynomials for high-order variations of a nonlinear control system. The theorem on adduction of a family of smooth vector fields to the canonical form proved in Section 4 is then applied to the construction of a nilpotent polynomial approximation for a control system. Finally, the relation between the attainable sets of an original system and an approximating one is established; it implies some conclusions on the local controllability of these systems.

Exponential mappings for contact sub-Riemannian structures

On sub-Riemannian manifolds any neighborhood of any point contains geodesics which are not length minimizers; the closures of the cut and the conjugate loci of any pointq containq. We study this phenomenon in the case of a contact distribution, essentially in the lowest possible dimension 3, where we extract differential invariants related to the singularities of the cut and the conjugate loci nearq and give a generic classification of these singularities.

Chronological algebras and nonstationary vector fields

A calculus is developed, reflecting the most general group-theoretic properties of flows, defined by nonstationary differential equations on manifolds.

Geometry of Jacobi Curves. I

Jacobi curves are deep generalizations of the spaces of “Jacobi fields” along Riemannian geodesics. Actually, Jacobi curves are curves in the Lagrange Grassmannians. In our paper we develop differential geometry of these curves which provides basic feedback or gauge invariants for a wide class of smooth control systems and geometric structures. Two principal invariants are the generalized Ricci curvature, which is an invariant of the parametrized curve in the Lagrange Grassmannian endowing the curve with a natural projective structure, and a fundamental form, which is a fourth-order differential on the curve. The so-called rank 1 curves are studied in more detail. Jacobi curves of this class are associated with systems with scalar controls and with rank 2 vector distributions.In the forthcoming second part of the paper we will present the comparison theorems (i.e., the estimates for the conjugate points in terms of our invariants( for rank 1 curves an introduce an important class of “flat curves.”

Feedback-invariant optimal control theory and differential geometry—I. Regular extremals

A feedback-invariant approach to smooth optimal control problems is considered. A Hamiltonian method of investigating regular extremals is developed, analogous to the differential-geometric method of investigating Riemannian geodesics in terms of the Levi-Civita connection and the curvature tensor.

On the Hausdorff volume in sub-Riemannian geometry

For a regular sub-Riemannian manifold we study the Radon-Nikodym derivative of the spherical Hausdorff measure with respect to a smooth volume. We prove that this is the volume of the unit ball in the nilpotent approximation and it is always a continuous function. We then prove that up to dimension 4 it is smooth, while starting from dimension 5, in corank 1 case, it is