Ph.D Thesis

Ph.D StudentZelenko Igor
SubjectInvariants of Curves in Lagrangian Grassmannian and
Differential Geometry of Smooth Control Systems
DepartmentDepartment of Mathematics
Supervisors PROFESSOR EMERITUS Alexander Yoffe
PROF. Andrei Agrachev


Our work is devoted to the construction and applications of feedback or gauge invariants  of a wide class of smooth control  systems and geometric structures. Using the Pontryagin Maximum Principle, one can construct, for a given extremal (even singular one) of the control system, a special family of Lagrangian subspaces, called Jacobi curve, in the appropriate symplectic space. Actually, Jacobi curve is generalization of the space of  "Jacobi fields'' along Riemannian geodesics. In this way one can reduce the problem of finding feedback invariants of control systems to the much more concrete problem of finding symplectic invariants of certain curves in Lagrangian Grassmannian (the set of all Lagrangian subspaces of some symplectic space).

We develop general theory of curves in the Lagrange Grassmannian. We introduce two principal symplectic invariants: the  generalized Ricci curvature, which is an invariant of the parametrized curve in Lagrange Grassmannian providing the curve with a natural projective structure, and a fundamental form, which is a degree four differential on the curve. In Riemannian geometry  this fundamental form is related to the Weyl conformal tensor. We construct  a complete system of symplectic invariants for the curve with rank 1 velocities. Jacobi curves of this class  are associated to systems with scalar controls and to rank 2 vector distributions. Also, we introduce and investigate the "straightest" curves of the given constant rank in the Lagrange Grassmannian, called flat curves. Basic symplectic invariants of a curve measure its deviation from the flat curve and thus play the role of curvatures. Further, we  give the estimates for the conjugate points of rank 1 curves in the Lagrangian Grassmannian of 4-dimensional symplectic space. The estimates are presented in the form of comparison theorems. Using these theorems one can estimate the  intervals of weak optimality of extremals of control system in terms of its Ricci curvature and fundamental form. Finally, we apply our general theory of  curves in the Lagrange Grassmannian to the equivalence problem of rank 2 vector distributions on n-dimensional manifold and to the problem of geodesic equivalence of control systems.