A basic property of families of vector fields is that their orbits are manifolds. This fact, known as the “orbit theorem,” marks a point of departure for geometric control theory, although from a more general mathematical perspective the theorem can also be seen as a fundamental result serving the needs of geometry, dynamical systems, mechanics, and control theory.

This chapter contains a proof of the orbit theorem, along with a self-contained treatment of the closely related integrability results, including the Frobenius integrability theorem and the Hermann-Nagano theorem concerning the orbits of families of real analytic fields. The latter theorem shows that the local structure of each orbit defined by a family of analytic vector fields is determined by the local properties of the vector fields in the family. This property of orbits, essential for geometric control theory, defines a distinguished class of families of vector fields, called “Lie-determined,” large enough to include families of real analytic vector fields, whose orbits admit easy descriptions in terms of Lie theoretical and algebraic criteria.

The basic theory developed in the first part of this chapter is directed to Lie groups, homogeneous spaces, and families of vector fields subordinated to a group action, partly to illustrate its use in the classic theory of Lie groups, but more importantly to establish the conceptual framework required for subsequent analysis of differential systems on Lie groups. The chapter ends with the fundamental properties of zero-time orbits required for the study of reachable sets in the next chapter.

Problems of optimal control, like the problems of its classic predecessor, mathematical physics, rely on the integration of Hamiltonian differential equations for their resolution. That remarkable discovery goes back to the work of R. W. Hamilton and C. G. Jacobi concerning the problems of classic mechanics in the 1830s. The content of their publications, subsequently known as the Hamilton-Jacobi theory, had profound impact on subsequent developments in mathematical physics and was the principal source of inspiration for the present theory of Hamiltonian systems. The maximum principle and contemporary optimal control theory are also anchored in the Hamilton-Jacobi theory, and the main issue before us is to understand those classic developments in modern geometric terms and make them accessible for problems of optimal control.

We shall begin this chapter with a related topic: The connection between symmetry and optimality. We shall first arrive at the appropriate definition for “symmetry,” which extends the classic theorem of E. Noether concerning the existence of extra integrals of motion. An extension of that theorem implies, in particular, that a right-invariant vector field is a symmetry for any left-invariant control problem, and consequently the Hamiltonian of the right-invariant vector field is an integral of motion for the extremal system induced by the optimal problem.

The existence of extra integrals of motion for a given Hamiltonian system makes a link with another classic topic: the theory of integrable Hamiltonian systems. That theory, which was discussed extensively by Poincaré (1892) in his treatise on celestial mechanics and later by Carathéodory (1935) in his book on the calculus of variations, is most naturally expressed through the geometry of Lagrangian submanifolds of cotangent bundles.

Geometric control theory provides the calculus of variations new perspectives that both unify its classic theory and outline new horizons toward which its theory extends. These perspectives grow from the theoretical foundations anchored in two important theorems not available to the classic theory of the calculus of variations.

The more immediate of these two theorems is the “maximum principle” of L. S. Pontryagin and his co-workers, obtained in the late 1950s. The maximum principle, a far-reaching generalization of Weierstrass's necessary conditions for strong minima, provides geometric conditions for a (strong) minimum of an integral criterion, called the “cost,” over the trajectories of a differential control system. These conditions are based on the topological fact that an optimal solution must terminate on the boundary of the extended reachable set formed by the competing curves and their integral costs.

An important novelty of Pontryagin's approach to problems of optimal control consists of liberating the variations along the optimal curves from the constricting condition that they must terminate at the given boundary data. Instead, he considers variations that are infinitesimally near the terminal point and that generate a convex cone of directions locally tangent to the reachable set at the terminal point defined by the optimal trajectory. As a consequence of optimality, the direction of decreasing cost cannot be contained in the interior of this cone. This observation leads to the “separation theorem,” which can be seen as a generalization of the classic Legendre transform in the calculus of variations, which ultimately produces the appropriate Hamiltonian function.

Continuing with the general theme begun in Chapter 5 of amalgamating the basic theory with additional mathematical structures, we shall now consider differential systems possessing group symmetries. Having in mind particular applications in geometry, mechanics, and control of mechanical systems, this chapter focuses on differential systems on Lie groups having either left or right invariance properties. We shall presently show that the basic geometric control theory described in earlier chapters adapts well to systems on Lie groups, and when enriched with additional geometric structure, it provides a substantial theoretical foundation from which various mathematical topics can be effectively pursued. The reader may find it useful to consider, first, several specific situations that have motivated our interest in much of the material in this chapter.

Motions of a rigid body The motions of a rigid body around a fixed point in a Euclidean space E3 can be viewed as paths in the group of rotations SO3(R). The correspondence between the motions and the paths is achieved through an orthonormal frame attached to the body, called a moving frame, and an orthonormal frame stationary in the ambient space. The stationary frame is called fixed or absolute. At each instant of time, the position of the body is described by a rotation defined by the displacement of the moving frame relative to the fixed frame.

Associated with each path in the rotation group is its angular velocity. We shall be interested in the motions of a rigid body whose angular velocities are constrained to belong to a fixed subset of ℝ3. Such situations typically occur in the presence of non-holonomic constraints.