For the purposes of this book, a “control system” is any system of differential equations in which control functions appear as parameters. Our qualitative theory of control systems begins with the important geometric observation that each control determines a vector field, and therefore a control system can also be viewed as a family of vector fields parametrized by controls. A trajectory of such a system is a continuous curve made up of finitely many segments of integral curves of vector fields in the family.

This geometric view of control systems fits closely the theoretical framework of Sophus Lie for integration of differential equations and points to the non-commutativity of vector fields as a fundamental issue of control theory. The geometric context quickly reveals the Lie bracket as the basic theoretical tool, and the corresponding theory, known as geometric control theory, becomes a subject intimately connected with the structural properties of the enveloping Lie algebras and their integral manifolds. For this reason, our treatment of the subject begins with differentiable manifolds, rather than with ℝn as is customary in the control-theory literature.

As natural a beginning as it may seem, particularly to the reader already familiar with differential geometry, this point of view is a departure from the usual presentation of control theory, which traditionally has been confined either to linear theory and the use of linear algebra or to control systems in ℝn, with an emphasis on optimality. The absence of geometric considerations and explicit mention of the Lie bracket in this literature can be attributed to the historical development of the subject.

The process of transferring one state into another along a trajectory of a given differential system such that the time of transfer is minimal is known as the minimal-time problem, and it is one of the basic concerns of optimal control theory.

Minimal-time problems go back to the beginning of the calculus of variations. John Bernoulli's solution of the brachistochrone problem in 1697 was based on Fermat's principle of least time, which postulates that light traverses any medium in the least possible time. According to Goldstine's account (1980) of the history of the calculus of variations, Fermat announced that principle in 1662 in his collected works by saying that “nature operates by means and ways that are the easiest and fastest,” and he further differentiated that statement from the statement that “nature always acts along shortest paths” by citing an example from Galileo concerning the paths of particles moving under the action of gravity. Since then, time-optimal problems have remained important sources of inspiration during the growth of the calculus of variations.

In spite of the extensive literature on the subject, control theorists in the early 1950s believed that the classic theory did not adequately confront optimal problems that involved inequalities and was not applicable to problems of optimal control. Their early papers on time-optimal control problems paved the way to the maximum principle as a necessary condition for optimality.

This chapter begins with linear time-optimal problems and provides a selfcontained characterization of their time-optimal trajectories.

The differential systems that are encountered in applications are fundamentally analytic and therefore share a common geometric base. At the same time, such systems have extra-mathematical properties that differentiate one system from another and account for the particular features of their solutions. The theory of such systems is a blend of the general, shared by all Lie-determined systems, and the particular, due to other mathematical structures, and this recognition provides much insight and understanding, even for systems motivated by narrow practical considerations.

In this chapter we shall expand the theory of linear systems for systems with single inputs initiated in Chapter 4. In contrast to the traditional presentation of this theory, which relies entirely on the use of linear algebra and functional analysis, we shall develop the basic theory using the geometric tools introduced in Chapter 3. The geometric point of view will reveal that much of the theory of linear systems follows from considerations independent of the linear properties of the system and therefore extends to larger classes of systems.

Our selection of topics in this theory is motivated partly by the striking nature of the mathematical results and partly by their relevance to the second part of this book, dealing with optimality. Our treatment of linear systems begins with their controllability properties. We shall show that the main theorem has natural Lie-theory interpretations in terms of the Lie saturate of the system. Second, we shall show that a linear controllable system can be decoupled by means of linear feedback into a finite number of independent scalar linear differential equations. This result is known as Brunovsky's normal form.

Much like the unidentified hero figure in classic folk tales who, after having won his position by virtue of his achievements, in the end reveals a hidden but distinguished parentage, so optimal control theory, recognized initially as an engineering subject, reveals, as it reaches maturity, a distinct relationship to classic forebears: the calculus of variations, differential geometry, and mechanics. This distinctive character of optimal control theory can be traced to the mathematical problems of the subject in the mid-1950s dealing with inequality constraints. Faced with the practical, time-optimal control problems of that period, mathematicians and engineers looked to the calculus of variations for answers, but soon discovered, through the papers of Bellman et al. (1956), LaSalle (1954), and Bushaw (1958) on the bang-bang controls, that the answers to their problems were outside the scope of the classic theory and would require different mathematical tools. That realization initiated a search for new necessary conditions for optimality suitable for control problems. That search, further intensified by the space program and the race to the moon, eventually led, in 1959, to the “maximum principle,” which answered the practical needs of that period. So strongly was the maximum principle linked to control problems involving bounds on the controls that its significance in a larger context – as an important extension of Weierstrass's excess function in the calculus of variations – went unnoticed for a long time after its discovery.

The treatment here of optimal control problems combines the rich theory of the calculus of variations with control-theory innovations. I believe, as does Young (1969), that a control-theory point of view is more natural for the calculus of variations.

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.