Control Theory from the Geometric Viewpoint

Frontmatter

1. Vector Fields and Control Systems on Smooth Manifolds

Abstract

We give just a brief outline of basic notions related to the smooth manifolds. For a consistent presentation, see an introductory chapter to any textbook on analysis on manifolds, e.g. [146].

Andrei A. Agrachev, Yuri L. Sachkov

2. Elements of Chronological Calculus

Abstract

We introduce an operator calculus that will allow us to work with nonlinear systems and flows as with linear ones, at least at the formal level. The idea is to replace a nonlinear object, a smooth manifold M,by a linear, although infinite-dimensional one: the commutative algebra of smooth functions on M (for details, see [19], [22]). For basic definitions and facts of functional analysis used in this chapter, one can consult e.g. [144].

Andrei A. Agrachev, Yuri L. Sachkov

3. Linear Systems

Abstract

In this chapter we consider the simplest class of control systems — linear systems

$$\dot x = Ax + c + \sum\limits_{i = 1}^m {{u_i}} {b_i},x \in {R^n},u = \left( {{u_1}, \cdots ,{u_m}} \right) \in {R^m},$$

(3.1)

where A is a constant real n × n matrix and c, b ₁,..., b _m are constant vectors in ℝ ⁿ .

Andrei A. Agrachev, Yuri L. Sachkov

4. State Linearizability of Nonlinear Systems

Abstract

The aim of this chapter is to characterize nonlinear systems

$$\dot q = {f_0}\left( q \right) + \sum\limits_{i = 1}^m {{u_i}} {f_i}\left( q \right),u = \left( {{u_1}, \cdots ,{u_m}} \right) \in {R^m},q \in M$$

(4.1)

that are equivalent, locally or globally, to controllable linear systems. That is, we seek conditions on vector fields f ₀, f ₁..., f _m that guarantee existence of a diffeomorphism (global Φ: M → ℝ ⁿ or local Φ: O _qo ⊂ M → O ₀ ⊂ ℝ ⁿ ) which transforms nonlinear system (4.1) into a controllable linear one (3.1).

Andrei A. Agrachev, Yuri L. Sachkov

5. The Orbit Theorem and its Applications

Abstract

Let F ⊂ Vec M be any set of smooth vector fields. In order to simplify notation, we assume that all fields from F are complete. Actually, all further definitions and results have clear generalizations to the case of noncomplete fields; we leave them to the reader.

Andrei A. Agrachev, Yuri L. Sachkov

6. Rotations of the Rigid Body

Abstract

In this chapter we consider rotations of a rigid body around a fixed point. That is, we study motions of a body in the three-dimensional space such that:

• distances between all points in the body remain fixed (rigidity), and
• there is a point in the body that stays immovable during motion (fixed point).

We consider both free motions (in the absence of external forces) and controlled motions (when external forces are applied in order to bring the body to a desired state).

Andrei A. Agrachev, Yuri L. Sachkov

7. Control of Configurations

Abstract

In this chapter we apply the Orbit Theorem to systems which can be controlled by the change of their configuration, i.e., of relative position of parts of the systems. A falling cat exhibits a well-known example of such a control. If a cat is left free over ground (e.g. if it falls from a tree or is thrown down by a child), then the cat starts to rotate its tail and bend its body, and finally falls to the ground exactly on its paws, regardless of its initial orientation over the ground. Such a behavior cannot be demonstrated by a mechanical system less skillful in turning and bending its parts (e.g. a dog or just a rigid body), so the crucial point in the falling cat phenomenon seems to be control by the change of configuration. We present a simple model of systems controlled in such a way, and study orbits in several simplest examples.

Andrei A. Agrachev, Yuri L. Sachkov

8. Attainable Sets

Abstract

In this chapter we study general properties of attainable sets. We consider families of vector fields F on a smooth manifold M that satisfy the property

$$Li{e_q}F = {T_q}M\forall q \in M.$$

(8.1)

In this case the system F is called bracket-generating, or full-rank. By the analytic version of the Orbit Theorem (Corollary 5.17), orbits of a bracket-generating system are open subsets of the state space M.

Andrei A. Agrachev, Yuri L. Sachkov

9. Feedback and State Equivalence of Control Systems

Abstract

Consider control systems of the form

$$\dot q = f(q,u),q \in M,u \in U$$

(9.1)

We suppose that not only M, but also U is a smooth manifold. For the right-hand side, we suppose that for all fixed u ∈ U, f(q, u) is a smooth vector field on M, and, moreover, the mapping \((u,q) \mapsto f(q,u)\) is smooth. Admissible controls are measurable locally bounded mappings \(t \mapsto u(t) \in U\) (for simplicity, one can consider piecewise continuous controls). If such a control u(t) is substituted to control system (9.1),one obtains a nonautonomous ODE

$$\dot q = f(q,u(t)),$$

(9.2)

with the right-hand side smooth in q and measurable, locally bounded in t. For such ODEs, there holds a standard theorem on existence and uniqueness of solutions, at least local. Solutions q(•) to ODEs (9.2) are Lipschitzian curves in M (see Subsect. 2.4.1).

Andrei A. Agrachev, Yuri L. Sachkov

10. Optimal Control Problem

Abstract

Consider a control system of the form

$$\dot q = fu(q),q \in M,u \in U \subset {R^m}.$$

(10.1)

Here M is, as usual, a smooth manifold, and U an arbitrary subset of ℝ^m. For the right-hand side of the control system, we suppose that:

$$q \mapsto fu(q)$$

(10.2)

is a smooth vector field on M for any fixed u ∈ U,

$$(q,u) \mapsto {f_u}(q)$$

(10.3)

is a continuous mapping for q ∈ M, u ∈ Ū, and moreover, in any local coordinates on M

$$(q,u) \mapsto \frac{{\partial {f_u}}}{{\partial q}}(q)$$

(10.4)

is a continuous mapping for q ∈ M, u ∈ Ū. Admissible controls are measurable locally bounded mappings \(u:t \mapsto u(t) \in U\)Substitute such a control u = u(t) for control parameter into system (10.1), then we obtain a nonautonomous ODE \(\dot q = fu(q)\). By the classical Carathéodory’s Theorem, for any point q ₀ ∈ M, the Cauchy problem

$$\dot q = {f_u}(q),q(0) = {q_0},$$

(10.5)

has a unique solution, see Subsect. 2.4.1. We will often fix the initial point q ₀ and then denote the corresponding solution to problem (10.5) as q _u (t).

Andrei A. Agrachev, Yuri L. Sachkov

11. Elements of Exterior Calculus and Symplectic Geometry

Abstract

In order to state necessary conditions of optimality for optimal control problems on smooth manifolds — Pontryagin Maximum Principle, see Chap. 12 — we make use of some standard technique of Symplectic Geometry. In this chapter we develop such a technique. Before this we recall some basic facts on calculus of exterior differential forms on manifolds. The exposition in this chapter is rather explanatory than systematic, it is not a substitute to a regular textbook. For a detailed treatment of the subject, see e.g. [146], [135], [137].

Andrei A. Agrachev, Yuri L. Sachkov

12. Pontryagin Maximum Principle

Abstract

In this chapter we prove the fundamental necessary condition of optimality for optimal control problems — Pontryagin Maximum Principle (PMP). In order to obtain a coordinate-free formulation of PMP on manifolds, we apply the technique of Symplectic Geometry developed in the previous chapter. The first classical version of PMP was obtained for optimal control problems in ℝ ⁿ by L. S. Pontryagin and his collaborators [15].

Andrei A. Agrachev, Yuri L. Sachkov

13. Examples of Optimal Control Problems

Abstract

In this chapter we apply Pontryagin Maximum Principle to solve concrete optimal control problems.

Andrei A. Agrachev, Yuri L. Sachkov

14. Hamiltonian Systems with Convex Hamiltonians

Abstract

A well-known theorem states that if a level surface of a Hamiltonian is convex, then it contains a periodic trajectory of the Hamiltonian system [142], [147]. In this chapter we prove a more general statement as an application of optimal control theory for linear systems.

Andrei A. Agrachev, Yuri L. Sachkov

15. Linear Time-Optimal Problem

Abstract

In this chapter we study the following optimal control problem:

$$\begin{array}{*{20}{c}} {\dot{x} = Ax + Bu,x \in {{R}^{n}},u \in U \subset {{R}^{m}},} \hfill \\ {x(0) = {{x}_{0}},x({{t}_{1}}) = {{x}_{1}},{{x}_{0}},{{x}_{1}} \in {{R}^{n}}fixed,} \hfill \\ {{{t}_{1}} \to \min ,} \hfill \\ \end{array}$$

(15.1)

where U is a compact convex polytope in ℝ ^m , and A and B are constant matrices of order n × n and n × m respectively. Such problem is called linear time-optimal problem.

Andrei A. Agrachev, Yuri L. Sachkov

16. Linear-Quadratic Problem

Abstract

In this chapter we study a class of optimal control problems very popular in applications, linear-quadratic problems. That is, we consider linear systems with quadratic cost functional:

$$\begin{array}{*{20}{c}} {\dot{x} = Ax = Bu,x \in {{R}^{n}},u \in {{R}^{m}},} \hfill \\ {x(0) = {\text{ }}{{x}_{0}},x({\text{ }}{{t}_{1}}) = {\text{ }}{{x}_{1}},{\text{ }}{{x}_{0}},{\text{ }}{{x}_{1}},{\text{ }}{{t}_{1}},fixed,} \hfill \\ {J(u) = \frac{1}{2}\int_{0}^{{{{t}_{1}}}} {\left\langle {Ru(t),u(t)} \right\rangle } + \left\langle {Px(t),u(t)} \right\rangle + \left\langle {Qx(t),x(t)} \right\rangle dt \to \min .} \hfill \\ \end{array}$$

(1)

.

Andrei A. Agrachev, Yuri L. Sachkov

17. Sufficient Optimality Conditions, Hamilton-Jacobi Equation, and Dynamic Programming

Abstract

Pontryagin Maximum Principle is a universal and powerful necessary optimality condition, but the theory of sufficient optimality conditions is not so complete. In this section we consider an approach to sufficient optimality conditions that generalizes fields of extremals of the Classical Calculus of Variations.

Andrei A. Agrachev, Yuri L. Sachkov

18. Hamiltonian Systems for Geometric Optimal Control Problems

Abstract

Consider a control system described by a finite set of vector fields on a manifold M:

$$\dot q = {f_{u|}}(q),u \in \{ 1, \ldots ,k\} ,q \in M.$$

(18.1)

We construct a parametrization of the cotangent bundle T*M adapted to this system. First, choose a basis in tangent spaces T _q M of the fields f _u (q) and their iterated Lie brackets:

$${T_q}M = span({f_1}(q), \ldots ,{f_n}(q)),$$

we assume that the system is bracket-generating. Then we have special coordinates in the tangent spaces:

$$\begin{array}{*{20}{c}} {\forall \nu \in {{T}_{q}}M\nu = \sum\limits_{{i = 1}}^{n} {{{\xi }_{i}}} {{f}_{i}}(q),} \hfill \\ {({{\xi }_{1}}, \ldots ,{{\xi }_{n}}) \in {{R}^{n}}.} \hfill \\ \end{array}$$

Andrei A. Agrachev, Yuri L. Sachkov

19. Examples of Optimal Control Problems on Compact Lie Groups

Abstract

Let M be a compact Lie group. The invariant scalar product (•, •) in the Lie algebra M = T _Id M defines a left-invariant Riemannian structure on M:

$${\langle qu,qv\rangle _q}_ = ^{def}\langle u,u\rangle ,u,v \in M,q \in M,qu,qv \in {T_q}M.$$

So in every tangent space T _q M there is a scalar product (•,•)q. For any Lipschitzian curve

$$q:[0,1] \to M$$

its Riemannian length is defined as integral of velocity:

$$\iota = {\text{ }}\int_0^1 {\left| {\dot q\left( t \right)} \right|} dt,\left| {\dot q} \right| = \sqrt {\langle \dot q,\dot q\rangle } .$$

Andrei A. Agrachev, Yuri L. Sachkov

20. Second Order Optimality Conditions

Abstract

In this chapter we obtain second order necessary optimality conditions for control problems. As we know, geometrically the study of optimality reduces to the study of boundary of attainable sets (see Sect. 10.2). Consider a control system

$$\dot q = {f_u}(q),q \in M,u \in U = \operatorname{int} U \subset {R^m},$$

(20.1)

where the state space M is, as usual, a smooth manifold, and the space of control parameters U is open (essentially, this means that we study optimal controls that do not come to the boundary of U, although a similar theory for bang-bang controls can also be constructed). The attainable set A _q0 (t_i)of system (20.1) is the image of the endpoint mapping

$${F_{{t_1}}}:u(\cdot ) \mapsto {q_o}^0\overrightarrow {\exp } {\text{ }}\int_0^{{t_1}} {{f_{u\left( t \right)}}} dt.$$

Andrei A. Agrachev, Yuri L. Sachkov

Jacobi Equation

Abstract

In Chap. 20 we established that the sign of the quadratic form λ _t Hess _ũ F _t is related to optimality of the extremal control ũ. Under natural assumptions, the second variation is negative on short segments. Now we wish to catch the instant of time where this quadratic form fails to be negative. We derive an ODE (Jacobi equation) that allows to find such instants (conjugate times). Moreover, we give necessary and sufficient optimality conditions in these terms.

Andrei A. Agrachev, Yuri L. Sachkov

Reduction

Abstract

In this chapter we consider a method for reducing a control-affine system to a nonlinear system on a manifold of a less dimension.

Andrei A. Agrachev, Yuri L. Sachkov

Curvature

Abstract

Consider a control system of the form

$$MathType!End!2!1!$$

(23.1)

where

$$MathType!End!2!1!$$

.

Andrei A. Agrachev, Yuri L. Sachkov

Rolling Bodies

Abstract

We apply the Orbit Theorem and Pontryagin Maximum Principle to an intrinsic geometric model of a pair of rolling rigid bodies. We solve the controllability problem: in particular, we show that the system is completely controllable if the bodies are not isometric. We also state an optimal control problem and study its extremals.

Andrei A. Agrachev, Yuri L. Sachkov

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