Control Theory from the Geometric Viewpoint

1 Vector Fields and Control Systems on Smooth Manifolds 1 1.1 Smooth Manifolds 1 1.2 Vector Fields on Smooth Manifolds 4 1.3 Smooth Differential Equations and Flows on Manifolds 8 1.4 Control Systems 12 2 Elements of Chronological Calculus 21 2.1 Points, Diffeomorphisms, and Vector Fields 21 2.2 Seminorms and $C^{\infty }(M)$-Topology 25 2.3 Families of Functionals and Operators 26 2.4 Chronological Exponential 28 2.5 Action of Diffeomorphisms on Vector Fields 37 2.6 Commutation of Flows 40 2.7 Variations Formula 41 2.8 Derivative of Flow with Respect to Parameter 43 3 Linear Systems 47 3.1 Cauchy's Formula for Linear Systems 47 3.2 Controllability of Linear Systems 49 4 State Linearizability of Nonlinear Systems 53 4.1 Local Linearizability 53 4.2 Global Linearizability 57 5 The Orbit Theorem and its Applications 63 5.1 Formulation of the Orbit Theorem 63 5.2 Immersed Submanifolds 64 5.3 Corollaries of the Orbit Theorem 66 5.4 Proof of the Orbit Theorem 67 5.5 Analytic Case 72 5.6 Frobenius Theorem 74 5.7 State Equivalence of Control Systems 76 6 Rotations of the Rigid Body 81 6.1 State Space 81 6.2 Euler Equations 84 6.3 Phase Portrait 88 6.4 Controlled Rigid Body: Orbits 90 7 Control of Configurations 97 7.1 Model 97 7.2 Two Free Points 100 7.3 Three Free Points 101 7.4 Broken Line 104 8 Attainable Sets 109 8.1 Attainable Sets of Full-Rank Systems 109 8.2 Compatible Vector Fields and Relaxations 113 8.3 Poisson Stability 116 8.4 Controlled Rigid Body: Attainable Sets 118 9 Feedback and State Equivalence of Control Systems 121 9.1 Feedback Equivalence 121 9.2 Linear Systems 123 9.3 State-Feedback Linearizability 131 10 Optimal Control Problem 137 10.1 Problem Statement 137 10.2 Reduction to Study of Attainable Sets 138 10.3 Compactness of Attainable Sets 140 10.4 Time-Optimal Problem 143 10.5 Relaxations 143 11 Elements of Exterior Calculus and Symplectic Geometry 145 11.1 Differential 1-Forms 145 11.2 Differential $k$-Forms 147 11.3 Exterior Differential 151 11.4

Andrei A. Agrachev
- Born in Moscow, Russia.
- Graduated: Moscow State Univ., Applied Math. Dept., 1974.
- Ph.D.: Moscow State Univ., 1977.
- Doctor of Sciences (habilitation): Steklov Inst. for Mathematics, Moscow, 1989.
- Invited speaker at the International Congress of Mathematicians ICM-94 in Zurich.
- Over 90 research papers on Control Theory, Optimization, Geometry (featured review of Amer. Math. Soc., 2002).
- Professional Activity: Inst. for Scientific Information, Russian Academy of Sciences, Moscow, 1977-1992; Moscow State Univ., 1989-1997; Steklov Inst. for Mathematics, Moscow, 1992-present; International School for Advanced Studies (SISSA-ISAS), Trieste, 2000-present.
- Current positions: Professor of SISSA-ISAS, Trieste, Italy
- and Leading Researcher of the Steklov Ins. for Math., Moscow, Russia

Yuri L. Sachkov
- Born in Dniepropetrovsk, Ukraine.
- Graduated: Moscow State Univ., Math. Dept., 1986.
- Ph.D.: Moscow State Univ., 1992.
- Over 20 research papers on Control Theory.
- Professional Activity: Program Systems Institute, Russian Academy of Sciences, Pereslavl-Zalessky, 1989-present;
- University of Pereslavl, 1993-present.
- Steklov Inst. for Mathematics, Moscow, 1998-1999;
- International School for Advanced Studies (SISSA-ISAS), Trieste, 1999-2001.
- Current positions: Senior researcher of Program Systems Institute, Pereslavl-Zalessky, Russia;
- Associate professor of University of Pereslavl, Russia.

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