## Control Theory from the Geometric ViewpointThis book presents some facts and methods of the Mathematical Control Theory treated from the geometric point of view. The book is mainly based on graduate courses given by the first coauthor in the years 2000-2001 at the International School for Advanced Studies, Trieste, Italy. Mathematical prerequisites are reduced to standard courses of Analysis and Linear Algebra plus some basic Real and Functional Analysis. No preliminary knowledge of Control Theory or Differential Geometry is required. What this book is about? The classical deterministic physical world is described by smooth dynamical systems: the future in such a system is com pletely determined by the initial conditions. Moreover, the near future changes smoothly with the initial data. If we leave room for "free will" in this fatalistic world, then we come to control systems. We do so by allowing certain param eters of the dynamical system to change freely at every instant of time. That is what we routinely do in real life with our body, car, cooker, as well as with aircraft, technological processes etc. We try to control all these dynamical systems! Smooth dynamical systems are governed by differential equations. In this book we deal only with finite dimensional systems: they are governed by ordi nary differential equations on finite dimensional smooth manifolds. A control system for us is thus a family of ordinary differential equations. The family is parametrized by control parameters. |

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### Contents

I | 1 |

II | 4 |

III | 8 |

IV | 12 |

V | 21 |

VII | 25 |

VIII | 26 |

IX | 28 |

LIX | 194 |

LX | 197 |

LXI | 199 |

LXII | 200 |

LXIII | 207 |

LXIV | 211 |

LXV | 212 |

LXVI | 213 |

X | 37 |

XI | 40 |

XII | 41 |

XIII | 43 |

XIV | 47 |

XVI | 49 |

XVII | 53 |

XIX | 57 |

XX | 63 |

XXI | 64 |

XXII | 66 |

XXIII | 67 |

XXIV | 72 |

XXV | 74 |

XXVI | 76 |

XXVII | 81 |

XXVIII | 84 |

XXIX | 88 |

XXX | 90 |

XXXI | 97 |

XXXIII | 100 |

XXXIV | 101 |

XXXV | 104 |

XXXVI | 109 |

XXXVII | 113 |

XXXVIII | 116 |

XXXIX | 118 |

XL | 121 |

XLI | 123 |

XLII | 131 |

XLIII | 137 |

XLIV | 138 |

XLV | 140 |

XLVI | 143 |

XLVIII | 145 |

XLIX | 147 |

L | 151 |

LI | 153 |

LII | 157 |

LIII | 167 |

LIV | 172 |

LV | 177 |

LVI | 179 |

LVII | 182 |

LVIII | 191 |

LXVII | 215 |

LXVIII | 218 |

LXIX | 223 |

LXX | 224 |

LXXI | 227 |

LXXII | 229 |

LXXIII | 235 |

LXXIV | 242 |

LXXV | 244 |

LXXVI | 247 |

LXXVII | 255 |

LXXVIII | 260 |

LXXIX | 265 |

LXXX | 267 |

LXXXI | 271 |

LXXXII | 284 |

LXXXIII | 293 |

LXXXIV | 297 |

LXXXV | 304 |

LXXXVI | 309 |

LXXXVII | 318 |

LXXXVIII | 321 |

LXXXIX | 333 |

XC | 334 |

XCI | 338 |

XCII | 342 |

XCIII | 343 |

XCIV | 346 |

XCV | 355 |

XCVI | 358 |

XCVII | 359 |

XCVIII | 363 |

XCIX | 373 |

C | 377 |

CI | 379 |

CII | 383 |

CIII | 384 |

CIV | 387 |

CV | 393 |

CVII | 395 |

399 | |

407 | |

409 | |

### Common terms and phrases

2-dimensional abnormal extremals admissible control Agrachev attainable sets boundary conditions bracket-generating Cauchy problem compact compute conjugate points consider const control parameters control system control u(t convex coordinates corank corresponding cotangent bundle covector defined Denote derivative diffeomorphism differential forms equality equivalent exists extremal trajectory field f finite flow follows formula fu(q geometrically optimal Hamiltonian system Hamiltonian vector field inequality initial integral invariant Jacobi equation Lagrangian subspace left-invariant Legendre condition Lemma Lie algebra Lie bracket Lie groups linear system linearly independent Lipschitzian matrix maximality condition neighborhood nonautonomous vector field obtain optimal control optimal control problem optimal trajectories optimality conditions Orbit Theorem point q Pontryagin Maximum Principle Proof Proposition prove q G M quadratic form Riemannian right-hand side rigid body satisfies segment smooth manifold smooth mapping solution sub-Riemannian symplectic tangent space tangent vector time-optimal problem VecM velocity VT dr