MATHEMATICS OF AUTOMATIC CONTROL
CHAPTER 1 THE CALCULUS OF VARIATIONS: A HISTORICAL PERSPECTIVE 11 THE BRACHISTOCHRONE PROBLEM 12 THE EULER–LAGRANGE EQUATION 13 SURFACES OF REVOLUTION OF MINIMUM AREA 14 THE LEGENDRE AND JACOBI CONDITIONS 15 THE GEOMETRY OF CONJUGATE POINTS AND ENVELOPES 16 FIELDS OF EXTREMALS AND THE WEIERSTRASS CONDITION 17 OPTIMAL SOLUTIONS FOR THE MINIMUM SURFACES OF REVOLUTION 18 OPTIMALITY OF THE CYCLOIDS FOR THE BRACHISTOCHRONE PROBLEM 19 THE HAMILTON–JACOBI EQUATION 110 FROM THE CALCULUS OF VARIATIONS TO OPTIMAL CONTROL 111 NOTES CHAPTER 2 THE PONTRYAGIN MAXIMUM PRINCIPLE: FROM NECESSARY CONDITIONS TO THE CONSTRUCTION OF AN OPTIMAL SOLUTION 21 LINEAR-QUADRATIC OPTIMAL CONTROL 22 OPTIMAL CONTROL PROBLEMS 221 CONTROL SYSTEMS 222 THE PONTRYAGIN MAXIMUM PRINCIPLE 23 THE SIMPLEST PROBLEM IN THE CALCULUS OF VARIATIONS IN RN 24 THE LINEAR-QUADRATIC REGULATOR REVISITED 241 A DERIVATION OF THE OPTIMAL CONTROL FROM THE MAXIMUM PRINCIPLE 242 TWO SCALAR EXAMPLES 25 TIME-OPTIMAL CONTROL FOR LINEAR TIME-INVARIANT SYSTEMS 26 TIME-OPTIMAL CONTROL FOR PLANAR LINEAR TIME-INVARIANT SYSTEMS: EXAMPLES 261 THE DOUBLE INTEGRATOR 262 A HYPERBOLIC SADDLE 263 AN UNSTABLE NODE 264 THE HARMONIC OSCILLATOR 27 EXTENSIONS OF THE MODEL: TWO EXAMPLES 271 AN ECONOMIC TRADING MODEL 272 THE MOON-LANDING PROBLEM 28 SINGULAR CONTROLS AND LIE DERIVATIVES 281 TIME-OPTIMAL CONTROL FOR A SINGLE-INPUT CONTROL-AFFINE NONLINEAR SYSTEM 282 THE SWITCHING FUNCTION AND SINGULAR CONTROLS 283 LIE DERIVATIVES AND THE LIE BRACKET 284 THE ORDER OF A SINGULAR CONTROL AND THE LEGENDRE–CLEBSCH CONDITIONS 285 MULTI-INPUT SYSTEMS AND THE GOH CONDITION 29 TIME-OPTIMAL CONTROL FOR NONLINEAR SYSTEMS IN THE PLANE 291 OPTIMAL BANG-BANG CONTROLS IN THE SIMPLE SUBCASES 292 FAST AND SLOW SINGULAR ARCS 293 OPTIMAL BANG-BANG TRAJECTORIES NEAR A SLOWSINGULAR ARC 210 INPUT SYMMETRIES AND CODIMENSION-2 CASES IN THE PLANE 2101 INPUT SYMMETRIES 2102 SATURATING SINGULAR ARCS 211 CHATTERING ARCS: THE FULLER PROBLEM 2111 THE FULLER PROBLEM AS A TIME-OPTIMAL CONTROL PROBLEM IN R3 2112 ELEMENTARY PROPERTIES OF EXTREMALS 2113 SYMMETRIES OF EXTREMALS 2114 A SYNTHESIS OF INVARIANT EXTREMALS 212 NOTES CHAPTER 3 REACHABLE SETS OF LINEAR TIME-INVARIANT SYSTEMS: FROM CONVEX SETS TO THE BANG-BANG THEOREM 31 ELEMENTARY THEORY OF CONVEX SETS 32 WEAK CONVERGENCE IN L1(I) 33 TOPOLOGICAL PROPERTIES OF REACHABLE SETS 34 THE GENERAL BANG-BANG THEOREM 35 BOUNDARY TRAJECTORIES AND SMALL-TIME LOCAL CONTROLLABILITY 36 THE BANG-BANG THEOREM FOR COMPACT POLYHEDRA 37 NOTES CHAPTER 4 THE HIGH-ORDER MAXIMUM PRINCIPLE: FROM APPROXIMATIONS OF REACHABLE SETS TO HIGH-ORDER NECESSARY CONDITIONS FOR OPTIMALITY 41 BOLTYANSKY APPROXIMATING CONES 42 PROOF OF THE PONTRYAGIN MAXIMUM PRINCIPLE 421 TANGENT VECTORS TO THE REACHABLE SET 422 CONSTRUCTION OF AN APPROXIMATING CONE 423 BOUNDARY TRAJECTORIES 424 NECESSARY CONDITIONS FOR OPTIMALITY 43 CONTROL SYSTEMS ON MANIFOLDS: DEFINITION AND EXAMPLES 431 SHORTEST PATHS ON A SPHERE 432 CONTROL OF A RIGID BODY 433 TRAJECTORY PLANNING FOR REDUNDANT ROBOTIC MANIPULATORS 44 THE HIGH-ORDER MAXIMUM PRINCIPLE 441 EMBEDDINGS AND POINT VARIATIONS 442 VARIATIONAL VECTOR AND COVECTOR FIELDS 443 C1-EXTENDABLE VARIATIONS 444 CONSTRUCTION OF AN APPROXIMATING CONE 45 EXPONENTIAL REPRESENTATIONS OF FLOWS 46 HIGH-ORDER NECESSARY CONDITIONS FOR OPTIMALITY 461 THE LEGENDRE–CLEBSCH CONDITION 462 THE KELLEY CONDITION 463 THE GOH CONDITION FOR MULTI-INPUT SYSTEMS 47 NOTES CHAPTER 5 THE METHOD OF CHARACTERISTICS: A GEOMETRIC APPROACH TO SUFFICIENT CONDITIONS FOR A LOCAL MINIMUM 51 THE VALUE FUNCTION AND THE HAMILTON–JACOBI–BELLMAN EQUATION 52 PARAMETERIZED FAMILIES OF EXTREMALS AND THE SHADOW-PRICE LEMMA 521 PARAMETERIZED FAMILIES OF EXTREMALS 522 THE SHADOW-PRICE LEMMA AND SOLUTIONS TO THE HAMILTON–JACOBI–BELLMAN EQUATION 523 THE FULLER PROBLEM REVISITED 53 NEIGHBORING EXTREMALS AND SUFFICIENT CONDITIONS FOR A LOCAL MINIMUM 531 A CANONICAL PARAMETERIZED FAMILY OF EXTREMALS 532 PERTURBATION FEEDBACK CONTROL AND REGULARITY OF THE FLOW 54 FOLD SINGULARITIES AND CONJUGATE POINTS 541 CLASSICAL ENVELOPES 542 THE HILBERT INVARIANT INTEGRAL AND CONTROL ENVELOPES 543 LYAPUNOV–SCHMIDT REDUCTION AND THE GEOMETRY OF FOLD SINGULARITIES 544 THE GEOMETRY OF THE FLOW AND THE GRAPH OF THE VALUE FUNCTION VE NEAR A FOLD SINGULAR POINT 55 SIMPLE CUSP SINGULARITIES AND CUT-LOCI 56 NOTES CHAPTER 6 SYNTHESIS OF OPTIMAL CONTROLLED TRAJECTORIES: FROM LOCAL TO GLOBAL SOLUTIONS 61 PARAMETERIZED FAMILIES OF BROKEN EXTREMALS 611 CONCATENATIONS OF PARAMETERIZED FAMILIES OF EXTREMALS 612 TRANSVERSAL CROSSINGS 613 TRANSVERSAL FOLDS 614 LOCAL ANALYSIS OF A FLOW OF BROKEN EXTREMALS 62 A MATHEMATICAL MODEL FOR TUMOR ANTIANGIOGENIC TREATMENT 621 PRELIMINARY ANALYSIS OF EXTREMALS 622 SINGULAR CONTROL AND SINGULAR ARCS 623 A FAMILY OF BROKEN EXTREMALS WITH SINGULAR ARCS 624 ANALYSIS OF THE CORRESPONDING FLOW AND VALUE FUNCTION 63 SUFFICIENT CONDITIONS FOR A GLOBAL MINIMUM: SYNTHESES OF OPTIMAL CONTROLLED TRAJECTORIES 64 NOTES CHAPTER 7 CONTROL-AFFINE SYSTEMS IN LOW DIMENSIONS: FROM SMALL-TIME REACHABLE SETS TO TIME-OPTIMAL SYNTHESES 71 BASIC TOPOLOGICAL PROPERTIES OF SMALL-TIME REACHABLE SETS 72 SMALL-TIME REACHABLE SETS IN DIMENSION 2 73 SMALL-TIME REACHABLE SETS IN DIMENSION 3 731 BOUNDARY TRAJECTORIES IN DIMENSION 3: LOBRY'S EXAMPLE 732 SMALL-TIME REACHABLE SETS UNDER CODIMENSION-0 ASSUMPTIONS 74 FROM BOUNDARY TRAJECTORIES IN DIMENSION 4 TO TIME-OPTIMAL CONTROL IN R3 741 BOUNDARY TRAJECTORIES IN DIMENSION 4 UNDER CODIMENSION-0 ASSUMPTIONS 742 CONSTRUCTION OF A LOCAL TIME-OPTIMAL SYNTHESIS TO AN EQUILIBRIUM POINT IN DIMENSION 3 75 THE CODIMENSION-1 CASE IN DIMENSION 4: SATURATING SINGULAR ARCS 76 NOTES APPENDIX A A REVIEW OF SOME BASIC RESULTS FROM ADVANCED CALCULUS A1 TOPOLOGY AND CONVERGENCE IN NORMED VECTOR SPACES A2 UNIFORM CONVERGENCE AND THE BANACH SPACE C(K) A3 DIFFERENTIABLE MAPPINGS AND THE IMPLICIT FUNCTION THEOREM A4 REGULAR AND SINGULAR VALUES: SARD'S THEOREM APPENDIX B ORDINARY DIFFERENTIAL EQUATIONS B1 EXISTENCE AND UNIQUENESS OF SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS B2 DEPENDENCE OF SOLUTIONS ON INITIAL CONDITIONS AND PARAMETERS APPENDIX C AN INTRODUCTION TO DIFFERENTIABLE MANIFOLDS C1 EMBEDDED SUBMANIFOLDS OF RK C2 MANIFOLDS: THE GENERAL CASE C3 TANGENT AND COTANGENT SPACES C4 VECTOR FIELDS AND LIE BRACKETS APPENDIX D SOME FACTS FROM REAL ANALYSIS D1 LEBESGUE MEASURE AND LEBESGUE MEASURABLE FUNCTIONS IN RN D2 THE LEBESGUE INTEGRAL IN RN D3 LP-SPACES D4 SOLUTIONS TO ORDINARY DIFFERENTIAL EQUATIONS WITH LEBESGUE MEASURABLE RIGHT-HAND SIDES