List of Figures ................................................ xi
Notation ..................................................... xiii
Preface ...................................................... xvii
Introduction .................................................... 1
I Second-Order Optimality Conditions for Broken Extremals in
the Calculus of Variations ...................................... 7
1 Abstract Scheme for Obtaining Higher-Order Conditions in
Smooth Extremal Problems with Constraints .................... 9
1.1 Main Concepts and Main Theorem .......................... 9
1.2 Proof of the Main Theorem .............................. 15
1.3 Simple Applications of the Abstract Scheme ............. 21
2 Quadratic Conditions in the General Problem of the
Calculus of Variations ...................................... 27
2.1 Statements of Quadratic Conditions for a Pontryagin
Minimum ................................................ 27
2.2 Basic Constant and the Problem of Its Decoding ......... 34
2.3 Local Sequences, Higher Order γ, Representation of
the Lagrange Function on Local Sequences with
Accuracy up to o(γ) .................................... 39
2.4 Estimation of the Basic Constant from Above ............ 54
2.5 Estimation of the Basic Constant from Below ............ 75
2.6 Completing the Proof of Theorem 2.4 ................... 102
2.7 Sufficient Conditions for Bounded Strong and Strong
Minima in the Problem on a Fixed Time Interval ........ 115
3 Quadratic Conditions for Optimal Control Problems with
Mixed Control-State Constraints ............................ 127
3.1 Quadratic Necessary Conditions in the Problem with
Mixed Control-State Equality Constraints on a Fixed
Time Interval ......................................... 127
3.2 Quadratic Sufficient Conditions in the Problem with
Mixed Control-State Equality Constraints on a Fixed
Time Interval ......................................... 138
3.3 Quadratic Conditions in the Problem with Mixed
Control-State Equality Constraints on a Variable
Time Interval ......................................... 150
3.4 Quadratic Conditions for Optimal Control Problems
with Mixed Control-State Equality and Inequality
Constraints ........................................... 164
4 Jacobi-Type Conditions and Riccati Equation for Broken
Extremals .................................................. 183
4.1 Jacobi-Type Conditions and Riccati Equation for
Broken Extremals in the Simplest Problem of the
Calculus of Variations ................................ 183
4.2 Riccati Equation for Broken Extremal in the General
Problem of the Calculus of Variations ................. 214
II Second-Order Optimality Conditions in Optimal Bang-Bang
Control Problems .............................................. 221
5 Second-Order Optimality Conditions in Optimal Control
Problems Linear in a Part of Controls ...................... 223
5.1 Quadratic Optimality Conditions in the Problem on
a Fixed Time Interval ................................. 223
5.2 Quadratic Optimality Conditions in the Problem on
a Variable Time Interval .............................. 237
5.3 Riccati Approach ...................................... 245
5.4 Numerical Example: Optimal Control of Production and
Maintenance ........................................... 248
6 Second-Order Optimality Conditions for Bang-Bang Control ... 255
6.1 Bang-Bang Control Problems on Nonfixed Time
Intervals ............................................. 255
6.2 Quadratic Necessary and Sufficient Optimality
Conditions ............................................ 259
6.3 Sufficient Conditions for Positive Definiteness of
the Quadratic Form Ω on the Critical Cone  .......... 266
6.4 Example: Minimal Fuel Consumption of a Car ............ 272
6.5 Quadratic Optimality Conditions in Time-Optimal
Bang-Bang Control Problems ............................ 274
6.6 Sufficient Conditions for Positive Definiteness of
the Quadratic Form Ω on the Critical Subspace for
Time-Optimal Control Problems ......................... 281
6.7 Numerical Examples of Time-Optimal Control Problems ... 286
6.8 Time-Optimal Control Problems for Linear Systems
with Constant Entries ................................. 293
7 Bang-Bang Control Problem and Its Induced Optimization
Problem .................................................... 299
7.1 Main Results .......................................... 299
7.2 First-Order Derivatives of x(tƒ,t0,x0,θ) with
Respect to t0,tƒ,x0, and θ. Lagrange Multipliers
and Critical Cones .................................... 305
7.3 Second-Order Derivatives of x(tƒ,t0,x0,θ) with
Respect t0,tƒ,x0,and θ ................................. 310
7.4 Explicit Representation of the Quadratic Form for
the Induced Optimization Problem ...................... 319
7.5 Equivalence of the Quadratic Forms in the Basic and
Induced Optimization Problem .......................... 333
8 Numerical Methods for Solving the Induced Optimization
Problem and Applications ................................... 339
8.1 The Arc-Parametrization Method ........................ 339
8.2 Time-Optimal Control of the Rayleigh Equation
Revisited ............................................. 344
8.3 Time-Optimal Control of a Two-Link Robot .............. 346
8.4 Time-Optimal Control of a Single Mode Semiconductor
Laser ................................................. 353
8.5 Optimal Control of a Batch-Reactor .................... 357
8.6 Optimal Production and Maintenance with
L1-Functional ......................................... 361
8.7 Van der Pol Oscillator with Bang-Singular Control ..... 365
Bibliography .................................................. 367
Index ......................................................... 377
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