Approximation and Optimization of Discrete and Differential Inclusions

9780123884282......Page 1
sdarticle......Page 2
Front matter
......Page 3
Copyright......Page 4
Dedication......Page 5
Preface......Page 6
Acknowledgments......Page 10
About the Author......Page 11
1.1 Introduction......Page 12
1.2 Some Basic Properties of Convex Sets......Page 14
1.3 Convex Cones and Dual Cones......Page 25
1.4 The Main Properties of Convex Functions......Page 36
1.5 Conjugate of Convex Function......Page 47
1.6 Directional Derivatives and Subdifferentials......Page 52
2.1 Introduction......Page 63
2.2 Locally Adjoint Mappings to Convex Multivalued Mappings......Page 65
2.3 The Calculus of Locally Adjoint Mappings......Page 74
2.4 Locally Adjoint Mappings in Concrete Cases......Page 82
2.5 Duality Theorems for Convex Multivalued Mappings......Page 96
3.1 Introduction......Page 103
3.2 Necessary Conditions for an Extremum in Convex Programming Problems......Page 107
3.3 Lagrangian and Duality in Convex Programming Problems......Page 117
3.4 Cone of Tangent Directions and Locally Tents......Page 127
3.5 CUA of Functions......Page 132
3.6 LAM in the Nonconvex Case......Page 137
3.7 Necessary Conditions for an Extremum in Nonconvex Problems......Page 143
4.1 Introduction......Page 152
4.2 Optimization of Ordinary Discrete Inclusions......Page 158
4.3 Polyhedral Optimization of Discrete and Differential Inclusions......Page 165
4.4 Polyhedral Adjoint Differential Inclusions and the Finiteness of Switching Numbers......Page 179
4.5 Bolza Problems for Differential Inclusions with State Constraints......Page 190
4.6 Optimal Control of Hereditary Functional-Differential Inclusions with Varying Time Interval and State Constraints......Page 197
Optimization of Neutral Discrete Inclusions (PD)......Page 199
Sufficient Condition for the Optimality of Neutral Differential Inclusions......Page 204
Sufficient Condition for Optimality in Problem (PV)......Page 211
4.7 Optimal Control of HODI of Bolza Type with Varying Time Interval......Page 215
5.1 Introduction......Page 226
5.2 Duality in Mathematical Programs with Equilibrium Constraints......Page 228
5.3 Duality in Problems Governed by Polyhedral Maps......Page 235
5.4 Duality in Problems Described by Convex Discrete Inclusions......Page 244
5.5 The Main Duality Results in Problems with Convex Differential Inclusions......Page 254
6.1 Introduction......Page 262
6.2 The Optimality Principle of Boundary-Value Problems for Discrete-Approximation and First-Order Partial Differential Inc.........Page 266
1 Necessary and Sufficient Conditions for an Extremum for Discrete Inclusions......Page 268
2 Approximation of the Continuous Problem and a Necessary Condition for the Discrete-Approximation Problem......Page 271
3 Sufficient Conditions of Optimality for Differential Inclusions......Page 275
4 Duality in the Problems of Optimal Control Described by First-Order Partial Differential Inclusions......Page 279
6.3 Optimal Control of the Cauchy Problem for First-Order Discrete and Partial Differential Inclusions......Page 283
1 Optimization Problem for Discrete Inclusions with Distributed Parameters......Page 285
2 Necessary and Sufficient Conditions for the Discrete-Approximation Problem Associated with the Continuous Problem......Page 287
3 Sufficient Conditions in the Cauchy Problem for First-Order Partial Differential Inclusions......Page 291
6.4 Optimal Control of Darboux-Type Discrete-Approximation and Differential Inclusions with Set-Valued Boundary Conditions .........Page 298
1 Necessary and Sufficient Conditions for the Discrete Inclusion Problem (PD)......Page 300
2 Approximation of a Continuous Problem (PC) and Optimization of a Discrete-Approximation Problem......Page 304
3 Optimization of Problem (PC)......Page 307
4 On Duality in the Convex Problem (PC)......Page 318
6.5 Optimal Control of the Elliptic-Type Discrete and Differential Inclusions with Dirichlet and Neumann Boundary Condition.........Page 321
1 Necessary and Sufficient Conditions for the Dirichlet Problem of Elliptic Discrete Inclusions......Page 324
2 Approximation of the Continuous Problem and Necessary Condition for the Discrete-Approximation Problem......Page 328
3 Sufficient Conditions for Optimality for Differential Inclusions of Elliptic Type......Page 332
4 Multidimensional Optimal Control Problems for Elliptic Differential Inclusions......Page 335
5 Sufficient Conditions for Generalized Dirichlet and Neumann Problems in Dynamic Optimization......Page 337
6 Optimization of the Multidimensional Problems (PMD) and (PMN)......Page 341
7 On Duality in Elliptic Differential Inclusions......Page 345
6.6 Optimization of Discrete-Approximation and Differential Inclusions of Parabolic Type and Duality......Page 347
1 Sufficient Conditions for Optimality for Differential Inclusions of Parabolic Type......Page 348
2 Duality in Differential Inclusions of Parabolic Type......Page 353
6.7 Optimization of the First Boundary Value Problem for Hyperbolic-Type Discrete-Approximation and Differential Inclusions......Page 356
1 Necessary and Sufficient Conditions for First Boundary Value Problems of Discrete Hyperbolic Inclusions......Page 359
2 Approximation of the Continuous Problem and Necessary Conditions for the Discrete-Approximation Problem......Page 362
3 Sufficient Conditions for Optimality for Differential Inclusions of Hyperbolic Type......Page 365
4 Multidimensional Optimal Control Problems for Hyperbolic Differential Inclusions......Page 368
5 Duality in Problems for Hyperbolic Differential Inclusions......Page 371
References......Page 373
Glossary of Notations......Page 385