5 Constrained Optimization and Equilibria . . . . . . . . . . . . . . . . . . 3
5.1 Necessary Conditions in Mathematical Programming . . . . . . . . . 3
5.1.1 Minimization Problems with Geometric Constraints . . . 4
5.1.2 Necessary Conditions under Operator Constraints . . . . . 9
5.1.3 Necessary Conditions under Functional Constraints . . . . 22
5.1.4 Suboptimality Conditions for Constrained Problems . . . 41
5.2 Mathematical Programs with Equilibrium Constraints . . . . . . . 46
5.2.1 Necessary Conditions for Abstract MPECs . . . . . . . . . . . 47
5.2.2 Variational Systems as Equilibrium Constraints . . . . . . . 51
5.2.3 Refined Lower Subdifferential Conditions
for MPECs via Exact Penalization . . . . . . . . . . . . . . . . . . . 61
5.3 Multiobjective Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.3.1 Optimal Solutions to Multiobjective Problems . . . . . . . . 70
5.3.2 Generalized Order Optimality . . . . . . . . . . . . . . . . . . . . . . . 73
5.3.3 Extremal Principle for Set-Valued Mappings . . . . . . . . . . 83
5.3.4 Optimality Conditions with Respect
to Closed Preferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.3.5 Multiobjective Optimization
with Equilibrium Constraints . . . . . . . . . . . . . . . . . . . . . . . 99
5.4 Subextremality and Suboptimality at Linear Rate . . . . . . . . . . . 109
5.4.1 Linear Subextremality of Set Systems . . . . . . . . . . . . . . . . 110
5.4.2 Linear Suboptimality in Multiobjective Optimization . . 115
5.4.3 Linear Suboptimality for Minimization Problems . . . . . . 125
5.5 Commentary to Chap. 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
XX Contents
6 Optimal Control of Evolution Systems in Banach Spaces . . 159
6.1 Optimal Control of Discrete-Time and Continuoustime
Evolution Inclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
6.1.1 Differential Inclusions and Their Discrete
Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
6.1.2 Bolza Problem for Differential Inclusions
and Relaxation Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
6.1.3 Well-Posed Discrete Approximations
of the Bolza Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
6.1.4 Necessary Optimality Conditions for Discrete-
Time Inclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
6.1.5 Euler-Lagrange Conditions for Relaxed Minimizers . . . . 198
6.2 Necessary Optimality Conditions for Differential Inclusions
without Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210
6.2.1 Euler-Lagrange and Maximum Conditions
for Intermediate Local Minimizers . . . . . . . . . . . . . . . . . . . 211
6.2.2 Discussion and Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
6.3 Maximum Principle for Continuous-Time Systems
with Smooth Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
6.3.1 Formulation and Discussion of Main Results . . . . . . . . . . 228
6.3.2 Maximum Principle for Free-Endpoint Problems . . . . . . . 234
6.3.3 Transversality Conditions for Problems
with Inequality Constraints . . . . . . . . . . . . . . . . . . . . . . . . . 239
6.3.4 Transversality Conditions for Problems
with Equality Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . 244
6.4 Approximate Maximum Principle in Optimal Control . . . . . . . . 248
6.4.1 Exact and Approximate Maximum Principles
for Discrete-Time Control Systems . . . . . . . . . . . . . . . . . . 248
6.4.2 Uniformly Upper Subdifferentiable Functions . . . . . . . . . 254
6.4.3 Approximate Maximum Principle
for Free-Endpoint Control Systems . . . . . . . . . . . . . . . . . . 258
6.4.4 Approximate Maximum Principle under Endpoint
Constraints: Positive and Negative Statements . . . . . . . . 268
6.4.5 Approximate Maximum Principle
under Endpoint Constraints: Proofs and
Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276
6.4.6 Control Systems with Delays and of Neutral Type . . . . . 290
6.5 Commentary to Chap. 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
7 Optimal Control of Distributed Systems . . . . . . . . . . . . . . . . . . . 335
7.1 Optimization of Differential-Algebraic Inclusions with Delays . . 336
7.1.1 Discrete Approximations of Differential-Algebraic
Inclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338
7.1.2 Strong Convergence of Discrete Approximations . . . . . . . 346
Contents XXI
7.1.3 Necessary Optimality Conditions
for Difference-Algebraic Systems . . . . . . . . . . . . . . . . . . . . 352
7.1.4 Euler-Lagrange and Hamiltonian Conditions
for Differential-Algebraic Systems . . . . . . . . . . . . . . . . . . . 357
7.2 Neumann Boundary Control
of Semilinear Constrained Hyperbolic Equations . . . . . . . . . . . . . 364
7.2.1 Problem Formulation and Necessary Optimality
Conditions for Neumann Boundary Controls . . . . . . . . . . 365
7.2.2 Analysis of State and Adjoint Systems
in the Neumann Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . 369
7.2.3 Needle-Type Variations and Increment Formula . . . . . . . 376
7.2.4 Proof of Necessary Optimality Conditions . . . . . . . . . . . . 380
7.3 Dirichlet Boundary Control
of Linear Constrained Hyperbolic Equations . . . . . . . . . . . . . . . . 386
7.3.1 Problem Formulation and Main Results
for Dirichlet Controls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387
7.3.2 Existence of Dirichlet Optimal Controls . . . . . . . . . . . . . . 390
7.3.3 Adjoint System in the Dirichlet Problem . . . . . . . . . . . . . 391
7.3.4 Proof of Optimality Conditions . . . . . . . . . . . . . . . . . . . . . 395
7.4 Minimax Control of Parabolic Systems
with Pointwise State Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . 398
7.4.1 Problem Formulation and Splitting . . . . . . . . . . . . . . . . . . 400
7.4.2 Properties of Mild Solutions
and Minimax Existence Theorem . . . . . . . . . . . . . . . . . . . . 404
7.4.3 Suboptimality Conditions for Worst Perturbations . . . . . 410
7.4.4 Suboptimal Controls under Worst Perturbations . . . . . . . 422
7.4.5 Necessary Optimality Conditions
under State Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427
7.5 Commentary to Chap. 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439
8 Applications to Economics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461
8.1 Models of Welfare Economics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461
8.1.1 Basic Concepts and Model Description . . . . . . . . . . . . . . . 462
8.1.2 Net Demand Qualification Conditions for Pareto
and Weak Pareto Optimal Allocations . . . . . . . . . . . . . . . 465
8.2 Second Welfare Theorem for Nonconvex Economies . . . . . . . . . . 468
8.2.1 Approximate Versions of Second Welfare Theorem . . . . . 469
8.2.2 Exact Versions of Second Welfare Theorem . . . . . . . . . . . 474
8.3 Nonconvex Economies with Ordered Commodity Spaces . . . . . . 477
8.3.1 Positive Marginal Prices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477
8.3.2 Enhanced Results for Strong Pareto Optimality . . . . . . . 479
8.4 Abstract Versions and Further Extensions . . . . . . . . . . . . . . . . . . 484
8.4.1 Abstract Versions of Second Welfare Theorem . . . . . . . . . 484
8.4.2 Public Goods and Restriction on Exchange . . . . . . . . . . . 490
8.5 Commentary to Chap. 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492
XXII Contents
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507
List of Statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573
Glossary of Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595
Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 599
