Part Ⅰ Finite Dimensional Control Problems 1
1 Calculus of Variations and Control Theory 3
1.1 Calculus of Variations：Surface of Revolution of Minimum Area 3
1.2 Interpretation of the Results 8
1.3 Mechanics and Calculus of Variations 9
1.4 Optimal Control：Fuel Optimal Landing of a Space Vehicle 11
1.5 Optimal Control Problems Described by Ordinary Differential Equations 12
1.6 Calculus of Variations and Optimal Control.Spike Perturbations 13
1.7 Optimal Control：Minimum Drag Nose Shape in Hypersonic Flow 17
1.8 Control of Functional Differential Equations：Optimal Forest Growth 18
1.9 Control of Partial Differential Equations 20
1.10 Finite Dimensional and Infinite Dimensional Control Problems 25
2 Optimal Control Problems Without Target Conditions 26
2.0 Elements of Measure and Integration Theory 26
2.1 Control Systems Described by Ordinary Differential Equations 42
2.2 Existence Theory for Optimal Control Problems 51
2.3 Trajectories and Spike Perturbations 60
2.4 Cost Functionals and Spike Perturbations 66
2.5 Optimal Control Problems without Target Condition：The Hamiltonian Formalism 67
2.6 Invariance of the Hamiltonian 71
2.7 The Linear-Quadratic Problem：Existence and Uniqueness of Optimal Controls 76
2.8 The Unconstrained Linear-Quadratic Problem：Feedback,the Riccati Equation 78
2.9 The Constrained Linear-Quadratic Problem 82
3 Abstract Minimization Problems：The Minimum Principle for the Time Optimal Problem 84
3.1 Abstract Minimization Problems 84
3.2 Ekeland's Variational Principle 87
3.3 The Abstract Time Optimal Problem 92
3.4 The Control Spaces 100
3.5 Continuity of the Solution Map 101
3.6 Continuity of the Solution Operator of the Variational Equation 102
3.7 The Minimum Principle for the Time Optimal Problem 104
3.8 Time Optimal Capture of a Wandering Particle 107
3.9 Time Optimal Stopping of an Oscillator 111
3.10 Higher Dimensional Problems 118
4 The Minimum Principle for General Optimal Control Problems 122
4.1 The Abstract Minimization Problem 122
4.2 The Minimum Principle for Problems with Fixed Terminal Time 125
4.3 Optimal Capture of a Wandering Particle in Fixed Time，Ⅰ 129
4.4 Singular Intervals and Singular Arcs 137
4.5 Optimal Capture of a Wandering Particle in Fixed Time，Ⅱ 138
4.6 The Minimum Principle for Problems with Variable Terminal Time 143
4.7 Fuel Optimal Soft Landing of a Space Vehicle 146
4.8 Fuel Optimal Soft Landing of a Space Vehicle 149
4.9 The Linear-Quadratic Problem and the Minimum Drag Nose Shape Problem 151
4.10 Nonlinear Programming Problems：The Kuhn-Tucker Theorem 159
Part Ⅱ Infinite Dimensional Control Problems 167
5 Differential Equations in Banach Spaces and Semigroup Theory 169
5.0 Banach Spaces and Their Duals.Linear Operators.Integration of Vector Valued Functions 169
5.1 Partial Differential Equations as Ordinary Differential Equations in Banach Spaces 189
5.2 Abstract Cauchy Problems in t≥0 192
5.3 Abstract Cauchy Problems in-∞＜t＜∞ 203
5.4 Evolution Equations 207
5.5 Semilinear Equations in Banach Spaces.Perturbation Theory 210
5.6 Wave Equations 226
5.7 Semilinear Wave Equations：Local Existence 234
5.8 Semilinear Equations in Banach Spaces：Global Existence 239
5.9 Semilinear Wave Equations：Global Existence 246
6 Abstract Minimization Problems in Hilbert Spaces 251
6.1 Control Systems：Continuity of the Solution Map 251
6.2 Patch Perturbations and Directional Derivatives 253
6.3 Continuity of the Solution Operator of the Variational Equation 262
6.4 Abstract Minimization Problems Again 263
6.5 The Minimum Principle for the Time Optimal Problem 273
6.6 The Minimum Principle for General Control Problems 279
6.7 Optimal Problems for Some Linear and Semilinear Equations 283
6.8 Semilinear Wave Equations Again 288
6.9 The Time Optimal Problem for a Semilinear Wave Equation，Ⅰ 291
6.10 Some Remarks on Adjoint Equations 293
6.11 Some Remarks on Controllability 298
6.12 The Time Optimal Problem for a Semilinear Wave Equation，Ⅱ 303
7 Abstract Minimization Problems in Banach Spaces 310
7.1 Some Geometry of Banach Spaces 310
7.2 Abstract Minimization Problems for the Last Time 316
7.3 The Minimum Principle in Banach Spaces 325
7.4 Fractional Powers of Infinitesimal Generators.Analytic Semigroups.Duality 329
7.5 Elliptic Operators in L2 Spaces 340
7.6 Elliptic Operators in Lp and C Spaces.Duality 343
7.7 Semilinear Abstract Parabolic Equations 347
7.8 Semilinear Abstract Parabolic Equations：Global Existence 359
7.9 Linear Abstract Parabolic Equations.Duality 364
7.10 Patch Perurbations and Directional Derivatives 377
8 Interpolation and Domains of Fractional Powers 385
8.1 Trace Spaces and Semigroups 385
8.2 Interpolation and Fractional Powers 393
8.3 Interpolation and Sobolev Spaces 399
8.4 Parabolic Equations 402
8.5 Fractional Powers and the Complex Interpolation Method 412
8.6 The Navier-Stokes Equations 418
9 Linear Control Systems 426
9.1 Linear Systems：The Minimum Principle 426
9.2 The Minimum Principle with Full Control 437
9.3 Bang-Bang Theorems and Approximate Controllability 444
9.4 Exact and Approximate Controllability 452
9.5 Controllability with Finite Dimensional Controls 458
9.6 Controllability and the Minimum Principle 467
10 Optimal Control Problems with State Constraints 474
10.1 Optimal Control Problems with State Constraints 474
10.2 Integration with Respect to Vector-Valued Measures 475
10.3 The Minimum Principle with State Constraints 490
10.4 Saturation of the State Constraint 498
10.5 Surface of Revolution of Minimum Area as a Control Problem 501
10.6 Other Applications 506
11 Optimal Control Problems with State Constraints 509
11.1 Abstract Parabolic Equations：The Measure-Driven Adjoint Variational Equation 509
11.2 Abstract Parabolic Equations：The Minimum Principle with State Constraints 517
11.3 Applications to Parabolic Distributed Parameter Systems 523
11.4 Parabolic Distributed Parameter Systems，Ⅰ 528
11.5 Parabolic Distributed Parameter Systems，Ⅱ 537
11.6 Linear Systems：The Minimum Principle with State Constraints 548
11.7 Control Problems for the Navier-Stokes Equations 558
11.8 Control Problems for the Navier-Stokes Equations：The Point Target Case 562
11.9 Convergence of Suboptimal Controls，Ⅰ 564
11.10 Convergence of Suboptimal Controls，Ⅱ 570
11.11 Parabolic Equations 575
11.12 The Navier-Stokes Equations 583
Part Ⅲ Relaxed Controls 601
12 Spaces of Relaxed Controls.Topology and Measure Theory 603
12.0 Weak Topologies in Linear Spaces 603
12.1 Existence Theory of Optimal Control Problems：Measure-Valued Controls 614
12.2 Spaces of Vector Valued Functions and Their Duals，Ⅰ 618
12.3 Finitely Additive Measures：Integration 628
12.4 Measures and Linear Functionals in Function Spaces 635
12.5 Spaces of Relaxed Controls 642
12.6 Approximation in Spaces of Measures and Spaces of Relaxed Controls 648
12.7 Topology and Measure Theory 654
12.8 The Filippov Implicit Function Theorem 659
12.9 Spaces of Vector Valued Functions and Their Duals，Ⅱ 666
13 Relaxed Controls in Finite Dimensional Systems 674
13.1 Installation of Relaxed Controls in Finite Dimensional Systems 674
13.2 Approximation of Relaxed Trajectories by Ordinary Trajectories 677
13.3 The Filippov Implicit Function Theorem in the Compact Case 679
13.4 Differential Inclusions 683
13.5 Existence Theorems for Relaxed Optimal Control Problems 685
13.6 Existence Theorems for Ordinary Optimal Control Problems 690
13.7 The Minimum Principle for Relaxed Optimal Control Problems 692
13.8 Noncompact Control Sets 699
14 Relaxed Controls in Infinite Dimensional Systems 709
14.1 Control Systems：Limits of Trajectories 709
14.2 Semilinear Systems Linear in the Control.Approximation by Extremal Trajectories 711
14.3 Installation of Relaxed Controls in Infinite Dimensional Systems 720
14.4 Differential Inclusions 725
14.5 Existence Theorems for Optimal Control Problems，Ⅰ 730
14.6 Existence Theorems for Optimal Control Problems，Ⅱ 738
14.7 Abstract Parabolic Equations，Ⅰ 742
14.8 Abstract Parabolic Equations，Ⅱ 748
14.9 Existence Under Compactness of the Nonlinear Term 754
14.10 Existence Without Compactness 759
References 773
Index 795