Preface |
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xi | |
Notation |
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xvii | |
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1 | (60) |
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1 | (3) |
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The Calculus of Variations |
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4 | (14) |
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Existence of Minimizers and Tonelli's Direct Method |
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18 | (3) |
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Sufficient Conditions and the Hamilton-Jacobi Equation |
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21 | (4) |
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25 | (5) |
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30 | (5) |
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35 | (4) |
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39 | (13) |
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Nonsmooth Optimal Control |
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52 | (4) |
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56 | (4) |
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60 | (1) |
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Measurable Multifunctions and Differential Inclusions |
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61 | (48) |
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61 | (1) |
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62 | (2) |
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Measurable Multifunctions |
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64 | (11) |
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Existence and Estimation of F-Trajectories |
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75 | (11) |
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Perturbed Differential Inclusions |
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86 | (5) |
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Existence of Minimizing F-Trajectories |
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91 | (3) |
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94 | (6) |
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The Generalized Bolza Problem |
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100 | (8) |
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108 | (1) |
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109 | (18) |
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109 | (1) |
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110 | (1) |
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111 | (4) |
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115 | (10) |
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125 | (2) |
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127 | (52) |
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127 | (1) |
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128 | (5) |
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133 | (6) |
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Difference Quotient Representations |
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139 | (5) |
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Nonsmooth Mean Value Inequalities |
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144 | (5) |
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Characterization of Limiting Subgradients |
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149 | (5) |
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Subgradients of Lipschitz Continuous Functions |
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154 | (7) |
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161 | (5) |
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Criteria for Lipschitz Continuity |
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166 | (4) |
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Relationships Between Normal and Tangent Cones |
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170 | (9) |
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179 | (22) |
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179 | (2) |
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A Marginal Function Principle |
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181 | (4) |
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Partial Limiting Subgradients |
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185 | (2) |
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187 | (3) |
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190 | (3) |
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Lagrange Multiplier Rules |
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193 | (4) |
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Notes for Chapters 4 and 5 |
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197 | (4) |
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201 | (32) |
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201 | (2) |
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203 | (5) |
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Derivation of the Maximum Principle from the Extended Euler Condition |
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208 | (6) |
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A Smooth Maximum Principle |
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214 | (14) |
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228 | (5) |
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The Extended Euler-Lagrange and Hamilton Conditions |
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233 | (52) |
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233 | (4) |
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Properties of the Distance Function |
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237 | (5) |
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Necessary Conditions for a Finite Lagrangian Problem |
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242 | (10) |
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The Extended Euler-Lagrange Condition: Nonconvex Velocity Sets |
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252 | (7) |
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The Extended Euler-Lagrange Condition: Convex Velocity Sets |
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259 | (5) |
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Dualization of the Extended Euler-Lagrange Condition |
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264 | (13) |
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The Extended Hamilton Condition |
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277 | (3) |
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280 | (5) |
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Necessary Conditions for Free End-Time Problems |
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285 | (36) |
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285 | (3) |
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Lipschitz Time Dependence |
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288 | (7) |
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295 | (2) |
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Measurable Time Dependence |
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297 | (4) |
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301 | (9) |
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310 | (3) |
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A Free End-Time Maximum Principle |
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313 | (5) |
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318 | (3) |
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The Maximum Principle for State Constrained Problems |
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321 | (40) |
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321 | (3) |
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324 | (5) |
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The Maximum Principle for Problems with State Constraints |
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329 | (5) |
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Derivation of the Maximum Principle for State Constrained Problems from the Euler-Lagrange Condition |
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334 | (5) |
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A Smooth Maximum Principle for State Constrained Problems |
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339 | (20) |
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359 | (2) |
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Differential Inclusions with State Constraints |
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361 | (36) |
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361 | (1) |
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A Finite Lagrangian Problem |
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362 | (6) |
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The Extended Euler-Lagrange Condition for State Constrained Problems: Nonconvex Velocity Sets |
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368 | (7) |
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Necessary Conditions for State Constrained Problems: Convex Velocity Sets |
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375 | (7) |
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Free Time Problems with State Constraints |
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382 | (5) |
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Nondegenerate Necessary Conditions |
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387 | (9) |
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396 | (1) |
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397 | (38) |
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397 | (6) |
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403 | (5) |
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Proof of The Generalized Tonelli Regularity Theorem |
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408 | (9) |
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Lipschitz Continuous Minimizers |
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417 | (5) |
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Autonomous Variational Problems with State Constraints |
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422 | (3) |
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425 | (3) |
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Lipschitz Continuous Controls |
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428 | (4) |
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432 | (3) |
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435 | (58) |
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435 | (7) |
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442 | (10) |
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The Value Function and Generalized Solutions of the Hamilton-Jacobi Equation |
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452 | (13) |
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Local Verification Theorems |
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465 | (9) |
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Adjoint Arcs and Gradients of the Value Function |
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474 | (9) |
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State Constrained Problems |
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483 | (4) |
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487 | (6) |
References |
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493 | (12) |
Index |
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505 | |