برنامه‌نویسی پویا و حساب تغییرات (ریاضیات در علم و مهندسی، جلد 21)

Dynamic Programming and the Calculus of Variations......Page 6 Copyright Page......Page 7 Contents......Page 16 Preface......Page 10 1. Introduction......Page 24 2. An Example of a Multistage Decision Process Problem......Page 25 3. The Dynamic Programming solution of the Example......Page 26 4. The Dynamic Programming Formalism......Page 30 5. Two Properties of the Optimal Value Function......Page 33 6. An Alternative Method of Solution......Page 36 8. A Property of Multistage Decision Processes......Page 38 9. Further Illustrative Examples......Page 39 10. Terminal Control Problems......Page 43 12. Solution of the Example......Page 44 13. Properties of the Solution of a Terminal Control Problem......Page 46 14. Summary......Page 47 1. Introduction......Page 48 3. Admissible Solutions......Page 49 5. Functionals......Page 50 7. Arc-Length......Page 51 8. The Simplest General Problem......Page 52 10. The Nature of Necessary Conditions......Page 53 11. Example......Page 54 14. The Absolute Minimum of a Functional......Page 55 15. A Relative Minimum of a Function......Page 56 17. A Weak Relative Minimum of a Functional......Page 57 18. Weak Variations......Page 59 19. The First and Second Variations......Page 61 20. The Euler-Lagrange Equation......Page 62 22. The Legendre Condition......Page 64 23. The Second Variation and the Second Derivative......Page 65 24. The Jacobi Necessary Condition......Page 66 26. Focal Point......Page 67 28. The Weierstrass Necessary Condition......Page 68 29. Example......Page 71 31. Transversality Conditions......Page 73 32. Corner Conditions......Page 74 33. Relative Summary......Page 75 34. Sufficient Conditions......Page 77 36. Other Problem Formulations......Page 78 37. Example of a Terminal Control Problem......Page 80 38. Necessary Conditions for the Problem of Mayer......Page 82 40. Two-Point Boundary Value Problems......Page 83 41. A Well-Posed Problem......Page 84 43. Computational Solution......Page 87 44. Summary......Page 88 References to Standard Texts......Page 89 1. Introduction......Page 92 2. Notation......Page 93 3. The Fundamental Partial Differential Equation......Page 94 4. A Connection with Classical Variations......Page 97 6. Two Kinds of Derivatives......Page 98 7. Discussion of the Fundamental Partial Differential Equation......Page 99 8. Characterization of the Optimal Policy Function......Page 101 9. Partial Derivatives along Optimal Curves......Page 102 10. Boundary Conditions for the Fundamental Equation: I......Page 104 11. Boundary Conditions: II......Page 106 12. An Illustrative Example—Variable End Point......Page 107 13. A Further Example—Fixed Terminal Point......Page 109 14. A Higher-Dimensional Example......Page 111 15. A Different Method of Analytic Solution......Page 112 16. An Example......Page 117 18. The Euler-Lagrange Equation......Page 120 19. A Second Derivation of the Euler-Lagrange Equation......Page 122 21. The Weierstrass Necessary Condition......Page 124 22. The Jacobi Necessary Condition......Page 126 23. Discussion of the Jacobi Condition......Page 129 25. An Illustrative Example......Page 130 26. Determination of Focal Points......Page 131 27. Example......Page 133 28. Discussion of the Example......Page 134 29. Transversality Conditions......Page 137 30. Second-Order Transversality Conditions......Page 139 31. Example......Page 141 32. The Weierstrass Erdmann Corner Conditions......Page 142 33. Summary......Page 143 34. A Rigorus Dynamic Programming Approach......Page 144 35. An Isoperimetric Problem......Page 148 36. The Hamilton-Jacobi Equation......Page 151 1. Introduction......Page 152 2. Statement of the Problem......Page 153 4. The Fundamental Partial Differential Equation......Page 155 6. Interpretation of the Fundamental Equation......Page 157 7. Boundary Conditions for the Fundamental Equation......Page 159 9. Two Necessary Conditions......Page 160 10. The Multiplier Rule......Page 161 12. The Weierstrass Necessary Condition......Page 163 14. The Second Partial Derivatives of the Optimal Value Function......Page 164 15. A Matrix Differential Equation of Riccati Type......Page 166 16. Terminal Values of the Second Partial Derivatives of the Optimal Value Function......Page 167 17. The Guidance Problem......Page 168 18. Terminal Transversality Conditions......Page 170 19. Initial Transversality Conditions......Page 173 21. A First Integral of the Solution......Page 174 22. The Variational Hamiltonian......Page 175 23. Corner Conditions......Page 176 24. An Example......Page 177 25. Second-Order Transversality Conditions......Page 179 26. Problem Discontinuities......Page 180 27. Optimization of Parameters......Page 182 28. A Caution......Page 184 29. Summary......Page 186 2. Control-Variable Inequality Constraints......Page 187 3. The Appropriate Multiplier Rule......Page 188 4. A Second Derivation of the Result of Section 3......Page 192 5. Discussion......Page 194 6. The Sign of the Control Impulse Response Function......Page 197 8. The Appropriate Modification of the Multiplier Rule......Page 198 9. The Conventional Notation......Page 199 10. A Second Derivation of the Result of Section 9......Page 200 11. Discussion......Page 201 12. The Sign of the New Multipier Function......Page 203 13. State-Variable Inequality Constraints......Page 204 14. The Optimal Value Function for a State-Constrained Problem......Page 205 15. Derivation of a Multiplier Rule......Page 206 16. Generalizations......Page 209 17. A Connection with Other Forms of the Results......Page 210 18. Summary......Page 211 2. Switching Manifolds......Page 213 3. A Problem That Is Linear in the Derivative......Page 214 4. Analysis of the Problem of Section 3......Page 215 5. Discussion......Page 218 6. A Problem with Linear Dynamics and Criterion......Page 219 7. Investigation of the Problem of Section 6......Page 220 8. Further Analysis of the Problem of Section 6......Page 221 9. Discussion......Page 226 10. Summary......Page 227 1. Introduction......Page 228 2. A Deterministic Problem......Page 230 3. A Stochastic Problem......Page 232 4. Discussion......Page 235 6. The Optimal Expected Value Function......Page 236 7. The Fundamental Recurrence Relation......Page 237 9. A Continuous Stochastic Control Problem......Page 238 10. The Optimal Expected Value Function......Page 240 11. The Fundamental Partial Differential Equation......Page 241 12. Discussion......Page 242 13. The Analytic Solution of an Example......Page 243 14. Discussion......Page 244 15. A Modification of an Earlier Problem......Page 245 17. A Poisson Process......Page 247 18. The Fundamental Partial Differential Equation for a Poisson Process......Page 248 20. A Numerical Problem......Page 249 21. The Appropriate Prior-Probability Density......Page 251 23. The Fundamental Recurrence Equation......Page 252 24. A Further Specialization......Page 254 25. Numerical Solution......Page 255 26. Discussion......Page 258 27. A Control Problem with Partially Observable States and with Deterministic Dynamics......Page 259 28. Discussion......Page 261 29. Sufficient Statistics......Page 262 31. A Warning......Page 263 32. Summary......Page 264 Bibliography......Page 265 Author Index......Page 268 Subject Index......Page 269