viii Contents Lecture 9. Weak Versus Strong Variations 67 Lecture 10. Weierstrass’s Necessary Condition 73 Lecture 11. The Transversality Conditions 81 Lecture 12. Hilbert’s Invariant Integral 91 Lecture 13. The Fundamental Sufficient Condition 101 Appendix 13: The Equations of an Envelope 108 Lecture 14. Jacobi’s Condition Revisited 111 Lecture 15. Isoperimetrical Problems 119 Appendix 15: Constrained Optimization 124 Lecture 16. Optimal Control Problems 127 Lecture 17. Necessary Conditions for Optimality 135 Appendix 17: The Calculus of Variations Revisited 146 Lecture 18. Time-Optimal Control 149 Lecture 19. A Singular Control Problem 159 Lecture 20. A Biological Control Problem 163 Lecture 21. Optimal Control to a General Target 167 Appendix 21: The Invariance of the Hamiltonian 180 Lecture 22. Navigational Control Problems 183 Lecture 23. State Variable Restrictions 195 Lecture 24. Optimal Harvesting 203 Afterword 219 Solutions or Hints for Selected Exercises 221 Bibliography 245 Index 249
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