Suresh P Sethi Optimal Control Theory Applications to Management Science and Economics Third Edition Optimal Control Theory Suresh P Sethi Optimal Control Theory Applications to Management Science and Economics Third Edition 123 Suresh P Sethi Jindal School of Management, SM30 University of Texas at Dallas Richardson, TX, USA ISBN 978-3-319-98236-6 ISBN 978-3-319-98237-3 (eBook) https://doi.org/10.1007/978-3-319-98237-3 Library of Congress Control Number: 2018955904 2nd edition: © Springer-Verlag US 2000 © Springer Nature Switzerland AG 2019 This work is subject to copyright All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed The use of general descriptive names, registered names, trademarks, service marks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland This book is dedicated to the memory of my parents Manak Bai and Gulab Chand Sethi Preface to Third Edition The third edition of this book will not see my co-author Gerald L Thompson, who very sadly passed away on November 9, 2009 Gerry and I wrote the first edition of the 1981 book sitting practically side by side, and I learned a great deal about book writing in the process He was also my PhD supervisor and mentor and he is greatly missed After having used the second edition of the book in the classroom for many years, the third edition arrives with new material and many improvements Examples and exercises related to the interpretation of the adjoint variables and Lagrange multipliers are inserted in Chaps 2– Direct maximum principle is now discussed in detail in Chap along with the existing indirect maximum principle from the second edition Chattering or relaxed controls leading to pulsing advertising policies are introduced in Chap An application to information systems involving chattering controls is added as an exercise The objective function in Sect 11.1.3 is changed to the more popular objective of maximizing the total discounted society’s utility of consumption Further discussion leading to obtaining a saddle-point path on the phase diagram leading to the long-run stationary equilibrium is provided in Sect 11.2 For this purpose, a global saddle-point theorem is stated in Appendix D.7 Also inserted in Appendix D.8 is a discussion of the Sethi-Skiba points which lead to nonunique stable equilibria Finally, a new Sect 11.4 contains an adverse selection model with continuum of the agent types in a principal-agent framework, which requires an application of the maximum principle Chapter 12 of the second edition is removed except for the material on differential games and the distributed parameter maximum principle The differential game material joins new topics of stochastic Nash differential games and Stackelberg differential games via their applications to marketing to form a new Chap 13 titled Differential Games As a result, Chap 13 of the second edition becomes Chap 12 The material on the distributed parameter maximum principle is now Appendix D.9 The exposition is revised in some places for better reading New exercises are added and the list of references is updated Needless to say, the errors in the second edition are corrected, and the notation is made consistent vii viii Preface to Third Edition Thanks are due to Huseyin Cavusoglu, Andrei Dmitruk, Gustav Feichtinger, Richard Hartl, Yonghua Ji, Subodha Kumar, Sirong Lao, Helmut Maurer, Ernst Presman, Anyan Qi, Andrea Seidl, Atle Seierstad, Xi Shan, Lingling Shi, Xiahong Yue, and the students in my Optimal Control Theory and Applications course over the years for their suggestions for improvement Special thanks go to Qi (Annabelle) Feng for her dedication in updating and correcting the forthcoming solution manual that went with the first edition I cannot thank Barbara Gordon and Lindsay Wilson enough for their assistance in the preparation of the text, solution manual, and presentation materials In addition, the meticulous copy editing of the entire book by Lindsay Wilson is much appreciated Anshuman Chutani, Pooja Kamble, and Shivani Thakkar are also thanked for their assistance in drawing some of the figures in the book Richardson, TX, USA June 2018 Suresh P Sethi Preface to Second Edition The first edition of this book, which provided an introduction to optimal control theory and its applications to management science to many students in management, industrial engineering, operations research and economics, went out of print a number of years ago Over the years we have received feedback concerning its contents from a number of instructors who taught it, and students who studied from it We have also kept up with new results in the area as they were published in the literature For this reason we felt that now was a good time to come out with a new edition While some of the basic material remains, we have made several big changes and many small changes which we feel will make the use of the book easier The most visible change is that the book is written in Latex and the figures are drawn in CorelDRAW, in contrast to the typewritten text and hand-drawn figures of the first edition We have also included some problems along with their numerical solutions obtained using Excel The most important change is the division of the material in the old Chap 3, into Chaps and in the new edition Chapter now contains models having mixed (control and state) constraints, current value formulations, terminal conditions and model types, while Chap covers the more difficult topic of pure state constraints, together with mixed constraints Each of these chapters contain new results that were not available when the first edition was published The second most important change is the expansion of the material in the old Sect 12.4 on stochastic optimal control theory and its becoming the new Chap 13 The new Chap 12 now contains the following advanced topics on optimal control theory: differential games, distributed parameter systems, and impulse control The new Chap 13 provides a brief introduction to stochastic optimal control problems It contains formulations of simple stochastic models in production, marketing and finance, and their solutions We deleted the old Chap 11 of the first edition on computational methods, since there are a number of excellent references now available on this topic Some of these references are listed in Sect 4.2 of Chap and Sect 8.3 of Chap ix x Preface to Second Edition The emphasis of this book is not on mathematical rigor, but rather on developing models of realistic situations faced in business and management For that reason we have given, in Chaps and 8, proofs of the continuous and discrete maximum principles by using dynamic programming and Kuhn-Tucker theory, respectively More general maximum principles are stated without proofs in Chaps 3, and 12 One of the fascinating features of optimal control theory is its extraordinarily wide range of possible applications We have covered some of these as follows: Chap covers finance; Chap considers production and inventory problems; Chap covers marketing problems; Chap treats machine maintenance and replacement; Chap 10 deals with problems of optimal consumption of natural resources (renewable or exhaustible); and Chap 11 discusses a number of applications of control theory to economics The contents of Chaps 12 and 13 have been described earlier Finally, four appendices cover either elementary material, such as the theory of differential equations, or very advanced material, whose inclusion in the main text would interrupt its continuity At the end of the book is an extensive but not exhaustive bibliography of relevant material on optimal control theory including surveys of material devoted to specific applications We are deeply indebted to many people for their part in making this edition possible Onur Arugaslan, Gustav Feichtinger, Neil Geismar, Richard Hartl, Steffen Jørgensen, Subodha Kumar, Helmut Maurer, Gerhard Sorger, and Denny Yeh made helpful comments and suggestions about the first edition or preliminary chapters of this revision Many students who used the first edition, or preliminary chapters of this revision, also made suggestions for improvements We would like to express our gratitude to all of them for their help In addition we express our appreciation to Eleanor Balocik, Frank (Youhua) Chen, Feng Cheng, Howard Chow, Barbara Gordon, Jiong Jiang, Kuntal Kotecha, Ming Tam, and Srinivasa Yarrakonda for their typing of the various drafts of the manuscript They were advised by Dirk Beyer, Feng Cheng, Subodha Kumar, Young Ryu, Chelliah Sriskandarajah, Wulin Suo, Houmin Yan, Hanqin Zhang, and Qing Zhang on the technical problems of using LATEX We also thank our wives and children—Andrea, Chantal, Anjuli, Dorothea, Allison, Emily, and Abigail—for their encouragement and understanding during the time-consuming task of preparing this revision Preface to Second Edition xi Finally, while we regret that lack of time and pressure of other duties prevented us from bringing out a second edition soon after the first edition went out of print, we sincerely hope that the wait has been worthwhile In spite of the numerous applications of optimal control theory which already have been made to areas of management science and economics, we continue to believe there is much more that remains to be done We hope the present revision will rekindle interest in furthering such applications, and will enhance the continued development in the field Richardson, TX, USA Pittsburgh, PA, USA January 2000 Suresh P Sethi Gerald L Thompson Index Dunn, J.C., 277, 491 Durrett, R., 367, 491 Dury, K., 485 Dynamic efficiency condition, 337 Dynamic programming, 32, 366, 433 E Economic applications, 335, 383 Economic interpretation, 40, 84, 175, 337 Educational policy, 25 Eigenvalues, 412, 413 Eigenvectors, 412, 413 El-Hodiri, M., 491 Eliashberg, J., 361, 491, 507, 539 Elliott, R.J., 383, 491 El Ouardighi, F., 491, 492 Elton, E., 159, 492 Elzinga, D.J., 159, 188, 455, 488 Ending correction, 196 Entry time, 130 Envelope theorem, 54, 253, 356 EOQ, 191 Epidemic control, 343 Equilibrium relation, 42 Erickson, G.M., 492 Erickson, L.E., 11, 404, 506 Euler, Euler equation, 421, 422, 427, 428 Euler-Lagrange equation, 422 Ewald, C.-O., 543 Excel, ix, 57–60 Exhaustible resource model, 111, 324 Exit time, 130 F Factorial power, 414 Fan, L.T., 360, 492, 506 551 Farley, J.U., 539 Fattorini, H.O., 492 Feedback Nash equilibrium, 388, 393 Feedback Nash stochastic differential game, 392 Feedback Stackelberg equilibrium, 403 Feedback Stackelberg stochastic differential game, 395 Feenstra, T.L., 492 Feichtinger, G., x, xiv, 11, 39, 70, 79, 85, 104, 109, 122, 132, 136, 140, 207, 225, 335, 351, 360, 361, 458, 460, 463, 464, 475, 484, 485, 487–496, 499–504, 509, 512, 513, 515–517, 519–522, 527–530, 533, 537, 539–542, 544 Feinberg, F.M., 235, 495, 496 Fel’dbaum, A.A., 36, 433, 438, 496 Feng, Q., 291, 336, 339, 383, 474, 478 Ferreira, M.M.A., 143, 496 Ferreyra, G., 496 Filar, J., 483 Filipiak, J., 496 Finite difference equations, 414 First-order linear equations, 409 First-order pure state constraints, 135 Fischer, T., 496 Fisher, A.C., 525 Fishery management, 392 Fishery model, 312, 389 Fishing mortality function, 392 552 Fixed-end-point problem, 77, 86, 98, 99, 113 Fleming, W.H., 366, 367, 383, 442, 496 Fletcher, R., 496 Fomin, S.V., 419, 420, 422, 427, 429, 430, 499 Fond, S., 496 Forecast horizons, 213 Forest fertilization model, 331 Forestry model, 111 Forest thinning model, 317, 321 Forgetting coefficient, Forster, B.A., 496 Fourgeaud, C., 496 Fraiman, N.M., 291, 490 Francis, P.J., 343, 496 Frankena, J.F., 496 Frankowska, H., 41, 485 Free-end-point problem, 86 Free terminal time problems, 93 Friedman, A., 385, 496, 497 Fromovitz, S., 269, 518 Fruchter, G.E., 404, 492, 497 Fuller, D., 325, 497 Full-rank condition, 23, 72, 130, 131 Fundamental lemma, 422 Funke, U.H., 497 Fă urnkranz-Prskawetz, A., 463, 513, 544 Fursikov, A.V., 497 Furst, E., 540 G Gaandolfo, G., 498 Gaimon, C., 291, 360, 361, 484, 485, 497, 498 Index Gamkrelidze, R.V., 10, 27, 32, 70, 96, 141, 234, 433, 498, 526 Gandolfo, G., 493 Gaskins, D.W Jr., 498 Gaugusch, J., 498 Gaussian, 442, 443 Gavrila, C., 499 Geismar, N., x Gelfand, I.M., 419, 420, 422, 427, 429, 430, 499 General discrete maximum principle, 276 Generalized bang-bang, 113, 167 Generalized Legendre-Clebsch condition, 189, 454–456 Geoffrion, A.M., 516 Gerchak, Y., 499 Gfrerer, H., 499 Gibson, J.E., 455, 508 Gihman, I.I., 375, 499 Gillessen, W., 519 Girsanov, I.V., 499 Glad, S.T., 499 Global Saddle Point Theorem, 343, 350, 456, 457 Goal Seek, 57, 59 Goh, B.-S., 135, 311, 499 Goh, C.J., 540 Goldberg, S., 414, 499 Golden Path, 108 Golden Rule, 108, 123 Goldstein, J.R., 488 Goldstine, H.H., 499 Găollmann, L., 499 Goodwill, 5, 226 Goodwill elasticity of demand, 229 Gopalsamy, K., 499 Gordon, H.S., 312, 313, 499 Gordon, M.J., 181, 499 Index Gordon’s formula, 181 Gould, J.P., 231, 257, 362, 499 Grass, D., 11, 70, 109, 122, 360, 458, 460, 484, 485, 491, 500, 545 Green’s theorem, 225, 237, 239, 245, 254, 257, 314, 344, 346, 363, 391 Grienauer, W., 495 Grimm, W., 500, 523 Gross, M., 500 Gruber, M., 159, 492 Gruver, W.A., 512 Gutierrez, G.J., 404, 505 H Hă amă alăainen, R.P., 392, 500 Hadley, G., 11, 70, 335, 500 Hahn, M., 500 Halkin, H., 32, 276, 500 amă alăainen, R.P., 392 Hă Hamilton, 10 Hamiltonian, 35, 41, 73, 271, 337, 437 Hamiltonian maximizing condition, 36, 39, 74, 97, 99, 118, 437 Hamilton-Jacobi-Bellman (HJB) equation, 32, 36, 366, 368, 371, 393 Hamilton-Jacobi equation, 371, 394 Han, M., 500 Hanson, M., 521 Hanssens, D.M., 500 Harris, F.W., 191, 500 Harris, H., 360, 500 Harrison, J.M., 379, 501 Hartberger, R.J., 32, 436, 491, 501 553 Hartl, R.F., x, 11, 32, 39, 70, 79, 85, 95, 104, 131, 132, 135–137, 140, 141, 149, 207, 213, 225, 281, 285, 351, 360, 361, 458, 460, 463, 464, 484, 485, 490, 491, 493, 495, 497, 499–505, 512, 516, 519, 535, 537, 545 Hartman, R., 458, 504 Haruvy, E., 235, 253, 504, 545 Harvey, A.C., 504 Haunschmied, J.L., 360, 484, 495, 504 Haurie, A., 11, 106, 213, 385, 392, 463, 479, 483, 500, 504, 505 Haussmann, U.G., 505 He, X., 396, 404, 505, 535 Heal, G.M., 324, 487, 505 Heaps, T., 283, 505 Heckman, J., 505 Heineke, J.M., 487 Hestenes, M.R., 10, 70, 505 HJB equation, 36, 376 HMMS model, 191 Ho, Y.-C., 39, 113, 141, 385, 388, 442, 450, 453, 481, 505, 523, 537 Hochman, E., 383, 527 Hofbauer, J., 505, 506 Hoffmann, K.H., 506 Hohn, F., 213, 520 Holly, S., 506 Holt, C.C., 191, 200, 202, 506 Holtzman, J.M., 276, 506 554 Index J Jabrane, A., 483 Jacobi, 10 Jacobson, D.H., 49, 455, 476, 507, 514 Jacquemin, A.P., 228, 229, 231, 507 Jagpal, S., 507 Jain, D., 404, 486 Jamshidi, M., 507 Jardine, A., 535 Jarrar, R., 404, 480, 507 I Jazwinski, A.H., 507 Ijiri, Y., 191, 205, 506 Jedidi, K., 361, 507 Ilan, Y., 474 Jennings, L.S., 507 Illustration of left and right limits, Jeuland, A.P., 490, 508 18 Ji, Y., 508 Imp, 19 Jiang, J., 508 Impulse control, 19, 113, 242, 243 Johar, M., 508 Impulse control model, 113 Johnson, C.D., 455, 508 Impulse stochastic control, 383 Jones, P., 508 Imputed value, 261 Jørgensen, S., x, 11, 311, 335, 361, Incentive compatibility, 353, 354 385, 396, Indirect adjoining method, 404, 405, 489–491, 137, 147 493, 495, 502, 508, 509 Indirect contribution, 41 Joseph, P.D., 452 Individual rationality, 353 Jump conditions, 135 Infinite horizon, 6, 103 Jump Markov processes, 383 Instantaneous profit rate, 29 Junction times, 130 Interior interval, 130 Intriligator, M.D., 336, 506, 507, K 520 Kaitala, V.T., 392, 494, 500, 509 Inventory control problem, 152 Kalaba, R.E., 476 Investment allocation, 65 Kalish, S., 360, 497, 509, 510 Ioffe, A.D., 507 Kall, P., 501, 522, 533 Isaacs, R., 10, 170, 385, 507 Kalman, R.E., 442, 448, 510 Isoperimetric constraint, 88, 252 Kalman-Bucy filter, 442, 447, 448 Itˆ o stochastic differential equation, Kalman filter, 366, 441–443, 447 366, 393 Kamien, M.I., 11, 290, Ivanilov, Y.P., 360, 546 293, 296, 335, 360, 510 Homogeneous function of degree k, 22 Horsky, D., 506 Hotelling, H., 324, 506 Hritonenko, N., 474 Hsu, V.N., 221, 485 Hung, N.M., 504 Hurst, Jr., E.G., 11, 70, 113, 477 Hwang, C.L., 506 Hyun, J.S., 500 Index Kamien-Schwartz model, 111, 291 Kaplan, E.H., 360, 530, 544 Kaplan, W., 510 Karatzas, I., 337, 367, 381–383, 510 Karray, S., 511 Karreman, H.F., 480 Kavadias, S., 485 Keeler, E., 511 Keller, H.B., 346, 511 Kemp, M.C., 11, 70, 335, 486, 500, 511 Kendrick, D.A., 511 Keon, J.W., 540 Keppo, J., 383, 478 Kern, D., 499 Khmelnitsky, E., 11, 511, 512, 517 Kilkki, P., 317, 318, 511 Kim, J-H.R., 455, 519 Kirakossian, G.T., 503 Kirby, B.J., 511 Kirk, D.E., 36, 511 Kiseleva, T., 458, 511 Klein, C.F., 512 Kleindorfer, G.B., 274, 512 Kleindorfer, P.R., 213, 274, 512, 533 Kleinschmidt, P., 502 Kneese, A.V., 480, 505, 521 Knobloch, H.W., 512 Knowles, G., 512 Kogan, K., 11, 478, 511, 512, 517 Kopel, M., 488 Kort, P.M., 11, 311, 335, 351, 360, 361, 404, 458, 460, 463, 484, 485, 491, 492, 495, 502–504, 509, 512, 542 Kortanek, K., 360, 485 555 Kotowitz, Y., 512 Kozlowski, J., 546 Krabs, W., 506 Kraft, D., 135, 481 Krarup, J., 491 Krauth, J., 361, 502 Kreindler, E., 512 Krelle, W., 476, 512 Krichagina, E., 512 Kriebel, C.H., 274, 512 Krishnamoorthy, A., 475, 505, 513 Krouse, C.G., 164, 513 Krutilla, J.V., 317, 480, 510 Kugelmann, B., 513 Kuhn, H.W., 535 Kuhn, M., 463, 513, 544 Kuhn-Tucker conditions, 260, 262, 269, 271 Kumar, P.R., 513 Kumar, S., x, 442, 508, 513 Kurawarwala, A.A., 513 Kurcyusz, S., 144, 513 Kurz, M., 11, 54, 70, 106, 228, 335, 336, 474 Kushner, H.J., 513 Kydland, F.E., 513 L Laffont, J.J., 352, 513 Lagrange, Lagrange form, 30 Lagrange multipliers, 69, 70, 73, 74, 82, 260 Lagrangian, 73 Lagrangian form, 70, 76, 114 Lagrangian maximum principle, 70 Lagunov, V.N., 513 Lakhani, C., 528 Lansdowne, Z.F., 246, 514 Lasdon, L.S., 514 556 Leban, R., 11, 335, 514 Leclair, S.R., 507 Lee, E.B., 164, 514 Lee, S.C., 252, 532 Lee, W.Y., 513 Left and right limits, 17 Legendre, Legendre’s conditions, 428 Legey, L., 360, 514 Lehoczky, J.P., 337, 381, 382, 510, 514, 532 Leibniz, Leitmann, G., 32, 311, 385, 419, 486, 499, 500, 504, 505, 514, 520, 537 Leizarowitz, A., 483 Leland, H.E., 514 Lele, M.M., 507, 514 Lele, P.T., 285, 474 Lenclud, B., 496 L´eonard, D., 11, 335, 514 Leondes, C.T., 481, 519, 543, 544 Lesourne, J., 11, 335, 383, 476, 514 Lev, B., 497 Levine, J., 515 Levinson, N.L., 487 Lewis, T.R., 515 L’Hˆopital’s rule, 342 Li, G., 515 Li, M., 479 Li, T., 515, 532 Lieber, Z., 213, 500, 512, 515 Lignell, J., 515 Lilien, G.L., 360, 510 Linear independence, 23 Linearly independent, 23 Linear Mayer form, 30, 280 Linear programming, 112, 113, 167, 168 Index Linear-quadratic case, 110 Linear-quadratic problems, 448 Line integral, 238 Lintner, J., 515 Lions, J.L., 113, 383, 476, 506, 515, 533 Lipschutz, S., 402, 537 Little, J.D.C., 515 Little-o notation, 17 Liu, J., 402, 537 Liu, P.-T., 474, 480, 488, 514–516 Liu, R.H., 477 Logarithmic Brownian Motion, 378 Long, H., 478 Long, N.V., 11, 335, 385, 396, 404, 405, 486, 490, 511, 514, 516 Long-run stationary equilibrium, 106 Loon, P.J.J.M., van, 542 Lou, H., 234, 516 Lou, S., 512, 516 Lucas, Jr., R.E., 360, 516 Luenberger, D.G., 269, 516 Luhmer, A., 494, 516 Lumped parameter systems, 460 Lundin, R.A., 213, 516 Luptacik, M., 351, 503, 516, 517, 527 Luus, R., 517 Lykina, V., 517, 525 Lynn, J.W., 291, 473 M Macki, J., 517 Magat, W.A., 517 Magill, M.J.P., 517 Mahajan, V., 490, 510, 515, 517 Maimon, O., 11, 511, 512, 517 Index Maintenance and replacement model, 111, 283, 284, 290 Majumdar, M., 517 Malanowski, K., 78, 137, 517, 518, 536 Malik, T., 535 Malliaris, A.G., 383, 518 Mangasarian, O.L., 56, 79, 267–269, 518 Manh-Hung, N., 518 Mantrala, M.K., 448, 521 MAPI (Machinery and Applied Products Institute), 283 MAPLE, 185 Marginal cost, 42, 227 Marginal cost equals marginal revenue, 42 Marginal return, 438 Marginal revenue, 42 Marinelli, C., 460, 518 Markovian Stackelberg equilibrium, 396 Markus, L., 514 Martingale problems, 383 Mart´ın-Herr´ an, G., 404, 476, 507, 511, 518 Mart´ın-Herr´ an, G., 404 Martirena-Mantel, A.M., 518 Marzano, F., 493 Mass´e, P., 283, 518 Mate, K., 506 Mathematica, 185, 395, 402, 406 Mathematical requirements, Mathewson, F., 512 Matrix Riccati equation, 448, 450 Matsuo, H., 513 Maurer, H., x, 135, 137, 144, 455, 499, 518, 519 Maximum, 429 Maximum likelihood estimate, 443 557 Maximum principle, 27, 39, 40, 50, 69, 70, 73, 74, 76, 82, 96, 99, 109, 114, 118, 119, 136, 137, 259, 433 May, R.M., 474 Mayer form, 30, 433, 437 Mayne, D.Q., 32, 135, 277, 481, 519, 523, 526 McCabe, J.L., 514 McCann, J.M., 517 McEneaney, W.M., 534 McGuire, T.W., 360, 532 McIntyre, J., 519 McNicoll, G., 360, 475 McShane, E.J., 10 Measurement noise, 441 Mechanism design, 352, 356 Meech, J.A., 507 Megretski, A., 534 Mehlmann, A., 360, 361, 385, 490, 494, 503, 519 Mehra, R.K., 360, 519 Mehrez, A., 519 Merton, R.C., 378, 520 Mesak, H.I., 520 Michel, P., 488, 496, 520 Miele, A., 237, 514, 520 Miller, M.H., 166, 520 Miller, R.E., 520 Milyutin, A.A., 132, 491 Minimax solution, 385, 386 Minimum fuel problem, 280 Minjarez-Sosa, J.A., 383, 477 Mirman, L.J., 515, 520 Mirrlees, J., 520 Miscellaneous applications, 360 Miscellany, 16 Mischenko, E.F., 10, 27, 32, 70, 96, 141, 433, 526 Misra, S., 513 558 Mitchell, A., 521 Mitra, T., 517 Mitter, S.K., 477, 523 Mittnik, S., 490 Mixed constraints, 69, 71, 79 Mixed inequality constraints, 3, 69, 70 Mixed optimization technique, 302 Modeling tricks, 111 Modigliani, F., 166, 191, 200, 202, 213, 506, 520 Moiseev, N.N., 521 Monahan, G.E., 521 Mond, B., 521 Mookerjee, V.S., 508 Moore, E.J., 540 Moore, J.B., 383, 442, 450, 474, 491 Morey, R.C., 517 Mortimort, D., 352, 513 Morton, A., 498 Morton, T.E., xiv, 213, 297, 302, 309, 516, 521, 532 Moser, E., 351, 460, 521 Moskowitz, H., 485 Motta, M., 521 Muller, E., 490, 515, 517, 521 Mulvey, J.M., 532 Munro, G.R., 311, 521 Murata, Y., 521 Murray, D.M., 277, 521 Muth, J.F., 191, 200, 202, 506 Muzicant, J., 521 N Naert, P.A., 404, 477, 481 Nahorski, Z., 521 Naik, P.A., 404, 448, 521, 522, 526 Nash differential games, 387 Nash solutions, 385 Index Nă aslund, B., 11, 70, 113, 283, 323, 331, 477, 522 Natural resources, 311, 383 Necessary condition, 37, 39, 269 Neck, R., 522 Needle-shaped variation, 434, 435 Neighborhood, 16 Nelson, R.T., 360, 522 Nepomiastchy, P., 360, 522 Nerlove, M., 226, 228, 522 Nerlove-Arrow model, 110 Nerlove-Love advertising model, 226 Neuman, C.P., 226, 541 Neustadt, L.W., 10, 518, 522 Newton, Nguyen, D., 522 Nishimura, K., 458, 488 Nissen, G., 477 Nonlinear programming, 259, 260, 268 Norm, 16, 17 Norstră om, C.J., xiv, 191, 208, 522 Norton, F.E., 448, 481 Notation, 11 Novak, A.J., 361, 460, 484, 485, 494, 495, 503, 504, 522 O Oakland, W.H., 360, 480 Oberle, H.J., 500, 522, 523 Objective function, 2, 29, 438 Oettli, W., 481 O guztă oreli, M.N., 523 Øksendal, B.K., 383 Øksendal, B.K., 480, 523 Okuguchi, K., 486 Olsder, G.J., 385, 396, 405, 475, 523 Index One-sector model, 338 Oniki, H., 523 Open access fishery, 313 Open-Loop Nash Solution, 388, 389, 405, 406 Open-loop Stackelberg solution, 396, 405 Optimal consumption of an initial investment, 115 Optimal control problem, 29 Optimal control theory, 1, 419 Optimal economic growth models, 335, 340 Optimal financing model, 111, 164, 186, 187 Optimal long-run stationary equilibrium, 108, 228 Optimal path, 29 Optimal thinning, 318 Optimal trajectory, 29 Order of the constraint, 130 Oren, S.S., 523 Osayimwese, I., 523 Ouardighi, F.E., 478 Ozga, S., 257, 523 Ozga model, 257 P Paiewonsky, B., 519 Palda, K.S., 226, 523 Palokangas, T., 525 Pantoja, J.F., 277, 523 Parametric linear programming, 113 Parlar, M., 499, 523, 524 Parrish, B., 473 Parsons, L.J., 500 Partial fractions, 372 Pasin, F., 491 Path of least time, 559 Pauwels, W., 524 Pekelman, D., 213, 283, 302, 524 Pepyne, D.L., 277, 524 Perera, S., 478 Perrakis, S., 188, 524 Pesch, H.J., 10, 513, 524 Pessimal solution, 182 Peterson, D.W., 78, 524, 525 Peterson, F.M., 525 Peterson, R.A., 517 Petrosjan, L., 491 Petrov, Iu.P., 525 Phase diagram, 340, 349 Phelps, E.S., 499 Pickenhain, S., 517, 518, 525 Pierskalla, W.P., 283, 525 Pindyck, R.S., 324, 383, 489, 525, 541 Pitchford, J.D., 496, 516, 525 Plail, M., 10, 524 Pliska, S.R., 379, 501 Pohjola, M., 525 Polak, E., 135, 276, 482, 519, 526 Pollution control model, 346, 351 Polyanin, A.D., 410, 526 Pontryagin, L.S., 10, 27, 32, 70, 96, 141, 433, 526 Powell, S.G., 523 Prasad, A., 235, 253, 392, 396, 404, 475, 504, 505, 513, 522, 526, 535, 545 Predator-prey relationships, 311 Prescott, E.C., 513 Presman, E., 511, 526 Price elasticity of demand, 227 Price shield, 214–216 Principle-agent framework, vii Principle of optimality, 33, 367, 380 560 Production function, 338 Production-inventory model, 4, 191, 192, 274 Production planning model, 191, 365 Production smoothing, 192 Product rule for differentiation, 16 Proth, J.-M., 213, 477, 485 Prskawetz, A., 495, 513, 527 Pulsing, 494, 495, 500, 515, 517, 520, 521 Pulsing policy, 235, 251 Pure constraints, 151 Pure state variable inequality constraints, 3, 125, 129 Pytlak, R., 135, 527 Q Quasiconcave function, 79 Quasiconvex function, 79 R Rajagopalan, S., 515 Raman, K., 383, 527 Rampazzo, F., 521 Ramsey, F.P., 85, 335, 362, 527 Rank of a matrix, 23 Rao, A.G., 504 Rao, R.C., 404, 527 Rapoport, A., 527 Rapp, B., 283, 527 Rausser, G.C., 383, 524, 527 Raviv, A., 360, 527 Ravn, H.F., 269, 521, 527 Ray, A., 528 Reachable set, 3, 72, 89, 114 Reeves, C.M., 496 Regional allocation of investment, 64 Reinganum, J.F., 528 Index Rekhi, I., 485 Rempala, R., 213, 528 Ren, Q., 545 Richard, S.F., 487, 528 Rinc´ on-Zapatero, J.P., 518 Ringbeck, J., 361, 528 Ripper, M., 360, 514 Rishel, R.W., 366, 367, 442, 496, 528 Roberts, S.M., 39, 528 Robinett, R.D., 277, 490 Robinson, B., 528 Robson, A.J., 529 Rockafellar, R.T., 529 Roxin, E.O., 480, 482, 515 Royal, A., 383, 477, 478 Rozovsky, B., 478 Rubel, O., 518 Russak, B., 529 Russell, D.L., 529 Ră ustem, B., 506 Ruusunen, J., 392, 500 Ryu, Y., x S Saddle point, 11, 22, 23, 260, 387, 457, 492, 529 Sage, A.P., 462, 529 Salukvadze, M.E., 529 Salvage value, 3, 29, 113 Samaratunga, C., 529 Samuelson, P.A., 529 Sargent, T.J., 516 Sarma, V.V.S., 291, 309, 473, 529 Sasieni, M., 529 Sat function, 19 Savin, S., 460, 518 Sawyer, A., 448, 521 Scalzo, R.C., 529 Schaefer, M.B., 312, 529 Index Schijndel, G.-J.C.Th.,van, 360, 542 Schilling, K., 529 Schmalensee, R., 515 Scholes, M., 378, 480 Schubert, U., 351, 517 Schultz, R L., 500 Schwartz, N.L., 11, 290, 293, 296, 335, 360, 510 Schwodiauer, G., 540 Scott, A.D., 311, 521 Second-order differential equations, 382 Second-order linear equations with constant coefficients, 410 Second-order variations, 428, 452 Seeger, A., 525 Segers, R., 491 Seidl, A., 351, 360, 458, 460, 484, 485, 521, 530, 544 Seidman, T.I., 460, 530 Seierstad, A., 11, 32, 53, 70, 73, 77–79, 104, 136, 149, 335, 530 Selten, R., 385, 530 Semmler, W., 489, 490 Sen, S.K., 488, 510 Sengupta, J.K., 540 Separation principle, 451, 452 Sethi, S.P., 11, 32, 54, 95, 131, 132, 135–137, 140, 141, 149, 159, 164, 166, 187, 191, 192, 213, 221, 225, 231, 235–237, 242, 246, 248, 251–253, 256, 257, 277, 281, 283, 289, 291, 293, 297, 302, 309, 314, 315, 321, 561 323, 324, 336, 337, 339, 343, 346, 351, 360, 361, 371, 375, 376, 380–383, 385, 388, 392, 396, 397, 404, 458, 460, 462, 463, 474–482, 485–490, 495, 503–505, 507, 508, 510–516, 522, 524, 526, 528–535, 539, 540, 545 Sethi-Morton model, 297, 309 Sethi-Skiba Points, vii, 109, 315, 441, 458–460, 464 Shadow price, 10, 40, 261 Shani, U., 361, 535 Shapiro, A., 535 Shapiro, C., 144, 535 Sharomi, O., 535 Shell, K., 335, 484, 535 Shi, R., 383, 477, 478 Shipman, J.S., 39, 528 Shreve, S.E., 337, 366, 367, 381–383, 479, 510 Shtub, A., 512 Siebert, H., 536 Silva, G.N., 536 Simaan, M., 536 Simon, H.A., 191, 200, 202, 452, 506, 536 Simon, L.S., 506 Simple cash balance problem, 159, 160, 187 Simplest variational problem, 420 Singh, M.G., 531, 536 Singhal, J., 536 Singhal, K., 536 Singhal, V., 498 Singular arcs, 454 562 Singular control, 48, 49, 110, 162, 163, 167, 176, 177, 345, 454, 455 Skiba, A.K., 458, 464, 536 Skorohod, A.V., 375, 499 Smith, B.L.R., 507 Smith, M., 507 Smith, R.L., 475 Smith, V.K., 510 Smith, V.L., 536 Snower, D.J., 536 Sole-owner fishery resource model, 111, 312 Soliman, M.A., 360, 539 Solow, R.M., 324, 536 Soner, H.M., 383, 496, 514 Sorger, G., x, 11, 297, 335, 385, 396, 404, 405, 481, 485, 490, 494, 505, 506, 516, 532, 533, 536, 537 Sothmann, B., 523 Southwick, L., 360, 537 Special topics, 441 Spence, M., 346, 511, 537 Speyer, J.L., 507 Spiegel, M.R., 402, 414, 537 Spremann, K., 537 Sprzeuzkouski, A.Y., 537 Spulber, P.F., 515, 520 Srinivasan, V., 226, 537 Sriskandarajah, C., x Staats, P.W., 343, 533 Stackelberg differential games, 385, 396, 404, 475, 478 Stalford, H., 514, 537 Standard adjoint variables, 80 Standard Hamiltonian, 80 Standard Lagrangian, 80 Index Standard multipliers, 80 Starr, A.W., 385, 388, 537 Starting correction, 196 State equation, 28 State trajectory, 2, 28 State variable, 2, 28 State vector, 28 Static efficiency condition, 337 Stein, R.B., 523 Steinberg, R., 491 Steindl, A., 460, 494, 516, 537 Steiner, P.O., 491 Stepan, A., 537 Stern, L.E., 537 Stiglitz, J.E., 538 Stirling numbers of the first kind, 416 Stirling numbers of the second kind, 415 Stochastic advertising problem, 375 Stochastic calculus, 367, 444 Stochastic manufacturing problems, 383 Stochastic optimal control, 365, 366 Stochastic production inventory model, 370 Stockout cost, Stoer, J., 481, 538 Stopping time, 380 Stoppler, S., 11, 488, 489, 538 Strauss, A., 517 Streitferdt, L., 542 Strengthened Jacobi condition, 429 Strengthened Legendre-Clebsch condition, 454 Strengthened Legendre condition, 429 Index Strictly concave function, 21, 79 Strictly convex function, 79 Strong forecast horizon, 213, 216, 219 Strong maximum, 430 Subsidy rate, 397 Sufficiency conditions, 53, 54, 79, 136, 269 Sulem, A., 538 Summary of transversality conditions, 89 Suo, W., x, 526, 534, 535 Surveys of applications, 10 Sutinen, J.G., 488, 516 Swan, G.W., 343, 538 Sweeney, D.J., 473, 538 Sweeney, J.L., 480, 505, 521 Switching curves, 99 Switching point, 171, 175, 178 Switching time, 102 Sydsæter, K., 11, 32, 53, 70, 73, 77–79, 104, 136, 149, 335, 530, 538 Synthesis of optimal controls, 97, 170 System noise, 442 Szego, G.P., 535 T Taboubi, S., 404, 476, 518 Takayama, A., 42, 335, 491, 538 Taksar, M.I., 512, 514, 533–535 Tan, K.C., 538 Tapiero, C.S., 11, 297, 309, 360, 383, 477, 491, 533, 538, 539 Taraysev, A., 525 Taylor, J.G., 360, 539 Teichroew, D., 11, 487 Teng, J.-T., 539, 540 563 Teo, K.L., 135, 462, 473, 507, 540 Terborgh, G., 283, 540 Terminal conditions, 38, 74, 86 Terminal inequality constraints, 72 Terminal time, 4, 29, 71, 75, 86, 89, 98, 113 Th´epot, J., 360, 515, 540 Thisse, J., 507 Thompson, G.L., 54, 159, 191, 205, 213, 253, 274, 283, 291, 302, 309, 360, 371, 388, 404, 460, 462, 463, 474, 488, 489, 498, 506, 512, 533, 534, 539, 540 Tidball, M., 540 Tihomirov, V.M., 507 Time-optimal control problem, 96, 97 Tintner, G., 540 Titli, A., 536 Tolwinski, B., 385, 505, 540 Total contribution, 41 Tou, J.T., 452 Toussaint, S., 541 TPBVP, 39, 40, 57, 58, 60, 64, 67, 221, 338 Tracz, G.S., 541 Tragler, G., 11, 70, 109, 122, 360, 458, 460, 475, 484, 495, 499, 500, 541, 545 Transition matrix, 436 Transversality conditions, 38, 75, 77, 86, 88, 89, 91, 99, 104, 105, 116, 121 Transversality conditions: special cases, 86 Treadway, A.B., 360, 541 564 Troch, I., 541 Tsachev, T., 495 Tsur, Y., 361, 535 Tsurumi, H., 252, 541 Tsurumi, Y., 252, 541 Tu, P.N.V., 11, 360, 541 Tuominen, M.P.T., 515 Turner, R.E., 226, 541 Turnovsky, S.J., 496, 516, 525, 541 Turnpike, 108, 196, 228, 248 Two person zero-sum games, 386 Two-point boundary value problem, 39, 40, 57, 64, 413 Two-reservoir system, 151 Tzafestas, S.G., 462, 493, 541 U Udayabhanu, V., 534 Uhler, R.S., 541 Utility of consumption, 7, 335, 336, 377 V Vaisanen, U., 317, 318, 511 Valentine, F.A., 10, 541 Value function, 33, 367 Van Hilten, O., 11, 335, 542 Van Loon, P.J.J.M., 11, 335, 542 Vanthienen, L., 213, 542 Varaiya, P.P., 360, 385, 442, 513, 514, 542 Variational equations, 435 Veinott, A.F., 474 Veliov, V.M., 462, 463, 488, 494, 495, 542 Venezia, I., 539 Verheyen, P.A., 360, 542 Verification theorem, 369, 396 Verma, B., 507 Index Vickson, R.G., 32, 131, 132, 135–137, 140, 141, 149, 325, 477, 497, 503, 523, 524, 542, 546 Vidal, R.V.V., 521 Vidale, M.L., 226, 235, 236, 542 Vidale-Wolfe advertising model, 111, 235, 375 Vidyasagar, M., 109, 542 Vilcassim, N.J., 404, 486 Villas-Boas, J.M., 542 Vincent, T.L., 311, 499 Vinokurov, V.R., 543 Vinter, R.B., 41, 132, 135, 143, 479, 480, 496, 527, 536, 543 Voelker, J.A., 283, 525 Vossen, G., 455, 519 Vousden, N., 360, 516, 543 W Wagener, F.O.O., 458, 511, 543 Wagner, H.M., 191, 297, 543 Wagner, M., 517, 525 Wagner-Whitin framework, 301 Wagner-Whitin solution, 306 Wan, F.Y., 536 Wan, Jr., H.Y., 487 Wang, C.-S., 360, 492 Wang, M., 478 Wang, P.K.C., 543 Wang, W.-K., 543 Warehousing constraint, 214 Warga, J., 543 Warnecke, H.J., 543 Warschat, J., 543 Weak forecast horizon, 213, 215 Weak maximum, 429 Weber, T.A., 335, 544 Weierstrass, 10 Index Weierstrass-Erdmann corner conditions, 428 Weierstrass necessary condition, 430, 432 Weinstein, M.C., 324, 544 Weitz, B., 527 Weizsă acker, C.C von, 543 Welam, U.P., 544 Well, K.H., 500 Wensley, R., 527 Westphal, L.C., 544 Wheat trading model with no short-selling, 208 Whitin, T.M., 191, 297, 543 Whittle, P., 544 Wickwire, K., 11, 343, 544 Wiegand, M., 144, 519 Wiener, N., 544 Wiener process, 367, 378 Wind, Y., 490, 510, 517, 539 Wirl, F., 484, 489, 494, 495, 504, 544 Wolfe, H.B., 226, 235, 236, 542 Wong, K.H., 135, 540 Wonham, W.M., 544 Wright, C., 351, 544 Wright, S.J., 277, 544 Wrzaczek, S., 360, 463, 513, 530, 544 Wunderlich, H.J., 543 X Xepapadeas, A., 351, 545 Y Yakowitz, S.J., 277, 521 Yan, H., x, 535, 545 Yang, J., 545 565 Yang, T.H., 135, 526 Yatsenko, Y., 474 Yeh, D.H.M., 508, 535 Yeung, D.K., 491 Yin, G., 383, 526, 532, 534, 545 Young, L.C., 419, 439, 545 Z Zabczyk, J., 528 Zaccour, G., 404, 480, 507, 509, 518, 537, 540, 545 Zaitsev, V.F., 410, 526 Zalkin, J.H., 525 Zarrop, M.B., 506 Zeckhauser, R.J., 324, 346, 511, 544 Zeidan, V., 545 Zeiler, I., 458, 545 Zelikin, M.I., 545 Zemel, A., 361, 535 Zemel, E., 521 Zhang, H., x, 383, 478, 526, 534, 535 Zhang, J., 352, 487 Zhang, Q., x, 383, 516, 526, 532–535, 545 Zhang, R., 504, 535, 545 Zhou, J., 504 Zhou, X., 529, 534, 535, 545 Ziemba, W.T., 532, 546 Zimin, I.N., 360, 546 Zi´ olko, M., 546 Zionts, S., 360, 534, 537 Zoltners, A.A., 508, 546 Zowe, J., 144, 513 Zuckermann, D., 539 Zwillinger, D., 197, 410, 546 ... Sethi Preface to Second Edition The first edition of this book, which provided an introduction to optimal control theory and its applications to management science to many students in management, .. .Optimal Control Theory Suresh P Sethi Optimal Control Theory Applications to Management Science and Economics Third Edition 123 Suresh P Sethi Jindal School of Management, SM30... indebted to Eleanor Balocik and Rosilita Jones for their patience and careful typing Although the applications of optimal control theory to management science are recent and many fascinating applications