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Optimization techniques with applications to aerospace systems
Book
the sum representational 1963.
method”
263
reviews
by N.N. Lyashko,
Izd-vo
Kievskogo
un-ta,
B.M. Budak 0. LEITMANN, (Ed.), optimization Acad. Press, space Systems,
Techniques with Applications New-York, London, 1962.
to Aero-
This book, which is already widely known amongst mathematical specialists as well as mechanics and engineers dealing with aerospace systems, is a clear example of how one of the oldest subjects in mathematical analysis, variational calculus, finds new spheres of application. Long ago variational calculus was applied in many spheres of science and technology (geometry, physics, mechanics, etc.). Now new ones have been added to these: the theory of aerospace systems, automatic control, mathematical economics, etc. These applications have proved most fruitful both for the new technological disciplines and for variational calculus itself. In various countries (particularly in the USSR and the USA) there have appeared new directions for variational calculus over the past few years - the theory of optimal processes and dynamic programming and these continue to be developed intensively. These new directions are of especial value from the point of view of their applications. In this respect they have turned out to be more profitable than classical variational calculus. The present book is basically concerned with these new developments in variational calculus and their applications to the theory of aerospace systems. The book makes effective uses of (1) classical methods of variational calculus, improved for problems of rocket technology; (2) the maximum principle of L.S. Pontryagin; (3) Bellman’s dynamic programming method; (4) gradient methods, The book consists of fourteen chapters written by leading American experts. The account is in the nature of a survey; with rare exceptions proofs are not given, but there are references to the rich bibliography at the end of each chapter. There are many Soviet works in the bibliography. It must be said that the bibliographies for certain articles are extensive and contain many very valuable sources (although these are also not very accessible), published in the form of reports of various departments. At times lack of care is evidenced in the selection of the bibliography of Soviet authors. For example, in the bibliography for Chapter ‘7. “The maximum principle of Pontryagin” (by R. Coney) the article by A.D. Aleksandrov, “Research on the maximum principle”, Izv. vyssh. zavedenii, Mat., No. 5, 6, 126-157, is included although it does not refer to Pontryagin’s principle.
284
Book
reviews
As far as possible the editor of the book has united the content of the various chapters, made cross references and compiled a detailed index. Nevertheless. the different chapters are not written with the same degree of thoroughness. Thus the first seven chapters, of greatest interest to mathematicians, are written in more detail and can be read with interest. Chapters S-10, also of great interest to mathematici~s, are written more concisely and with numerous references to other published works. The last chapters, devoted to the mechanics of aerospace systems, are basically intended for engineers. Let us discuss briefly the contents of the book. Chapter 1 (by T.N. Edelbaum) is devoted to the general theory of maxima and minima for functions of many variables and for functionals. In Chapter 2 (F.D. Foulkner) certain numerical methods (which the author calls direct methods) for determining optimal rocket trajectories are described. Chapter 3 (A. Miele) is devoted to the method of finding the extremumof integrals of linear functions with the aid of Green’s theorem. Chapter 4 (A. Hiele) is called “Variational calculus in applied aerodynamics and the mechanics of flight”. This long chapter, whose contents are reflected in its title, is of the greatest interest both to mathematicians and to engineers. The article ends with a most valuable bibliograp~ (158 titles). XIIChapter 5 (G. Leitmann) there is a description of methods for solving variational problems with constraints on the variable CODtrols. based on classical methods of variational calculus. The very detailed and interesting Chapter 8 (XJ. Kelley) is concerned with the gradient method. In Chapter 7 (E. Kopp) the essence of Pontryagin’s maximumprinciple is explained and the connection between it and the method of dynamic programming and the classical variational calculus is elucidated. In Chapter 8 (R. Bellman) there is a brief explanation of the computational potentialities of the dynamic programming method and Chapter 9 (R. Kalaba) is devoted to numerical methods for the solution of problems of deterministic and adaptive control processes. Chapter 10 (CM. Kashmar, E.L. Peterson) is devoted to the so-called inclusion method for the solution of problems of the optimization of control processes, which consists in the fact that the given problem with boundary conditions at the start and the end of the interval is transformed to a more general variational problem with conditions at the start of the interval only. Existing methods (such as dynamic programming) are applied to the latter. Finally, Chapter 11 (D.F. Lawden). Chapter 12 (J. Breakwell), Chapter 13 (G. Leitmann). Chapter 14 (R.W. Bussard) are devoted to various problems of the mechanics of aerospace systems and are of a more applied nature.
Book review
As a whole, the book, which contains important surveys of many applied questions of variational calculus, is carefully written, edited and beautifully produced. It is a valuable textbook for a large Circle of readers, mathematicians, mechanics and engineers.
M.K. Kerimov B.V. DEAN, M.W. SASIENI and S.K. QUPTA, Mathematics for Modern knagement, John Wiley and Sons, Inc., New York-London, 1963. This book is a mathematics textbook for first year student8 specializing in business management. It could be useful for mathematics teacher8 in economics and engineering educational institutes as an example of the successful combination of the necessary mathematical information and the special problems of simulating economic phenomena. Comparatively little attention is paid in the book to question8 of Classical analysis. but space ha8 been found for such important questions for the economist as linear and dynamic programming, the basic concepts of probability theory, mathematical models of economic phenomena. There is a characteristically large number of examples (of which many are numerical), graphs and exercises. The main heading8 give a clear idea of the structure of the book. Part 1. Fundamental concepts (sets and relations; functional representation; series, limits and continuity; analytic geometry in a plane). Part 2. Deterministic systems (the derivative; the optimization QrOblem; Taylor's theorem and its applications; deterministic economic models). Part 3. Stochastic systems (integration; differential equations; probability theory and stochastic models). Part 4. Linear systems (linear algebra; linear programming - the simplex method and the transportation problem). Part 5. The mathematics of finance (Problem8 associated with the analysis of capital investment; questions of insurance and the problem of replacement of equipment involving the use of probability theory).
I.V. Romanovekii
the sum representational 1963.
method”
263
reviews
by N.N. Lyashko,
Izd-vo
Kievskogo
un-ta,
B.M. Budak 0. LEITMANN, (Ed.), optimization Acad. Press, space Systems,
Techniques with Applications New-York, London, 1962.
to Aero-
This book, which is already widely known amongst mathematical specialists as well as mechanics and engineers dealing with aerospace systems, is a clear example of how one of the oldest subjects in mathematical analysis, variational calculus, finds new spheres of application. Long ago variational calculus was applied in many spheres of science and technology (geometry, physics, mechanics, etc.). Now new ones have been added to these: the theory of aerospace systems, automatic control, mathematical economics, etc. These applications have proved most fruitful both for the new technological disciplines and for variational calculus itself. In various countries (particularly in the USSR and the USA) there have appeared new directions for variational calculus over the past few years - the theory of optimal processes and dynamic programming and these continue to be developed intensively. These new directions are of especial value from the point of view of their applications. In this respect they have turned out to be more profitable than classical variational calculus. The present book is basically concerned with these new developments in variational calculus and their applications to the theory of aerospace systems. The book makes effective uses of (1) classical methods of variational calculus, improved for problems of rocket technology; (2) the maximum principle of L.S. Pontryagin; (3) Bellman’s dynamic programming method; (4) gradient methods, The book consists of fourteen chapters written by leading American experts. The account is in the nature of a survey; with rare exceptions proofs are not given, but there are references to the rich bibliography at the end of each chapter. There are many Soviet works in the bibliography. It must be said that the bibliographies for certain articles are extensive and contain many very valuable sources (although these are also not very accessible), published in the form of reports of various departments. At times lack of care is evidenced in the selection of the bibliography of Soviet authors. For example, in the bibliography for Chapter ‘7. “The maximum principle of Pontryagin” (by R. Coney) the article by A.D. Aleksandrov, “Research on the maximum principle”, Izv. vyssh. zavedenii, Mat., No. 5, 6, 126-157, is included although it does not refer to Pontryagin’s principle.
284
Book
reviews
As far as possible the editor of the book has united the content of the various chapters, made cross references and compiled a detailed index. Nevertheless. the different chapters are not written with the same degree of thoroughness. Thus the first seven chapters, of greatest interest to mathematicians, are written in more detail and can be read with interest. Chapters S-10, also of great interest to mathematici~s, are written more concisely and with numerous references to other published works. The last chapters, devoted to the mechanics of aerospace systems, are basically intended for engineers. Let us discuss briefly the contents of the book. Chapter 1 (by T.N. Edelbaum) is devoted to the general theory of maxima and minima for functions of many variables and for functionals. In Chapter 2 (F.D. Foulkner) certain numerical methods (which the author calls direct methods) for determining optimal rocket trajectories are described. Chapter 3 (A. Miele) is devoted to the method of finding the extremumof integrals of linear functions with the aid of Green’s theorem. Chapter 4 (A. Hiele) is called “Variational calculus in applied aerodynamics and the mechanics of flight”. This long chapter, whose contents are reflected in its title, is of the greatest interest both to mathematicians and to engineers. The article ends with a most valuable bibliograp~ (158 titles). XIIChapter 5 (G. Leitmann) there is a description of methods for solving variational problems with constraints on the variable CODtrols. based on classical methods of variational calculus. The very detailed and interesting Chapter 8 (XJ. Kelley) is concerned with the gradient method. In Chapter 7 (E. Kopp) the essence of Pontryagin’s maximumprinciple is explained and the connection between it and the method of dynamic programming and the classical variational calculus is elucidated. In Chapter 8 (R. Bellman) there is a brief explanation of the computational potentialities of the dynamic programming method and Chapter 9 (R. Kalaba) is devoted to numerical methods for the solution of problems of deterministic and adaptive control processes. Chapter 10 (CM. Kashmar, E.L. Peterson) is devoted to the so-called inclusion method for the solution of problems of the optimization of control processes, which consists in the fact that the given problem with boundary conditions at the start and the end of the interval is transformed to a more general variational problem with conditions at the start of the interval only. Existing methods (such as dynamic programming) are applied to the latter. Finally, Chapter 11 (D.F. Lawden). Chapter 12 (J. Breakwell), Chapter 13 (G. Leitmann). Chapter 14 (R.W. Bussard) are devoted to various problems of the mechanics of aerospace systems and are of a more applied nature.
Book review
As a whole, the book, which contains important surveys of many applied questions of variational calculus, is carefully written, edited and beautifully produced. It is a valuable textbook for a large Circle of readers, mathematicians, mechanics and engineers.
M.K. Kerimov B.V. DEAN, M.W. SASIENI and S.K. QUPTA, Mathematics for Modern knagement, John Wiley and Sons, Inc., New York-London, 1963. This book is a mathematics textbook for first year student8 specializing in business management. It could be useful for mathematics teacher8 in economics and engineering educational institutes as an example of the successful combination of the necessary mathematical information and the special problems of simulating economic phenomena. Comparatively little attention is paid in the book to question8 of Classical analysis. but space ha8 been found for such important questions for the economist as linear and dynamic programming, the basic concepts of probability theory, mathematical models of economic phenomena. There is a characteristically large number of examples (of which many are numerical), graphs and exercises. The main heading8 give a clear idea of the structure of the book. Part 1. Fundamental concepts (sets and relations; functional representation; series, limits and continuity; analytic geometry in a plane). Part 2. Deterministic systems (the derivative; the optimization QrOblem; Taylor's theorem and its applications; deterministic economic models). Part 3. Stochastic systems (integration; differential equations; probability theory and stochastic models). Part 4. Linear systems (linear algebra; linear programming - the simplex method and the transportation problem). Part 5. The mathematics of finance (Problem8 associated with the analysis of capital investment; questions of insurance and the problem of replacement of equipment involving the use of probability theory).
I.V. Romanovekii