Berkovitz, Leonard David 1924
Overview
Works:  23 works in 83 publications in 1 language and 1,502 library holdings 

Genres:  Biography 
Roles:  Author, Other 
Publication Timeline
.
Most widely held works by
Leonard David Berkovitz
Optimal control theory by
Leonard David Berkovitz(
Book
)
19 editions published between 1974 and 2010 in English and Undetermined and held by 588 WorldCat member libraries worldwide
This book is an introduction to the mathematical theory of optimal control of processes governed by ordinary differential eq tions. It is intended for students and professionals in mathematics and in areas of application who want a broad, yet relatively deep, concise and coherent introduction to the subject and to its relati ship with applications. In order to accommodate a range of mathema cal interests and backgrounds among readers, the material is arranged so that the more advanced mathematical sections can be omitted wi out loss of continuity. For readers primarily interested in appli tions a recommended minimum course consists of Chapter I, the sections of Chapters II, III, and IV so recommended in the introductory sec tions of those chapters, and all of Chapter V. The introductory sec tion of each chapter should further guide the individual reader toward material that is of interest to him. A reader who has had a good course in advanced calculus should be able to understand the defini tions and statements of the theorems and should be able to follow a substantial portion of the mathematical development. The entire book can be read by someone familiar with the basic aspects of Lebesque integration and functional analysis. For the reader who wishes to find out more about applications we recommend references [2], [13], [33], [35], and [50], of the Bibliography at the end of the book
19 editions published between 1974 and 2010 in English and Undetermined and held by 588 WorldCat member libraries worldwide
This book is an introduction to the mathematical theory of optimal control of processes governed by ordinary differential eq tions. It is intended for students and professionals in mathematics and in areas of application who want a broad, yet relatively deep, concise and coherent introduction to the subject and to its relati ship with applications. In order to accommodate a range of mathema cal interests and backgrounds among readers, the material is arranged so that the more advanced mathematical sections can be omitted wi out loss of continuity. For readers primarily interested in appli tions a recommended minimum course consists of Chapter I, the sections of Chapters II, III, and IV so recommended in the introductory sec tions of those chapters, and all of Chapter V. The introductory sec tion of each chapter should further guide the individual reader toward material that is of interest to him. A reader who has had a good course in advanced calculus should be able to understand the defini tions and statements of the theorems and should be able to follow a substantial portion of the mathematical development. The entire book can be read by someone familiar with the basic aspects of Lebesque integration and functional analysis. For the reader who wishes to find out more about applications we recommend references [2], [13], [33], [35], and [50], of the Bibliography at the end of the book
Convexity and optimization in R [superscript n] by
Leonard David Berkovitz(
Book
)
17 editions published between 2001 and 2002 in English and Undetermined and held by 240 WorldCat member libraries worldwide
"This book presents the mathematics of finite dimensional constrained optimization problems. It provides a basis for the further mathematical study of convexity, of more generalized optimization problems, and of numerical algorithms for the solution of finite dimensional optimization problems. For readers who do not have the requisite background in real analysis, the author provides a chapter covering this material. The text features abundant exercises and problems designed to lead the reader to a fundamental understanding of the material." "A detailed bibliography is included for further study and an index offers quick reference. Suitable as a text for both graduate and undergraduate students in mathematics and engineering, this accessible text is written from extensively classtested notes."Jacket
17 editions published between 2001 and 2002 in English and Undetermined and held by 240 WorldCat member libraries worldwide
"This book presents the mathematics of finite dimensional constrained optimization problems. It provides a basis for the further mathematical study of convexity, of more generalized optimization problems, and of numerical algorithms for the solution of finite dimensional optimization problems. For readers who do not have the requisite background in real analysis, the author provides a chapter covering this material. The text features abundant exercises and problems designed to lead the reader to a fundamental understanding of the material." "A detailed bibliography is included for further study and an index offers quick reference. Suitable as a text for both graduate and undergraduate students in mathematics and engineering, this accessible text is written from extensively classtested notes."Jacket
Nonlinear optimal control theory by
Leonard David Berkovitz(
Book
)
13 editions published between 2011 and 2013 in English and held by 80 WorldCat member libraries worldwide
"Preface This book is an introduction to the mathematical theory of optimal control of processes governed by ordinary differential and certain types of differential equations with memory. The book is intended for students, mathematicians, and those who apply the techniques of optimal control in their research. Our intention is to give a broad, yet relatively deep, concise and coherent introduction to the subject. We have dedicated an entire chapter for examples. We have dealt with the examples pointing out the mathematical issues that one needs to address. The first six chapters can provide enough material for an introductory course in optimal control theory governed by differential equations. Chapters 3, 4, and 5 could be covered with more or less details in the mathematical issues depending on the mathematical background of the students. For students with background in functional analysis and measure theory Chapter 7 can be added. Chapter 7 is a more mathematically rigorous version of the material in Chapter 6. We have included material dealing with problems governed by integrodifferential and delay equations. We have given a unified treatment of bounded state problems governed by ordinary, integrodifferential, and delay systems. We have also added material dealing with the HamiltonJacobi Theory. This material sheds light on the mathematical details that accompany the material in Chapter 6"
13 editions published between 2011 and 2013 in English and held by 80 WorldCat member libraries worldwide
"Preface This book is an introduction to the mathematical theory of optimal control of processes governed by ordinary differential and certain types of differential equations with memory. The book is intended for students, mathematicians, and those who apply the techniques of optimal control in their research. Our intention is to give a broad, yet relatively deep, concise and coherent introduction to the subject. We have dedicated an entire chapter for examples. We have dealt with the examples pointing out the mathematical issues that one needs to address. The first six chapters can provide enough material for an introductory course in optimal control theory governed by differential equations. Chapters 3, 4, and 5 could be covered with more or less details in the mathematical issues depending on the mathematical background of the students. For students with background in functional analysis and measure theory Chapter 7 can be added. Chapter 7 is a more mathematically rigorous version of the material in Chapter 6. We have included material dealing with problems governed by integrodifferential and delay equations. We have given a unified treatment of bounded state problems governed by ordinary, integrodifferential, and delay systems. We have also added material dealing with the HamiltonJacobi Theory. This material sheds light on the mathematical details that accompany the material in Chapter 6"
A dynamic programming approach to the nonparametric problem in the calculus of variations by
Leonard David Berkovitz(
Book
)
2 editions published in 1964 in English and held by 15 WorldCat member libraries worldwide
This memorandum presents an approach to the nonparametric simple problem of the calculus of variations that is in the spirit of dynamic programming. Under the assumptions that the functions that define minimizing curves exhibit a certain regularity in their behavior, differentiability properties of the optimal value function W are determined. Then the fundamental partial differential equation of dynamic programming is obtained, and the necessary conditions of the calculus of variations, with the exception of Jacobi's condition, are deduced as immediate corollaries. (Author)
2 editions published in 1964 in English and held by 15 WorldCat member libraries worldwide
This memorandum presents an approach to the nonparametric simple problem of the calculus of variations that is in the spirit of dynamic programming. Under the assumptions that the functions that define minimizing curves exhibit a certain regularity in their behavior, differentiability properties of the optimal value function W are determined. Then the fundamental partial differential equation of dynamic programming is obtained, and the necessary conditions of the calculus of variations, with the exception of Jacobi's condition, are deduced as immediate corollaries. (Author)
A variational approach to differential games by
Leonard David Berkovitz(
Book
)
5 editions published between 1960 and 1961 in English and held by 14 WorldCat member libraries worldwide
An investigation, derived from the calculus of variations, ofa class of differential games having pure strategy solutions. These games are related to two Bolza problems with differential inequalities as added side constraints. Necessary conditions that must hold along an optimal path are derived from the theory of the related Bolza problems. These conditions are (1) a multiplier ruls, together with transversality conditions and jump contitions; (2) a local minmax condition that is related to the Weierstrass cindition; and (3) an analogue of the Clebsch condition. The continuity and differentiability properties of the value of the game are derived. It is shown that, wherever the value is differentiable, it satisfies an analogue of the HamiltonJacobi equation. Sufficient conditions are given interms of the notion of a field and of a local minmax condition
5 editions published between 1960 and 1961 in English and held by 14 WorldCat member libraries worldwide
An investigation, derived from the calculus of variations, ofa class of differential games having pure strategy solutions. These games are related to two Bolza problems with differential inequalities as added side constraints. Necessary conditions that must hold along an optimal path are derived from the theory of the related Bolza problems. These conditions are (1) a multiplier ruls, together with transversality conditions and jump contitions; (2) a local minmax condition that is related to the Weierstrass cindition; and (3) an analogue of the Clebsch condition. The continuity and differentiability properties of the value of the game are derived. It is shown that, wherever the value is differentiable, it satisfies an analogue of the HamiltonJacobi equation. Sufficient conditions are given interms of the notion of a field and of a local minmax condition
Allocation of two types of aircraft in tactical air war : a gametheoretic analysis by
Leonard David Berkovitz(
Book
)
3 editions published between 1959 and 1960 in English and held by 9 WorldCat member libraries worldwide
An analysis, as a twosided war game, of the problem of allocating two types of aircraft (bombers and fighters) among three different air tasks (counter air, air defense, and support of ground operations) in a multistrike campaign. It is assumed that a bomber can be used in either the counterair or groundsupport operations, while a fighter can be used in either the airdefense or groundsupport roles. That is, bombers and fighters have the one task of ground support in common. Optimal employment during the last strikes of the campaign consists in a concentration of all resources on support of ground operations. Optimal employment during the early strikes of the campaign requires randomization by both sides
3 editions published between 1959 and 1960 in English and held by 9 WorldCat member libraries worldwide
An analysis, as a twosided war game, of the problem of allocating two types of aircraft (bombers and fighters) among three different air tasks (counter air, air defense, and support of ground operations) in a multistrike campaign. It is assumed that a bomber can be used in either the counterair or groundsupport operations, while a fighter can be used in either the airdefense or groundsupport roles. That is, bombers and fighters have the one task of ground support in common. Optimal employment during the last strikes of the campaign consists in a concentration of all resources on support of ground operations. Optimal employment during the early strikes of the campaign requires randomization by both sides
Advances in game theory by
Melvin Dresher(
Book
)
1 edition published in 1971 in English and held by 8 WorldCat member libraries worldwide
The book description for the forthcoming "Advances in Game Theory. (AM52)" is not yet available
1 edition published in 1971 in English and held by 8 WorldCat member libraries worldwide
The book description for the forthcoming "Advances in Game Theory. (AM52)" is not yet available
Variational methods in problems of control and programming by
Leonard David Berkovitz(
Book
)
2 editions published in 1961 in English and held by 8 WorldCat member libraries worldwide
A paper which shows how a fairly general control problem, or programming problem, with constrints can be reduced to a special type of classical Bolza problem in the calculus of variations. Necessary conditions from the Bolza problem are translated into necessary conditions for optimal control. It is seen from these conditions that Pontryagin's maximum principle is a translation of the usual Weierstrass condition, and is applicable to a wider class of problems than that considreed by Pontryagin. The differentiability and continuity properties of the value of the control are estabilished under reasonable hypotheses on the synthesis, and it is shown that the value satisfies the HamiltonJacobi equation. As a consequence, a rigorous proof of a functional equation of Bellman is obtained that is valid for a much wider class of problems than hertofore considered. A sufficiency theorem for synthesis of control is also given
2 editions published in 1961 in English and held by 8 WorldCat member libraries worldwide
A paper which shows how a fairly general control problem, or programming problem, with constrints can be reduced to a special type of classical Bolza problem in the calculus of variations. Necessary conditions from the Bolza problem are translated into necessary conditions for optimal control. It is seen from these conditions that Pontryagin's maximum principle is a translation of the usual Weierstrass condition, and is applicable to a wider class of problems than that considreed by Pontryagin. The differentiability and continuity properties of the value of the control are estabilished under reasonable hypotheses on the synthesis, and it is shown that the value satisfies the HamiltonJacobi equation. As a consequence, a rigorous proof of a functional equation of Bellman is obtained that is valid for a much wider class of problems than hertofore considered. A sufficiency theorem for synthesis of control is also given
The equivalence of some necessary conditions for optimal control in problems with bounded state variables by
Leonard David Berkovitz(
Book
)
2 editions published in 1963 in English and held by 7 WorldCat member libraries worldwide
The problem of optimal control is considered as earlier studied separately by Gamkrelidze, Berkovitz, and Dreyfus wherein a constraint is placed on the state variables. The three have studied the problem from different viewpoints: Gamkrelidze accounts for the constraint by modifying Pontryagin's maximum principle arguments; Berkovitz applies directly the classical theory developed for the problem of Bolza; and Dreyfus utilizes a dynamic programming approach. The results deduced by the latter approach, in some respects, appear to be unrelated to the Gamkrelidze and Berkovitz results, which are in agreement. It is demonstrated that the two sets of results are actually related. (Author)
2 editions published in 1963 in English and held by 7 WorldCat member libraries worldwide
The problem of optimal control is considered as earlier studied separately by Gamkrelidze, Berkovitz, and Dreyfus wherein a constraint is placed on the state variables. The three have studied the problem from different viewpoints: Gamkrelidze accounts for the constraint by modifying Pontryagin's maximum principle arguments; Berkovitz applies directly the classical theory developed for the problem of Bolza; and Dreyfus utilizes a dynamic programming approach. The results deduced by the latter approach, in some respects, appear to be unrelated to the Gamkrelidze and Berkovitz results, which are in agreement. It is demonstrated that the two sets of results are actually related. (Author)
A survey of certain aspects of the mathematics of control problems by
Leonard David Berkovitz(
Book
)
2 editions published in 1962 in English and held by 7 WorldCat member libraries worldwide
2 editions published in 1962 in English and held by 7 WorldCat member libraries worldwide
A differential game without pure strategy solutions on an open set by
Leonard David Berkovitz(
Book
)
3 editions published in 1962 in English and held by 6 WorldCat member libraries worldwide
An example is given of a differential game with integral payoff and continuous terminal payoff that does not have a pure strategy solution. The methods and results of a previous approach (A Variational Approach to Differential Games, Rand Corp., Memorandum RM2772, 1961) are used to establish the nonexistence of pure strategies. (Author)
3 editions published in 1962 in English and held by 6 WorldCat member libraries worldwide
An example is given of a differential game with integral payoff and continuous terminal payoff that does not have a pure strategy solution. The methods and results of a previous approach (A Variational Approach to Differential Games, Rand Corp., Memorandum RM2772, 1961) are used to establish the nonexistence of pure strategies. (Author)
On control problems with bounded state variables by
Leonard David Berkovitz(
Book
)
2 editions published in 1962 in English and held by 6 WorldCat member libraries worldwide
Id ntifiers: Problem solving. Control problems are studied in which constraints that are independent of the control are imposed on the state variables. Necessary conditions that hold along an optimal trajectory are derived in a relatively simple fashion from known results in the calculus of variations. These results had been obtained by Gambrelidze in a different fashion. (Author)
2 editions published in 1962 in English and held by 6 WorldCat member libraries worldwide
Id ntifiers: Problem solving. Control problems are studied in which constraints that are independent of the control are imposed on the state variables. Necessary conditions that hold along an optimal trajectory are derived in a relatively simple fashion from known results in the calculus of variations. These results had been obtained by Gambrelidze in a different fashion. (Author)
A game theory analysis of tactical air war by
Leonard David Berkovitz(
Book
)
1 edition published in 1959 in English and held by 3 WorldCat member libraries worldwide
1 edition published in 1959 in English and held by 3 WorldCat member libraries worldwide
Edward James McShane : May 10, 1904June 1, 1989(
Book
)
1 edition published in 2001 in English and held by 3 WorldCat member libraries worldwide
1 edition published in 2001 in English and held by 3 WorldCat member libraries worldwide
Convexity and Optimization in Rn by
Leonard David Berkovitz(
)
1 edition published in 2003 in English and held by 2 WorldCat member libraries worldwide
1 edition published in 2003 in English and held by 2 WorldCat member libraries worldwide
On differential games with integral payoff(
Book
)
1 edition published in 1955 in English and held by 1 WorldCat member library worldwide
Methods of the calculus of variations were employed to obtain necessary conditions that differential games of a certain type have a saddlepoint. By strengthening the necessary conditions, sufficient conditions for the existence of a saddlepoint and a method of constructing the saddlepoint were derived. (Author)
1 edition published in 1955 in English and held by 1 WorldCat member library worldwide
Methods of the calculus of variations were employed to obtain necessary conditions that differential games of a certain type have a saddlepoint. By strengthening the necessary conditions, sufficient conditions for the existence of a saddlepoint and a method of constructing the saddlepoint were derived. (Author)
A multimove allocation game by
Leonard David Berkovitz(
Book
)
1 edition published in 1958 in English and held by 1 WorldCat member library worldwide
1 edition published in 1958 in English and held by 1 WorldCat member library worldwide
A penalty function proof of maximum principle by
Leonard David Berkovitz(
Book
)
1 edition published in 1975 in English and held by 1 WorldCat member library worldwide
1 edition published in 1975 in English and held by 1 WorldCat member library worldwide
The implications of some gametheoretic analyses for war gaming by
Leonard David Berkovitz(
Book
)
1 edition published in 1960 in English and held by 1 WorldCat member library worldwide
1 edition published in 1960 in English and held by 1 WorldCat member library worldwide
A MULTIMOVE INFINITE GAME WITH LINEAR PAYOFF(
)
1 edition published in 1958 in English and held by 0 WorldCat member libraries worldwide
This paper analyzes a multimove infinite game with linear payoff function. The game had its origin in the consideration of a military problem, but is presented here solely for its mathematical interest. It is symmetric in every respect except that the initial conditions of the two players are different. On each move, each player allocates his resources to tasks that might be described roughly as attacking, defending, and scoring. His resources for the next move are diminished by the amount that his opponent's attack exceeds his own defense, while his score cumulates from move to move. The value of the game and the optimal strategies for the players are rigorously derived in the present paper. It is shown that one player has a pure optimal strategy and the other player must randomize
1 edition published in 1958 in English and held by 0 WorldCat member libraries worldwide
This paper analyzes a multimove infinite game with linear payoff function. The game had its origin in the consideration of a military problem, but is presented here solely for its mathematical interest. It is symmetric in every respect except that the initial conditions of the two players are different. On each move, each player allocates his resources to tasks that might be described roughly as attacking, defending, and scoring. His resources for the next move are diminished by the amount that his opponent's attack exceeds his own defense, while his score cumulates from move to move. The value of the game and the optimal strategies for the players are rigorously derived in the present paper. It is shown that one player has a pure optimal strategy and the other player must randomize
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Air warfareMathematical models Automatic control Calculus of variations Canada College administrators College teachers Control theory Convex sets Editors Educational assistance Functions of bounded variation Functions of real variables Games of strategy (Mathematics) Game theory Industrial managementMathematical models Mathematical optimization Mathematicians Mathematics McShane, E. J.(Edward James), Nonlinear control theory Nonlinear theories Research Scientists United States War games War gamesMathematical models
Alternative Names
Berkovitz, L.D.
Berkovitz, L. D. 1924
Berkovitz, Leonard D.
Berkovitz, Leonard David
Berkovitz, Leonard David 1924
Leonard D. Berkovitz American mathematician
Leonard D. Berkovitz Amerikaans wiskundige (19242009)
Leonard D. Berkovitz matemático estadounidense
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