Optimization Theory and Applications. by Lamberto CesariReview by: E. O. RoxinSIAM Review, Vol. 26, No. 3 (Jul., 1984), pp. 441-443Published by: Society for Industrial and Applied MathematicsStable URL: http://www.jstor.org/stable/2031292 .

Accessed: 14/06/2014 17:06

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact support@jstor.org.

.

Society for Industrial and Applied Mathematics is collaborating with JSTOR to digitize, preserve and extendaccess to SIAM Review.

http://www.jstor.org

This content downloaded from 62.122.77.48 on Sat, 14 Jun 2014 17:06:10 PMAll use subject to JSTOR Terms and Conditions

BOOK REVIEWS 441

Thus the present book is truly original, presenting a novel set of ideas of considerable importance. Unfortunately, its execution is disappointing. The book seems more a succession of examples than a unified treatise, exhibiting considerable repetition but not much depth. Various small points of wording, notation, mathematical precision, and graphics have been carelessly handled. The design of the whole looks rather like a somewhat hasty concatenation of Vichnevetsky's earlier papers.

A more serious weakness in this work is that it communicates very little perspective about its subject matter. The authors spend too much time with the particular example of the space-centered, semi-discrete model of u1 = cur, drawing specific conclusions that obscure the generality of what they are doing. It is particularly distressing that they repeatedly fail to connect their ideas to related topics in the numerical analysis literature. The following are some topics intimately related to the themes of this book, but which the authors completely fail to mention:

* "Modified equations" for asymptotic analysis of difference formulas; * Quantitative mathematical results on numerical oscillations about discontinuities; * Periodic dispersion relations as used in solid state physics for analyzing sound and light vibrations in crystals;

* "Absorbing" or "radiation" boundary conditions for artificial boundaries; * Stability for discrete models of initial boundary value problems.

In failing to apply their ideas to shed new light on established subjects such as these, Vichnevetsky and Bowles have missed the opportunity to make their book a truly major contribution to numerical analysis. In failing to even give appropriate references, they have limited its value as a text.

To summarize, I encourage anyone working with computational methods for partial differential equations to take a look at this book for a valuable exposure to the fascinating phenomena of numerical wave propagation. But in this book the "waves" style of Fourier analysis of numerical methods has not yet reached its culmination.

LLOYD N. TREFETHEN Courant Institute of Mathematical Sciences

Optimization Theory and Applications. By LAMBERTO CESARI. Springer-Verlag, New York, 1983. xiv + 542 pp. $68.00. ISBN 0-387-90676-2. Modern optimal control theory was developed in the early 1950's as can be seen from

the dates of the published pioneering research (Bellman et al. (1956) [1], Bushaw (thesis 1953, paper in the Lefschetz series [9], 1958)[2], Flugge Lotz (1953)[3], Gamkrelidze (1958)[4], Hestenes (1950)[5], Krasovskii (1957)[6], LaSalle (1959)[7], Lefschetz (1950)[9], Pontryagin et al. (1960)[10], Wazewski (1961)[13]). In the 1960's it became sufficiently formalized so that complete books on the subject started to appear. It was, of course, realized that optimal control theory is a part of the calculus of variations, a subject almost as old as the infinitesimal calculus itself, but the relation between the classical and the modern theory was not at all clear at the beginning. Indeed, the classical calculus of variations treats mainly "interior" maxima and minima, while the modern version puts much more emphasis on those extrema which lie on the boundary of some constraint set. Therefore equations become inequalities, the classical results have to be changed appropriately and the need arises to use different methods of approach. A book representing these new methods in all their power and elegance is Lee and Markus [8].

The gap between the classical and the modern approach was clearly bridged in the research literature, but most books on the subject still remained on one or the other side of

This content downloaded from 62.122.77.48 on Sat, 14 Jun 2014 17:06:10 PMAll use subject to JSTOR Terms and Conditions

442 BOOK REVIEWS

this gap. The book under review fills precisely this need. It gives a thorough and detailed presentation of both the classical and modern formulations, the different methods and the corresponding results. It carefully discusses their relations and successfully accomplishes presenting the whole as a unified body of knowledge.

At the start is the "classical" calculus of variations. Necessary and sufficient conditions of different type characterizing extrema are presented (Euler, Jacobi, Weier- strass, Du Bois, Reymond). In the great tradition of V. Volterra [12] and L. Tonelli [1 1], convexity and lower semicontinuity of functionals are discussed. The order of presentation is as follows: first the statement of the main results and a pertinent basic discussion; then examples of applications, very interesting in themselves, which are solved using the preceding results; finally, the complete proofs. This order is very convenient to avoid overwhelming the reader with technicalities before he firmly grasps the ideas. The examples treated are particularly useful, as they are discussed thoroughly (for example, in the problem of a surface of revolution of minimal area, the conditions for the solution to be smooth or not are carefully investigated). Applications to physics are well represented: Fermat's principle in optics is proved and the Lagrangian formulation of the laws of mechanics are presented in detail (here the "Hamiltonian" appears in the classical formulation, while it will be found again within the framework of the modern optimal control theory).

Next come the Mayer, Lagrange and Bolza problems (variational problems with differential equations as constraints), and with them the modern approach to optimal control. Again a list of good examples is given and discussed.

After giving all the material on necessary and sufficient conditions, the second half of the book is mainly devoted to existence problems for optimal solutions. An introduction (Chapter 8) presents the implicit function and the elementary closure theorems within the context of general linear or Banach spaces. Topics like upper semicontinuity for set-valued functions are carefully presented.

After giving Filippov's theorem on existence of optimal controls, the book presents the parallel formulation by "orientor fields" (also known as "contingent equations" or, nowadays, as "differential inclusions"). After that comes the generalization, by the author, of assumptions guaranteeing existence, where the convexity hypotheses are relaxed and substituted by the "property Q." The subject becomes highly technical, considering different types of assumptions, all within the framework of general functional analysis.

Existence theorems without convexity assumptions, duality and upper semicontinu- ity of set-valued functions, and approximations of usual and of generalized solutions are the titles of the last three chapters.

The reader may feel somewhat disappointed to find that topics which are quite near to the main subject, as for example optimal control of systems governed by functional differential equations, or by partial differential equations, or by stochastic differential equations, are not covered. But the reason for this omission is quite clear: in order to cover them with similar thoroughness, another volume of about 500 pages would be needed, and this was obviously not the plan.

In the opinion of the reviewer, this book is an excellent addition to the existing literature. As a reference manual for the subject, it is very useful, as it presents different kinds of approaches and their relations, which are often quite confusing. But this book is 'not addressed only to the specialists. It could be used very well for a graduate level course, provided a judicious selection is made on which sections to cover. Of course, the whole body of material in all its details could not be covered even in a one-year course. But this is

This content downloaded from 62.122.77.48 on Sat, 14 Jun 2014 17:06:10 PMAll use subject to JSTOR Terms and Conditions

BOOK REVIEWS 443

precisely the attractiveness: it leaves room for personal taste and changes in selecting topics for such a course or seminar.

REFERENCES

[1] R. BELLMAN, I. GLICKSBERG AND 0. GROSS, On the "bang-bang" control problem, Quart. Appl. Math., 14 (1956), pp. 11-18.

[2] D. W. BUSHAW, Optimal discontinuous forcing terms (thesis, 1953), Paper in Contributions to the Theory of Nonlinear Oscillations IV, 29-52, Annals of Math. Studies, Princeton Univ. Press, Princeton, NJ, 1958.

[3] I. FLUGGE LOTZ, Discontinuous Automatic Control, Princeton Univ. Press, Princeton, NJ, 1953. [4] R. V. GAMKRELIDZE, The theory of time optimal processes for linear systems, Izv. Akad. Nauk USSR,

Ser. Math, 22 (1958) pp. 449-474 (In Russian.) [5] M. R. HESTENES, A general problem in the calculus of variations with applications to paths of least time,

RAND Corp. Report No. 100, 1950. [6] N. N. KRASOVSKII, On the theory of optimal controls, Avtomat. Telemek., 18 (1957), pp. 960-970. (In

Russian.) [7] J. P. LASALLE, Time optimal control systems, Proc. Nat. Acad. Sci. U. S., 45 (1959), pp. 537-577. [8] E. B. LEE and L. MARKUS, Foundations of Optimal Control Theory, John Wiley, New York 1967. [9] S. LEFSCHETZ, ed., Contributions to the Theory of Nonlinear Oscillation, I (1950), II, III, IV, V (1960),

Annals of Math. Studies, vols. 20,29, 36, 41,45, Princeton Univ. Press, Princeton, NJ. [10] L. S. PONTRYAGIN, V. G. BOLTYANSKII, R. V. GAMKRELIDZE AND E. F. MISHCHENKO, The Mathematical

Theory of Optimal Processes, Russian 1956, Engl. transl., Wiley Interscience, New York, 1962. [11] L. TONELLI, Fondamenti di calcolo delle variazioni, vols. 1-2 Zanichelli, Roma, 1921-23. [12] V. VOLTERRA, Leqons sur lesfonctions de ligne, Gauthier Villars, Paris, 1913. [13] T. WAZEWSKI, Systemes de commande et equations au contingent, Bull. Acad. Polon. Sci., 9, (1961), pp.

151-155.

E. 0. ROXIN University of Rhode Island

Nonlinear Time-Discrete Systems. By M. GOSSEL. Springer-Verlag, Berlin, 1982. 109 pp. $8.00. Paper. ISBN 3-540-11914-0. The purpose of this book is to extend some of the basic principles of linear systems

theory to nonlinear systems. It is well known that linear systems can be identified by knowing only their impulse response. Critical to this is that linear systems satisfy a superposition principle with respect to addition (for example, a linear combination of two solutions of a linear discrete or differential system is again a solution). There exists no such general phenomenon in the analysis of nonlinear systems. Realizing that most models applied to phenomena in engineering and the physical sciences are linear in nature despite the fact that the phenomena may be nonlinear, scientists might benefit greatly from a theory on nonlinear systems which will reflect some of the important aspects of linear systems such as the superposition principles. Attempts have already been made in differential equations by W. Ames and others, but the work is in its infancy.

In this book a development of the theory of nonlinear superposition is begun for time-discrete systems. Automata theory, including linear and nonlinear automata is the setting for the analysis of the time-discrete systems. A brief review of automata theory is given with a particularly detailed treatment, including examples of the superposition principle of linear time-dependent and nontime-dependent linear automata. This superpo- sition principle is then extended to a pair of operations (0, V) (the usual superposition principle occurs when 0 = V = +) for general automata (not necessarily linear ones). Such automata are said to be (0, V)-superponable and it is shown how the input-output behavior of (0, V)-superponable automata can be described by means of generalized

This content downloaded from 62.122.77.48 on Sat, 14 Jun 2014 17:06:10 PMAll use subject to JSTOR Terms and Conditions