Springer Series in Soviet Mathematics – serie

This book is devoted to an investigation of control problems which can be described by ordinary differential equations and be expressed in terms of game theoretical notions. In these terms, a strategy is a control based on the feedback principle which will assure a definite equality for the controlled process which is subject to uncertain factors such as a move or a controlling action of the opponent. Game Theoretical Control Problems contains definitions and formalizations of differential games, existence for equilibrium and extensive discussions of optimal strategies. Formal definitions and statements are accompanied by suitable motivations and discussions of computational algorithms. The book is addessed to mathematicians, engineers, economists and other users of control theoretical and game theoretical notions.

Lev Semenovic Pontrjagin (1908) is one of the outstanding figures in 20th century mathematics. In a long career he has made fundamental con­ tributions to many branches of mathematics, both pure and applied. He has received every honor that a grateful government can bestow. Though in no way constrained to do so, he has through the years taught mathematics courses at Moscow State University. In the year 1975 he set himself the task of writing a series of books on secondary school and beginning university mathematics. In his own words, "I wished to set forth the foundations of higher mathematics in a form that would have been accessible to myself as a lad, but making use of all my experience as a scientist and a teacher, ac­ cumulated over many years. " The present volume is a translation of the first two out of four moderately sized volumes on this theme planned by Pro­ fessor Pontrjagin. The book begins at the beginning of modern mathematics, analytic ge­ ometry in the plane and 3-dimensional space. Refinements about limits and the nature of real numbers come only later. Many concrete examples are given; these may take the place of formal exercises, which the book does not provide. The book continues with careful treatment of differentiation and integration, of limits, of expansions of elementary functions in power se­ ries.

The proposed book is one of a series called "A Course of Higher Mathematics and Mathematical Physics" edited by A. N. Tikhonov, V. A. Ilyin and A. G. Sveshnikov. The book is based on a lecture course which, for a number of years now has been taught at the Physics Department and the Department of Computational Mathematics and Cybernetics of Moscow State University. The exposition reflects the present state of the theory of differential equations, as far as it is required by future specialists in physics and applied mathematics, and is at the same time elementary enough. An important part of the book is devoted to approximation methods for the solution and study of differential equations, e.g. numerical and asymptotic methods, which at the present time play an essential role in the study of mathematical models of physical phenomena. Less attention is paid to the integration of differential equations in elementary functions than to the study of algorithms on which numerical solution methods of differential equations for computers are based.

This book presents Maslov's canonical operator method for finding asymptotic solutions of pseudo differential equations. The classical WKB method, so named in honor of its authors: Wentzel, Kramers and Brillouin, was created for finding quasi classical approximations in quantum mechanics. The simplicity, obviousness and "physicalness" of this method quickly made it popular: specialists in mathematical physics accepted it unequivocally as one of the weapons in their arsenal. The number of publications which are connected with the WKB method in one way or another can probably no longer be counted. The alternative name of the WKB method in diffraction problem- the ray method or the method of geometric optics - indicates that the approximations in the WKB method are constructed by means of rays. More precisely, the first approximation of the WKB method is constructed by means of rays (isolating the singular part), after which the usual methods of the (regular) theory of perturbations are applied. However, the ray method is not applicable at the points of space where the rays focus or form a caustic. Mathematically this fact expresses itself in the fact that the amplitude of the waves at such points become infinite.

The method of normal forms is usually attributed to Poincaré although some of the basic ideas of the method can be found in earlier works of Jacobi, Briot and Bouquet. In this book, A.D.Bruno gives an account of the work of these mathematicians and further developments as well as the results of his own extensive investigations on the subject. The book begins with a thorough presentation of the analytical techniques necessary for the implementation of the theory as well as an extensive description of the geometry of the Newton polygon. It then proceeds to discuss the normal form of systems of ordinary differential equations giving many specific applications of the theory. An underlying theme of the book is the unifying nature of the method of normal forms regarding techniques for the study of the local properties of ordinary differential equations. In the second part of the book it is shown, for a special class of equations, how the method of normal forms yields classical results of Lyapunov concerning families of periodic orbits in the neighborhood of equilibrium points of Hamiltonian systems as well as the more modern results concerning families of quasiperiodic orbits obtained by Kolmogorov, Arnold and Moser. The book is intended for mathematicians, theoretical mechanicians, and physicists. It is suitable for advanced undergraduate and graduate students.

Homogeneous cosmological models, self-similar motion of self-gravitating gas and motion of gas with homogeneous deformation have important applica­ tions in the theory of evolution of the universe. In particular they can be applied to the theory of explosions of stars, formation of galaxies, pulsation of alternating stars etc. The equations of general relativity and Newtonian gas dynamics in the cases mentioned above are reduced to systems of a finite (but quite large) number of ordinary differential equations. In the last two decades these multi-dimensional dynamical systems were and still are being analyzed by means of traditional analytic and numerical methods. Important dynamical modes of some solutions were thus established. These include oscillatory modes of the space-time metric near a cosmological singularity, self-similar motion of self-gravitating gas with a shock wave and an expanding cavity inside (as in an explosion of a star), collapse of an ellipsoid of self-gravitating dust into a disc and others. However the multi­ dimensional dynamical systems in question are so complex, that a complete analysis of all dynamical modes of the solutions by means of well-known tra­ ditional analytic methods does not seem feasible. Therefore the development of effective methods of qualitative analysis of multi-dimensional dynamical systems and their application to the problems of astrophysics and gas dynamics previ­ ously unsolved by traditional methods becomes especially urgent.

Let me begin by explaining the meaning of the title of this book. In essence, the book studies boundary value problems for linear partial differ­ ential equations in a finite domain in n-dimensional Euclidean space. The problem that is investigated is the question of the dependence of the nature of the solvability of a given equation on the way in which the boundary conditions are chosen, i.e. on the supplementary requirements which the solution is to satisfy on specified parts of the boundary. The branch of mathematical analysis dealing with the study of boundary value problems for partial differential equations is often called mathematical physics. Classical courses in this subject usually consider quite restricted classes of equations, for which the problems have an immediate physical context, or generalizations of such problems. With the expanding domain of application of mathematical methods at the present time, there often arise problems connected with the study of partial differential equations that do not belong to any of the classical types. The elucidation of the correct formulation of these problems and the study of the specific properties of the solutions of similar equations are closely related to the study of questions of a general nature.

Today, the theory of random processes represents a large field of mathematics with many different branches, and the task of choosing topics for a brief introduction to this theory is far from being simple. This introduction to the theory of random processes uses mathematical models that are simple, but have some importance for applications. We consider different processes, whose development in time depends on some random factors. The fundamental problem can be briefly circumscribed in the following way: given some relatively simple characteristics of a process, compute the probability of another event which may be very complicated; or estimate a random variable which is related to the behaviour of the process. The models that we consider are chosen in such a way that it is possible to discuss the different methods of the theory of random processes by referring to these models. The book starts with a treatment of homogeneous Markov processes with a countable number of states. The main topic is the ergodic theorem, the method of Kolmogorov's differential equations (Secs. 1-4) and the Brownian motion process, the connecting link being the transition from Kolmogorov's differential-difference equations for random walk to a limit diffusion equation (Sec. 5).