Initial-value problems in potential theory

Potential theory characteristically leads to partial differential equations under boundary conditions. It is shown that these boundary-value problems can be converted directly into equivalent initial-value problems. These initial-value problems are shown to lead to a number of standard and nonstandard numerical procedures.This research was supported by the National Institutes of Health under Grants Nos. GM-16197-01 and GM-16437-01 and by the Atomic Energy Commission under Contract No. AT(11-1)-113, Project No. 19.     

Identification of parameters in nonlinear boundary-value problems

This paper presents a method for identification of parameters in nonlinear boundary-value problems. The successive approximations technique proposed uses the theory of Lagrange multipliers and the Newton-Raphson method. This method does not require storage of functions and is quadratically convergent. Numerical results are presented.This research was sponsored by the National Institutes of Health, Grant No. GM-16197-01. Computing assistance was obtained from the Health Sciences Computing Facility, University of California at Los Angeles, NIH Grant No. FR-3.     

Exact solution of Pontryagin's equations of optimal control—Part 1

In the treatment of constrained optimal control processes, it is customary to employ the Pontryagin maximum principle, which requires the solution of a two-point boundary-value problem. Various economic, mechanical, and biological control processes are of this type, including optimization of hemodialysis. Generally speaking, two-point boundary-value problems are more difficult to treat computationally than initial-value or Cauchy problems. In this paper, a Cauchy system is derived for a class of optimal control processes, and it is then shown that the solution of the Cauchy problem satisfies the Pontryagin equations.This research was supported by the National Science Foundation, Grant No. GF-294, and the National Institutes of Health, Grants Nos. GM-16197-01 and GM-16437-01.     

Identification of linear systems using long periods of observation

A new method is presented for identifying parameters in a linear differential system arising, e.g., from compartment models in drug kinetics. The linearity of the system is used to produce a series of recurrence relations that help reduce the computational load. The method is especially useful when a long period of observation is used to identify the parameters. Numerical experiments are described.This research was sponsored by the National Institutes of Health, Grant No. GM-16197-01. Computing assistance was obtained from the Health Sciences Computing Facility, sponsored by NIH Grant No. FR-3.     

A sequential method for nonlinear filtering: Numerical implementation and comparisons

Exact equations are presented for sequentially updating the optimal solution for a discrete-time analog of the basic Sridhar nonlinear filtering problem as the process length increases and new observations are obtained. A tabular method is described for implementing numerically the sequential filtering equations. The accuracy and efficiency of the tabular method are illustrated by means of several numerical examples.Dedicated to R. SridharThe work of R. Kalaba and L. Tesfatsion was partially supported by the National Science Foundation under Grant No. ENG-77-28432 and by the National Institutes of Health under Grant No. GM-23732-03.     

Variational equations for the eigenvalues and eigenvectors of nonsymmetric matrices

The tracking of eigenvalues and eigenvectors for parameterized matrices is of major importance in optimization and stability problems. In the present paper, we consider a one-parameter family of matrices with distinct eigenvalues. A complete system of differential equations is developed for both the eigenvalues and the right and left eigenvectors. The computational feasibility of the differential system is demonstrated by means of a numerical example.The work of R. Kalaba and L. Tesfatsion was partially supported by the National Science Foundation under Grant No. ENG-77-28432 and by the National Institutes of Health under Grant No. GM-23732-03.     

Transcendental geometric programs

This paper treats a class of posynomial-like functions whose variables may appear also as exponents or in logarithms. It is shown that the resulting programs, called transcendental geometric programs, retain many useful properties of ordinary geometric programs, although the new class of problems need not have unique minima and cannot, in general, be transformed into convex programs. A duality theory, analogous to geometric programming duality, is formulated under somewhat more restrictive conditions. The dual constraints are not all linear, but the notion ofdegrees of difficulty is maintained in its geometric programming sense. One formulation of the dual program is shown to be a generalization of the chemical equilibrium problem where correction factors are added to account for nonideality. Some of the computational difficulties in solving transcendental programs are discussed briefly.This research was partially supported by the National Institute of Health Grant No. GM-14789; Office of Naval Research under Contract No. N00014-75-C-0276; National Science Foundation Grant No. MPS-71-03341 A03; and the US Atomic Energy Commission Contract No. AT(04-3)-326 PA #18.     

Invariant imbedding and the numerical determination of Green's functions

A Cauchy system for a Green's function is derived. This is investigated numerically, and applications to the determination of eigenvalues and stochastic differential operators are sketched.This research was supported by the National Institutes of Health under Grants Nos. GM-16437-02 and GM-01724-04.     

A stability theorem for symmetrically rational counterplanning

In the absence of contrary information, it would seem prudent for a competitor to attribute to his opponents the same level of rationality that he himself employs. In the context of a general, linear-quadratic, nonzero-sum, two-person game, it is shown that a counterplanning procedure consistent with this principle of symmetrical rationality always converges to the unique Nash equilibrium for the game.The work of R. Kalaba and L. Tesfatsion was partially supported by the National Science Foundation under Grant No. ENG-77-28432 and by the National Institutes of Health under Grant No. GM-23732-03.     

A second-order method for the general nonlinear programming problem

This paper presents a multiplier-type method for nonlinear programming problems with both equality and inequality constraints. Slack variables are used for the inequalities. The penalty coefficient is adjusted automatically, and the method converges quadratically to points satisfying second-order conditions.The work of the first author was supported by NSF RANN and JSEP Contract No. F44620-71-C-0087; the work of the second author was supported by the National Science Foundation Grant No. ENG73-08214A01 and US Army Research Office Durham Contract No. DAHC04-73-C-0025.     

Two-point boundary value problems in relativistic dynamics

For gyro systems of relativistic type, we obtain solvability conditions for the two-point boundary value problem. We use the geodesic modeling method, in which the original problem is reduced to studying the existence of isotropic geodesic curves of the Kaluts-O. Klein Lorentz metric joining two fibers of a bundle over the configuration manifold of the system. As an example, we consider problems on the motion of charged test particles in an arbitrary electromagnetic field and in the outer Reissner-Nordstrem space-time in the field of a charged black hole and some external electromagnetic field.Translated fromMatematicheskie Zametki, Vol. 59, No. 3, pp. 437–449, March, 1996.This research was partially supported by the Russian Foundation for Basic Research under grant No. 94-01-00492a and by the International Science Foundation under grant No. NP4000.     

The number of periodic solutions of polynomial differential equations

We estimate the number of periodic solutions for special classes ofnth-order ordinary differential equations with variable coefficients. Translated fromMatematicheskie Zametki, Vol. 64, No. 5, pp. 720–727, November, 1998. The author thanks Yu. S. Il'yashenko for setting the problems, permanent advice, and overall support. The author is also thankful to D. A. Panov for numerous discussions. This research was supported by the CRDF Foundation under grant MR1-220, by the INTAS Foundation under grant No. 93-05-07, and by the Russian Foundation for Basic Research under grant No. 95-01-01258.     

An extended Gauss–Seidel method for a class of multi-valued complementarity problems

The complementarity problem is one of the basic topics in nonlinear analysis; however, the methods for solving complementarity problems are usually developed for problems with single-valued mappings. In this paper we examine a class of complementarity problems with multi-valued mappings and propose an extension of the Gauss–Seidel algorithm for finding its solution. Its convergence is proved under off-diagonal antitonicity assumptions. Applications to Walrasian type equilibrium problems and to nonlinear input–output problems are also given. In this work, the authors were supported by Brescia University grant PRIN - 2006: “Oligopolistic models and order monotonicity properties”, the third author was also supported by the joint RFBR–NNSF grant, project No. 07-01-92101.     

Stability of generalized solutions to equations of one-dimensional motion of viscous heat-conducting gases

Nonhomogeneous initial boundary value problems for a specific quasilinear system of equations of composite type are studied. The system describes the one-dimensional motion of a viscous perfect polytropic gas. We assume that the initial data belong to the spacesL () orL ₂() and the problems under consideration have generalized solutions only. For such solutions, a theorem on strong stability is proved, i.e., estimates for the norm of the difference of two solutions are expressed in terms of the sums of the norms of the differences of the corresponding data. Uniqueness of generalized solutions is a simple consequence of this theorem.Translated fromMatematicheskie Zametki, Vol. 63, No. 6, pp. 835–846, June, 1998.This research was supported by the Russian Foundation for Basic Research under grants No. 96-01-00621 and No. 97-01-00214, and by the INTAS Foundation under grant No. 93-27-16.     

Simple search games on an infinite circular cylinder

Differential search games with simple motions of objects on an infinite circular cylinder are approached geometrically, using sets whose structure depends on time. Sufficient conditions for detection with the corresponding strategies are given. New kinds of problems are proposed, their connections with the initial parameters are investigated. Translated fromMatematicheskie Zametki, Vol. 58, No. 5, pp. 762–772, November, 1995. This work was partially supported by the Russian Foundation for Basic Research under grant No. 242 95-01-01471.     

The compatibility condition for the embedding problem related top-extensions of groups

Embedding problems with a group and its Sylow p-subgroup over p-extensions of fields are considered. The Faddeev-Hasse compatibility condition for these problems is studied. It is proved that the compatibility conditions for the problems under consideration are equivalent if the kernel of the problem is a supersolvable group or a Sylow p-subgroup is an invariant subgroup. Bibliography: 3titles.Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 236, 1997, pp. 100–105.The present paper was supported by the Russian Foundation for Basic Research, grant No. 96-01-00S54.     

Strictly pseudoconvex domains and algebraic varieties

In the paper, we consider applications of strictly pseudoconvex domains to the problems of algebraicity and rationality. We give a new proof of the Kodaira theorem on the algebraicity of a surface and we also prove a multidimensional version of this theorem. Theorems analogous to the Hodge index theorem and the Lefschetz theorem about (1, 1)-classes are obtained for strictly pseudoconvex domains. Conjectures on the geometry of strictly pseudoconvex domains on algebraic surfaces are formulated. Translated fromMatematicheskie Zametki, Vol. 60, No. 3, pp. 414–422, September, 1996. This research was supported by the Russian Foundation for Basic Research under grant No. 93-01-00225 and by the International Science Foundation under grant No. 508.     

Stability Estimates in Inverse Problems for Hyperbolic Equations

We discuss here a method for investigating inverse problems for hyperbolic equations based on combining an asymptotic expansion of solutions to the equations in a neighborhood of a characteristic surface, the connections between coefficients of the expansion and the unknown coefficients of the equation and the estimates of a solution to the Cauchy problem in terms of the data on a time-like surface. We give some applications of this method to a number of inverse problems. Stability results to the inverse problems are given. This work was partly supported by the Russian Foundation for Basic Research (Grant No. 05-01-00171). Received: December 2005     

On the existence of Nash strategies and solutions to coupled riccati equations in linear-quadratic games

The existence of linear Nash strategies for the linear-quadratic game is considered. The solvability of the coupled Riccati matrix equations and the stability of the closed-loop matrix are investigated by using Brower's fixed-point theorem. The conditions derived state that the linear closed-loop Nash strategies exist, if the open loop matrixA has a sufficient degree of stability which is determined in terms of the norms of the weighting matrices. WhenA is not necessarily stable, sufficient conditions for existence are given in terms of the solutions of auxiliary problems using the same procedure.This work was supported in part by the Joint Services Electronics Program (US Army, US Navy, and US Air Force) under Contract No. DAAG-29-78-C-0016, in part by the National Science Foundation under Grant No. ENG-74-20091, and in part by the Department of Energy, Electric Energy Systems Division, under Contract No. US-ERDA-EX-76-C-01-2088.     

The leading term of the spectral asymptotics for the Kohn-Laplace operator in a bounded domain

We prove the Weyl asymptotic formula for the number of eigenvalues of the Kohn-Laplace operator on a Heisenberg group and write out the leading term of asymptotics. The method of study is based on estimates of the Green function for the Dirichlet problem for the corresponding parabolic operator and makes use of the classical Hardy-Littlewood Tauberian theorem.Translated fromMatematicheskie Zametki, Vol. 64, No. 4, pp. 493–505, October, 1998.This research was supported by the Russian Foundation for Basic Research under grants No. 96-01-00443 and No. 96-01-00503.