Controlled Markov processes on the infinite planning horizon: Weighted and overtaking cost criteria

Stochastic control problems for controlled Markov processes models with an infinite planning horizon are considered, under some non-standard cost criteria. The classical discounted and average cost criteria can be viewed as complementary, in the sense that the former captures the short-time and the latter the long-time performance of the system. Thus, we study a cost criterion obtained as weighted combinations of these criteria, extending to a general state and control space framework several recent results by Feinberg and Shwartz, and by Krass et al. In addition, a functional characterization is given for overtaking optimal policies, for problems with countable state spaces and compact control spaces; our approach is based on qualitative properties of the optimality equation for problems with an average cost criterion.Research partially supported by the Engineering Foundation under grant RI-A-93-10, in part by the National Science Foundation under grant NSF-INT 9201430, and in part by a grant from the AT&T Foundation.Research partially supported by the Air Force Office of Scientific Research under Grant F49620-92-J-0045, and in part by the National Science Foundation under Grant CDR-8803012.     

Use of the Lagrange method in nonlinear analysis

The problem of the spectrum of a nonlinear system and its relation to ellipsoid transform widths are studied.Translated fromMatematicheskie Zametki, Vol. 59, No. 3, pp. 334–342, March, 1996.This research was partially supported by the Russian Foundation for Basic Research under grant No. 93-01-00237 and by the International Science Foundation under grant No. MP1000.     

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.     

Bilinear programming and structured stochastic games

One-step algorithms are presented for two classes of structured stochastic games, namely, those with additive rewards and transitions and those which have switching controllers. Solutions to such classes of games under the average reward criterion can be derived from optimal solutions to appropriate bilinear programs. The validity of using bilinear programming as a solution method follows from two preliminary theorems, the first of which is a complete classification of undiscounted stochastic games with optimal stationary strategies. The second of these preliminary theorems relaxes the conditions of the classification theorem for certain classes of stochastic games and provides the basis for the bilinear programming results. Analogous results hold for the discounted stochastic games with the above special structures.This research was supported in part by the Air Force Office of Scientific Research and by the National Science Foundation under Grant No. ECS-850-3440.     

Solutions of the heat equation in domains with singularities

We study the asymptotics of solutions to the Dirichlet problem for the heat equation in time-dependent domains with singular points.Translated fromMatematicheskie Zametki, Vol. 64, No. 2, pp. 163–179, August, 1998.This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-00504 and by INTAS under grant No. 93-351.     

Conditions of topological conjugacy of gradient-like diffeomorphisms on irreducible 3-manifolds

In this paper, we obtain the topological classification of gradient-like diffeomorphisms and the conditions of topological conjugacy of Morse-Smale diffeomorphisms with finite sets of heteroclinic trajectories on three-dimensional manifolds.Translated fromMatematicheskie Zametki, Vol. 59, No. 1, pp. 73–80, January, 1996.This research was partially supported by the Russian Foundation for Basic Research under grant No. 93-01-01-407, by the International Science Foundation under grant R99000, and by the Foundation Cultural Initiative.     

Some conic bundless

A representation of the anticanonical K3 surface of a singular pencil of conics is described. This generalizes the well-known Shokurov theorem.Translated fromMatematicheskie Zametki, Vol. 63, No. 6, pp. 903–910, June, 1998.The author is greatly indebted to V. A. Iskovskikh, Yu. G. Prokhorov, and I. A. Chel'tsov for fruitful discussions.This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-00820 and by INTAS under grant No. 93-2805.00-00-00.     

A weakened asymptotic monogeny condition

Functions of one complex variable that satisfy a weakened asymptotic monogeny condition are proved to be holomorphic. Translated fromMatematicheskie Zametki, Vol. 60, No. 6, pp. 902–911, December, 1996. The author thanks Proffessor Dolzhenko for discussing the results of the paper. The research was supported by the Russian Foundation for Basic Research under grant No. 93-01-00236.     

Linear differential operators with unbounded operator coefficients and semigroups of bounded operators

Associated with a family of evolution operators in a complex Banach space is a linear unbounded operator, which is studied with the aid of a semigroup of difference operators and a difference operator in a sequence space. Some formulas for the spectra of the linear operators in question (in particular, for abstract hyperbolic differential operators) and the spectrum mapping theorem for the semigroup of difference operators are obtained. Translated fromMatematicheskie Zametki, Vol. 59, No. 6, pp. 811–820, June, 1996. This research was supported by the Russian Foundation for Basic Research under grant No. 95-01-00032 and by the International Science Foundation under grant No. NZA000 and grant No. NZA300.     

Separately continuous selectors

The existence of separately continuous selectors of a separately lower and upper semicontinuous many-valued multivariate map is proved, provided its domain is a compact set and its range is a convex closed subset of a metrizable convex compact set. Translated fromMatematicheskie Zametki, Vol. 63, No. 2, pp. 209–216, February, 1998. This research was partially supported by the Russian Foundation for Basic Research under grant No. 93-01-00264.     

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.     

Conditions for the spatial flatness and spatial injectivity of an indecomposable CSL algebra of finite width

This paper is concerned with the connection between the geometric properties of the latticeL of subspaces of a Hilbert spaceH and homological properties (flatness and injectivity) ofH regarded as a natural module over the reflexive algebra AlgL that consists of all operators leaving invariant each element of the latticeL. It follows from these results that the cohomology groups with coefficients inB(H) are trivial for a broad class of reflexive algebras. Translated fromMatemalicheskie Zametki, Vol. 63, No. 1, pp. 9–20, January, 1998. The author gladly expresses his sincere gratitude to A. Ya. Khelemskii and Yu. V. Selivanov for their assistance in this work. In particular, A. Ya. Khelemskii indicated how to simplify the proof of Theorem 1. This research was supported by the Russian Foundation for Basic Research under grant No. 93-01-00156, by the International Science Foundation under grant No. M95000, and by the INTAS fund under grant No. 93-1376.     

Optimal guaranteed control of linear systems under disturbances

This paper is devoted to the problem of the minimax control of a dynamical system with quadratic performance functional under external disturbances and geometric control constraints. The optimal guaranteed control strategy is obtained in explicit form.Translated fromMatematicheskie Zametki, Vol. 60, No. 2, pp. 198–205, August, 1996.This research was supported by the Russian Foundation for Basic Research under grant No. 95-01-000771a.     

Integrable boundary conditions for many-component burgers equations

Infinite series of boundary conditions that are consistent with even-order higher symmetries and ensure the integrability of a Burgers type equation are constructed. Translated fromMatematicheskie Zametki, Vol. 60, No. 6, pp. 888–901, December, 1996. This research was supported by the Russian Foundation for Basic Research under grant No. 93-011-16008 and by the International Science Foundation under grant Nos. MLY00 and RK2000.     

Modules over quantum polynomials

We describe the structure of finitely generated modules over general quantum Laurent polynomials and prove that Artin modules over general quantum Laurent polynomials are cyclic. Translated fromMatematicheskie Zametki, Vol. 59, No. 4, pp. 497–503, April, 1996. This research was partially supported by the Russian Foundation for the Basic Research under grant No. 93-011-1544 and by the INTAS program under grant No. 93-2618.     

On continuous extension of locally homeomorphic simplicial maps of ℝ2 into itself byσ-processes

It is shown that the Klein bottle with two points removed can be embedded in the compactification of ℝ² by a finite tree. Translated fromMatematicheskie Zametki, Vol. 59, No. 6, pp. 821–831, June, 1996. I wish to express sincere gratitude to A. G. Vitushkin for posing the problem and guidance. This research was supported by the Russian Foundation for Basic Research under grant No. 93-01-00225 and by the Ministry of Scientific and Technological Research in the Mathematical Sciences of Italy.     

Extension by zero of functions of several variables

We obtain sufficient conditions on a domainG ⊂ ℝn for functions defined onG to be extendable by zero to the entire space ℝn with smoothness preserved in an integral norm. Translated fromMatematicheskie Zametki, Vol. 64, No. 3, pp. 351–365, September, 1998. This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-00243 and by program “Leading Science Schools” under grant No. 96-15-96102.     

An estimate of a Dirichlet series with real coefficients on a ray

Best possible conditions are found under which the logarithm of the absolute value of an entire Dirichlet series with real coefficients rarely changing sign increases on the positive ray in the same way as in the whole plane. This result essentially strengthens a similar result due to Pavlov concerning the growth of Taylor series on the positive ray. Translated fromMatematicheskie Zametki, Vol. 60, No. 6, pp. 810–823, December, 1996. This research was supported by the Russian Foundation for Basic Research under grant No. 93-01-00259.     

Asymptotics ast → ∞ of the solutions of nonlinear equations with nonsmall initial perturbations

A new method is proposed for finding asymptotics ast → ∞ of the solutions of the Cauchy problem for nonlinear evolution equations with nonsmall initial data. Translated fromMatematicheskie Zametki, Vol. 59, No. 6, pp. 855–864, June, 1996. This research was supported by the Russian Foundation for Basic Research under grant No. 93-011-134 and by the International Science Foundation under grant No. BX000.     

Average optimality inequality for continuous-time Markov decision processes in Polish spaces

In this paper, we study the average optimality for continuous-time controlled jump Markov processes in general state and action spaces. The criterion to be minimized is the average expected costs. Both the transition rates and the cost rates are allowed to be unbounded. We propose another set of conditions under which we first establish one average optimality inequality by using the well-known “vanishing discounting factor approach”. Then, when the cost (or reward) rates are nonnegative (or nonpositive), from the average optimality inequality we prove the existence of an average optimal stationary policy in all randomized history dependent policies by using the Dynkin formula and the Tauberian theorem. Finally, when the cost (or reward) rates have neither upper nor lower bounds, we also prove the existence of an average optimal policy in all (deterministic) stationary policies by constructing a “new” cost (or reward) rate. Research partially supported by the Natural Science Foundation of China (Grant No: 10626021) and the Natural Science Foundation of Guangdong Province (Grant No: 06300957).