**Preface to "Singularly Perturbed Problems: Asymptotic Analysis and Approximate Solution"**

This book collects papers which were published in the Special Issue "Singularly Perturbed Problems: Asymptotic Analysis and Approximate Solution" of the *Axioms* journal. These papers represent various aspects of singular perturbation theory and its applications.

In their contribution, Margarita Besova and Vasiliy Kachalov develop the axiomatic approach in the analytic theory of singular perturbations in the frame of topological algebras. This allows the authors to present the main concepts of the singular perturbation analytical theory with maximal generality.

Dana Bibulova, Burkhan Kalimbetov and Valeriy Safonov consider a singularly perturbed integral–differential equation with a rapidly oscillating inhomogeneity and with a rapidly decreasing kernel of the integral operator of Fredholm type. For this equation, the authors construct and justify the regularized (in the sense of S.A. Lomov) asymptotic solution.

The contribution by Abduhafiz Bobodzhanov, Valeriy Safonov and Vasiliy Kachalov is devoted to the analysis of one singularly perturbed integral equation with weakly and rapidly varying kernels. The authors study the influence of the weakly varying integral kernel on the asymptotic solution of this equation. Additionally, using the method of holomorphic regularization, the authors consider the problem of constructing pseudo-analytic solutions of singularly perturbed problems.

The paper by Yuli D. Chashechkin and Artem A. Ochirov applies the theory of singular perturbation to study the propagation of two-dimensional periodic perturbations, including capillary and gravitational surface waves, in a viscous continuously stratified fluid.

Vasile Dragan studies a stochastic linear–quadratic optimal control problem, the dynamics of ˘ which are described by a system of singularly perturbed Ito differential equations with two fast ˆ time scales. The author derives the proper stabilizing asymptotic solution from the algebraic Riccati equation associated with this problem. Using this asymptotic solution, the author designs the suboptimal control with the gain matrices not depending upon the small parameters.

In their contribution, Alexander Eliseev and Tatjana Ratnikova consider a singularly perturbed Cauchy problem for a two-dimensional differential equation with a "simple" turning point. Using the regularization method of S.A. Lomov, the authors construct and justify the asymptotic solution to the problem under consideration. Additionally, under proper additional conditions on the considered problem, the authors show that the series, representing the asymptotic solution, converges and that its sum represents the exact solution to the singularly perturbed Cauchy problem.

The contribution by Valery Y. Glizer considers a singularly perturbed linear time-dependent controlled system with multiple point-wise delays and distributed delays in the state and control variables. It is assumed that the delays are small, in the order of a small positive multiplier for a part of the derivatives in the system. Both types of the considered system, standard and nonstandard, are analyzed. For each of these types, two much simpler, parameter-free subsystems (the slow and fast ones) are associated with the original system. The author establishes that the proper kinds of controllability of the slow and fast subsystems yield the complete Euclidean space controllability of the original system for all sufficiently small values of the singular perturbation parameter.

Burkhan Kalimbetov and Valeriy Safonov consider a system with rapidly oscillating coefficients. This system includes an integral operator with an exponentially varying kernel. The authors develop an algorithm for the regularization method (in the sense of S.A. Lomov) for this system. They also analyze the influence of the integral term on the asymptotic behavior of the solution to the original system.

The paper by Galina Kurina considers an initial value problem for a class of singularly perturbed systems in the case where the matrix of the coefficients for the state variable is singular (the critical case). The author applies the orthogonal projector method for the construction and justification the asymptotic solution to the considered problem.

In their paper, Galina Kurina and Margarita Kalashnikova apply the direct scheme method for the asymptotic solution of a weakly nonlinearly perturbed linear–quadratic optimal control problem with three-tempo state variables. Asymptotic expansions for the optimal control, optimal trajectory and optimal value of the minimized functional of the considered problem are derived and justified. Monotonic (non-increasing) behavior is established for the asymptotic expansion of the optimal value of the functional with respect to the order of this expansion.

Tatiana Ratnikova studies the singularly perturbed Cauchy problem for a parabolic equation in the case of violation of stability conditions of the limit–operator spectrum. This case is due to the presence of a "simple" turning point in the equation. Using the Lomov's regularization method, the author constructs a uniform asymptotic solution to the considered problem and proves the asymptotic convergence of the regularized series.

The contribution by Olga Tsekhan is devoted to undertaking an analysis of complete controllability of a linear time-invariant singularly perturbed system, with multiple commensurate non-small delays in the slow state variables. An extension of the Chang-type time-scale separation of a singularly perturbed system to the considered time delay system is carried out. Based on this time-scale separation of the original system, sufficient conditions are obtained for its complete controllability. These conditions are independent of the parameter of singular perturbation, while they provide the complete controllability of the original system for all sufficiently small values of this parameter.

Vladimir Turetsky and Valery Y. Glizer consider a finite-horizon zero-sum linear–quadratic differential game, modeling a pursuit–evasion problem. In the game's cost functional, the cost of the control of the minimizing player is much smaller than the cost of the control of the maximizing player and the cost of the state variable. This smallness is due to a positive small multiplier (a small parameter) for the quadratic form of the minimizing player's control in the cost functional. Parameter-free sufficient conditions for the existence of the game's solution, valid for all sufficiently small values of the parameter, are presented. The boundedness with respect to the small parameter of the time realizations of the players' optimal state feedback controls, along the corresponding game's trajectory, is established. The best achievable game value from the minimizing player's viewpoint is derived. A relation is established between solutions of the original game and the game that is obtained from the original one by replacing the small parameter with zero.

The paper by Robert Vrabel considers the problem of asymptotic behavior of the solutions for one class of non-resonant, singularly perturbed linear Neumann boundary value problems. The approach, proposed by the author for analysis of asymptotic behavior of the solution to such problems, is based on the study of an integral equation associated with this problem.

I express my sincere gratitude to all of the authors for their contributions to this book.

**Valery Y. Glizer** *Editor*
