Singularly Perturbed Problems: Asymptotic Analysis and Approximate Solution

A special issue of Axioms (ISSN 2075-1680). This special issue belongs to the section "Mathematical Analysis".

Deadline for manuscript submissions: closed (30 June 2022) | Viewed by 26090

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Special Issue Editor

The Galilee Research Center for Applied Mathematics, ORT Braude College of Engineering, Karmiel, Israel
Interests: asymptotic methods; differential games; generalized functions; hybrid systems; optimal control; robust control; singular optimal control problems and singular differential games; singularly perturbed problems; stochastic difference and differential equations; systems theory; time delay systems
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Special Issue Information

Dear Colleagues,

This issue is a continuation of our previous Special Issue on "Singularly Perturbed Problems". We will provide an opportunity to present recent developments in theory and various theoretical and  real-life applications of singularly perturbed problems, their asymptotic analysis and approximate (analytical/numerical) solutions. This Special Issue will address the following non-exhaustive list of topics:

Singularly perturbed ordinary differential equations (initial/boundary-value problems); Singularly perturbed partial differential equations (initial/boundary-value problems); Singularly perturbed  ordinary/partial  differential equations with deviating arguments (initial/boundary-value problems); Singularly perturbed integro-differential equations (initial/boundary-value problems); Singularly perturbed stochastic differential equations (initial/boundary-value problems);  Periodic solutions of singularly perturbed differential equations; Singularly perturbed difference equations (initial/boundary-value problems); Control problems with singularly perturbed continuous/discrete dynamics; Differential/difference games with singularly perturbed dynamics; Qualitative analysis of singularly perturbed equations; Miscellaneous applications of singularly perturbed problems; etc.

It should be noted that the Special Issue is open to receiving further ideas, apart from the aforementioned topics.

We hope that this initiative will be attractive to experts in the theory of singular perturbations, and its theoretical and real-life Applications. We encourage you to submit your current research to be included in the Special Issue.

Prof. Dr. Valery Y. Glizer
Guest Editor

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Keywords

  • Singularly perturbed ordinary differential equations
  • Singularly perturbed partial differential equations
  • Singularly perturbed differential equations with deviating arguments
  • Singularly perturbed integro-differential equations
  • Singularly perturbed stochastic differential equations
  • Singularly perturbed periodic differential equations
  • Singularly perturbed difference equations
  • Singularly perturbed control problems
  • Singularly perturbed dynamic games
  • Asymptotic analysis
  • Approximate analytical/numerical solution
  • Applications of singularly perturbed problems

Published Papers (14 papers)

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Research

31 pages, 440 KiB  
Article
Justification of Direct Scheme for Asymptotic Solving Three-Tempo Linear-Quadratic Control Problems under Weak Nonlinear Perturbations
Axioms 2022, 11(11), 647; https://doi.org/10.3390/axioms11110647 - 16 Nov 2022
Cited by 3 | Viewed by 979
Abstract
The paper deals with an application of the direct scheme method, consisting of immediately substituting a postulated asymptotic solution into a problem condition and determining a series of control problems for finding asymptotics terms, for asymptotics construction of a solution of a weakly [...] Read more.
The paper deals with an application of the direct scheme method, consisting of immediately substituting a postulated asymptotic solution into a problem condition and determining a series of control problems for finding asymptotics terms, for asymptotics construction of a solution of a weakly nonlinearly perturbed linear-quadratic optimal control problem with three-tempo state variables. For the first time, explicit formulas for linear-quadratic optimal control problems, from which all terms of the asymptotic expansion are found, are justified, and the estimates of the proximity between the asymptotic and exact solutions are proved for the control, state trajectory, and minimized functional. Non-increasing of the minimized functional, if a next approximation to the optimal control is used, following from the proposed algorithm of the asymptotics construction, is also established. Full article
21 pages, 2113 KiB  
Article
Periodic Waves and Ligaments on the Surface of a Viscous Exponentially Stratified Fluid in a Uniform Gravity Field
Axioms 2022, 11(8), 402; https://doi.org/10.3390/axioms11080402 - 15 Aug 2022
Cited by 4 | Viewed by 1220
Abstract
The theory of singular perturbations in a unified formulation is used, for the first time, to study the propagation of two-dimensional periodic perturbations, including capillary and gravitational surface waves and accompanying ligaments in the [...] Read more.
The theory of singular perturbations in a unified formulation is used, for the first time, to study the propagation of two-dimensional periodic perturbations, including capillary and gravitational surface waves and accompanying ligaments in the 104<ω<103s1 frequency range, in a viscous continuously stratified fluid. Dispersion relations for flow constituents are given, as well as expressions for phase and group velocities for surface waves and ligaments in physically observable variables. When the wave-length reaches values of the order of the stratification scale, the liquid behaves as homogeneous. As the wave frequency approaches the buoyancy frequency, the energy transfer rate decreases: the group velocity of surface waves tends to zero, while the phase velocity tends to infinity. In limiting cases, the expressions obtained are transformed into known wave dispersion expressions for an ideal stratified or actually homogeneous fluid. Full article
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7 pages, 302 KiB  
Article
Non-Resonant Non-Hyperbolic Singularly Perturbed Neumann Problem
Axioms 2022, 11(8), 394; https://doi.org/10.3390/axioms11080394 - 11 Aug 2022
Cited by 1 | Viewed by 916
Abstract
In this brief note, we study the problem of asymptotic behavior of the solutions for non-resonant, singularly perturbed linear Neumann boundary value problems εy+ky=f(t), y(a)=0, [...] Read more.
In this brief note, we study the problem of asymptotic behavior of the solutions for non-resonant, singularly perturbed linear Neumann boundary value problems εy+ky=f(t), y(a)=0, y(b)=0, k>0, with an indication of possible extension to more complex cases. Our approach is based on the analysis of an integral equation associated with this problem. Full article
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24 pages, 1346 KiB  
Article
Cheap Control in a Non-Scalarizable Linear-Quadratic Pursuit-Evasion Game: Asymptotic Analysis
Axioms 2022, 11(5), 214; https://doi.org/10.3390/axioms11050214 - 05 May 2022
Cited by 1 | Viewed by 1525
Abstract
In this work, a finite-horizon zero-sum linear-quadratic differential game, modeling a pursuit-evasion problem, was considered. In the game’s cost function, the cost of the control of the minimizing player (the minimizer/the pursuer) was much smaller than the cost of the control of the [...] Read more.
In this work, a finite-horizon zero-sum linear-quadratic differential game, modeling a pursuit-evasion problem, was considered. In the game’s cost function, the cost of the control of the minimizing player (the minimizer/the pursuer) was much smaller than the cost of the control of the maximizing player (the maximizer/the evader) and the cost of the state variable. This smallness was expressed by a positive small multiplier (a small parameter) of the square of the L2-norm of the minimizer’s control in the cost function. Parameter-free sufficient conditions for the existence of the game’s solution (the players’ optimal state-feedback controls and the game value), valid for all sufficiently small values of the parameter, were presented. The boundedness (with respect to the small parameter) of the time realizations of the optimal state-feedback controls along the corresponding game’s trajectory was established. The best achievable game value from the minimizer’s viewpoint was derived. A relation between solutions of the original cheap control game and the game that was obtained from the original one by replacing the small minimizer’s control cost with zero, was established. An illustrative real-life example is presented. Full article
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14 pages, 298 KiB  
Article
Regularized Asymptotic Solutions of a Singularly Perturbed Fredholm Equation with a Rapidly Varying Kernel and a Rapidly Oscillating Inhomogeneity
Axioms 2022, 11(3), 141; https://doi.org/10.3390/axioms11030141 - 18 Mar 2022
Cited by 3 | Viewed by 1337
Abstract
This article investigates an equation with a rapidly oscillating inhomogeneity and with a rapidly decreasing kernel of an integral operator of Fredholm type. Earlier, differential problems of this type were studied in which the integral term was either absent or had the form [...] Read more.
This article investigates an equation with a rapidly oscillating inhomogeneity and with a rapidly decreasing kernel of an integral operator of Fredholm type. Earlier, differential problems of this type were studied in which the integral term was either absent or had the form of a Volterra-type integral. The presence of an integral operator and its type significantly affect the development of an algorithm for asymptotic solutions, in the implementation of which it is necessary to take into account essential singularities generated by the rapidly decreasing kernel of the integral operator. It is shown in tise work that when passing the structure of essentially singular singularities changes from an integral operator of Volterra type to an operator of Fredholm type. If in the case of the Volterra operator they change with a change in the independent variable, then the singularities generated by the kernel of the integral Fredholm-type operators are constant and depend only on a small parameter. All these effects, as well as the effects introduced by the rapidly oscillating inhomogeneity, are necessary to take into account when developing an algorithm for constructing asymptotic solutions to the original problem, which is implemented in this work. Full article
9 pages, 244 KiB  
Article
Singularly Perturbed Cauchy Problem for a Parabolic Equation with a Rational “Simple” Turning Point
Axioms 2020, 9(4), 138; https://doi.org/10.3390/axioms9040138 - 27 Nov 2020
Cited by 3 | Viewed by 1436
Abstract
The aim of the research is to develop the regularization method. By Lomov’s regularization method, we constructed a uniform asymptotic solution of the singularly perturbed Cauchy problem for a parabolic equation in the case of violation of stability conditions of the limit-operator spectrum. [...] Read more.
The aim of the research is to develop the regularization method. By Lomov’s regularization method, we constructed a uniform asymptotic solution of the singularly perturbed Cauchy problem for a parabolic equation in the case of violation of stability conditions of the limit-operator spectrum. The problem with a “simple” turning point is considered in the case, when the eigenvalue vanishes at t=0 and has the form tm/na(t). The asymptotic convergence of the regularized series is proved. Full article
12 pages, 776 KiB  
Article
Regularization Method for Singularly Perturbed Integro-Differential Equations with Rapidly Oscillating Coefficients and Rapidly Changing Kernels
Axioms 2020, 9(4), 131; https://doi.org/10.3390/axioms9040131 - 13 Nov 2020
Cited by 10 | Viewed by 1610
Abstract
In this paper, we consider a system with rapidly oscillating coefficients, which includes an integral operator with an exponentially varying kernel. The main goal of the work is to develop an algorithm for the regularization method for such systems and to identify the [...] Read more.
In this paper, we consider a system with rapidly oscillating coefficients, which includes an integral operator with an exponentially varying kernel. The main goal of the work is to develop an algorithm for the regularization method for such systems and to identify the influence of the integral term on the asymptotic behavior of the solution of the original problem. Full article
11 pages, 257 KiB  
Article
Axiomatic Approach in the Analytic Theory of Singular Perturbations
Axioms 2020, 9(1), 9; https://doi.org/10.3390/axioms9010009 - 16 Jan 2020
Cited by 1 | Viewed by 1818
Abstract
Introduced by S.A. Lomov, the concept of a pseudoanalytic (pseudoholomorphic) solution laid the foundation for the development of the singular perturbation analytical theory. In order for this concept to work in case of linear problems, an apparatus for the theory of exponential type [...] Read more.
Introduced by S.A. Lomov, the concept of a pseudoanalytic (pseudoholomorphic) solution laid the foundation for the development of the singular perturbation analytical theory. In order for this concept to work in case of linear problems, an apparatus for the theory of exponential type vector spaces was developed. When considering nonlinear singularly perturbed problems, an algebraic approach is currently used. This approval is based on the properties of algebra homomorphisms for holomorphic functions with various numbers of variables, as a result of which it is possible to obtain pseudoholomorphic solutions. In this paper, formally singularly perturbed equations are considered in topological algebras, which allows the authors to formulate the main concepts of the singular perturbation analytical theory from the standpoint of maximal generality. Full article
12 pages, 273 KiB  
Article
Regularized Solution of Singularly Perturbed Cauchy Problem in the Presence of Rational “Simple” Turning Point in Two-Dimensional Case
Axioms 2019, 8(4), 124; https://doi.org/10.3390/axioms8040124 - 01 Nov 2019
Cited by 2 | Viewed by 1627
Abstract
By Lomov’s S.A. regularization method, we constructed an asymptotic solution of the singularly perturbed Cauchy problem in a two-dimensional case in the case of violation of stability conditions of the limit-operator spectrum. In particular, the problem with a ”simple” turning point was considered, [...] Read more.
By Lomov’s S.A. regularization method, we constructed an asymptotic solution of the singularly perturbed Cauchy problem in a two-dimensional case in the case of violation of stability conditions of the limit-operator spectrum. In particular, the problem with a ”simple” turning point was considered, i.e., one eigenvalue vanishes for t = 0 and has the form t m / n a ( t ) (limit operator is discretely irreversible). The regularization method allows us to construct an asymptotic solution that is uniform over the entire segment [ 0 , T ] , and under additional conditions on the parameters of the singularly perturbed problem and its right-hand side, the exact solution. Full article
19 pages, 307 KiB  
Article
Complete Controllability Conditions for Linear Singularly Perturbed Time-Invariant Systems with Multiple Delays via Chang-Type Transformation
Axioms 2019, 8(2), 71; https://doi.org/10.3390/axioms8020071 - 03 Jun 2019
Cited by 8 | Viewed by 2564
Abstract
The problem of complete controllability of a linear time-invariant singularly-perturbed system with multiple commensurate non-small delays in the slow state variables is considered. An approach to the time-scale separation of the original singularly-perturbed system by means of Chang-type non-degenerate transformation, generalized for the [...] Read more.
The problem of complete controllability of a linear time-invariant singularly-perturbed system with multiple commensurate non-small delays in the slow state variables is considered. An approach to the time-scale separation of the original singularly-perturbed system by means of Chang-type non-degenerate transformation, generalized for the system with delay, is used. Sufficient conditions for complete controllability of the singularly-perturbed system with delay are obtained. The conditions do not depend on a singularity parameter and are valid for all its sufficiently small values. The conditions have a parametric rank form and are expressed in terms of the controllability conditions of two systems of a lower dimension than the original one: the degenerate system and the boundary layer system. Full article
10 pages, 284 KiB  
Article
Projector Approach to Constructing Asymptotic Solution of Initial Value Problems for Singularly Perturbed Systems in Critical Case
Axioms 2019, 8(2), 56; https://doi.org/10.3390/axioms8020056 - 08 May 2019
Cited by 4 | Viewed by 2586
Abstract
Under some conditions, an asymptotic solution containing boundary functions was constructed in a paper by Vasil’eva and Butuzov (Differ. Uravn. 1970, 6(4), 650–664 (in Russian); English transl.: Differential Equations 1971, 6, 499–510) for an initial value problem for weakly non-linear differential equations with [...] Read more.
Under some conditions, an asymptotic solution containing boundary functions was constructed in a paper by Vasil’eva and Butuzov (Differ. Uravn. 1970, 6(4), 650–664 (in Russian); English transl.: Differential Equations 1971, 6, 499–510) for an initial value problem for weakly non-linear differential equations with a small parameter standing before the derivative, in the case of a singular matrix A ( t ) standing in front of the unknown function. In the present paper, the orthogonal projectors onto k e r A ( t ) and k e r A ( t ) (the prime denotes the transposition) are used for asymptotics construction. This approach essentially simplifies understanding of the algorithm of asymptotics construction. Full article
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27 pages, 344 KiB  
Article
Euclidean Space Controllability Conditions for Singularly Perturbed Linear Systems with Multiple State and Control Delays
Axioms 2019, 8(1), 36; https://doi.org/10.3390/axioms8010036 - 21 Mar 2019
Cited by 4 | Viewed by 2588
Abstract
A singularly perturbed linear time-dependent controlled system with multiple point-wise delays and distributed delays in the state and control variables is considered. The delays are small, of order of a small positive multiplier for a part of the derivatives in the system. This [...] Read more.
A singularly perturbed linear time-dependent controlled system with multiple point-wise delays and distributed delays in the state and control variables is considered. The delays are small, of order of a small positive multiplier for a part of the derivatives in the system. This multiplier is a parameter of the singular perturbation. Two types of the considered singularly perturbed system, standard and nonstandard, are analyzed. For each type, two much simpler parameter-free subsystems (the slow and fast ones) are associated with the original system. It is established in the paper that 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 parameter of singular perturbation. Illustrative examples are presented. Full article
22 pages, 385 KiB  
Article
On the Linear Quadratic Optimal Control for Systems Described by Singularly Perturbed Itô Differential Equations with Two Fast Time Scales
Axioms 2019, 8(1), 30; https://doi.org/10.3390/axioms8010030 - 05 Mar 2019
Cited by 7 | Viewed by 2267
Abstract
In this paper a stochastic optimal control problem described by a quadratic performance criterion and a linear controlled system modeled by a system of singularly perturbed Itô differential equations with two fast time scales is considered. The asymptotic structure of the stabilizing solution [...] Read more.
In this paper a stochastic optimal control problem described by a quadratic performance criterion and a linear controlled system modeled by a system of singularly perturbed Itô differential equations with two fast time scales is considered. The asymptotic structure of the stabilizing solution (satisfying a prescribed sign condition) to the corresponding stochastic algebraic Riccati equation is derived. Furthermore, a near optimal control whose gain matrices do not depend upon small parameters is discussed. Full article
20 pages, 321 KiB  
Article
Asymptotic and Pseudoholomorphic Solutions of Singularly Perturbed Differential and Integral Equations in the Lomov’s Regularization Method
Axioms 2019, 8(1), 27; https://doi.org/10.3390/axioms8010027 - 01 Mar 2019
Cited by 10 | Viewed by 2190
Abstract
We consider a singularly perturbed integral equation with weakly and rapidly varying kernels. The work is a continuation of the studies carried out previously, but these were focused solely on rapidly changing kernels. A generalization for the case of two kernels, one of [...] Read more.
We consider a singularly perturbed integral equation with weakly and rapidly varying kernels. The work is a continuation of the studies carried out previously, but these were focused solely on rapidly changing kernels. A generalization for the case of two kernels, one of which is weakly, and the other rapidly varying, has not previously been carried out. The aim of this study is to investigate the effects introduced into the asymptotics of the solution of the problem by a weakly varying integral kernel. In the second part of the work, the problem of constructing exact (more precise, pseudo-analytic) solutions of singularly perturbed problems is considered on the basis of the method of holomorphic regularization developed by one of the authors of this paper. The power series obtained with the help of this method for the solutions of singularly perturbed problems (in contrast to the asymptotic series constructed in the first part of this paper) converge in the usual sense. Full article
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