Relevant bibliographies by topics / Linear singularly perturbed problem

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Academic literature on the topic 'Linear singularly perturbed problem'

Author: Grafiati
Published: 3 June 2025
Last updated: 13 July 2025

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Journal articles on the topic "Linear singularly perturbed problem"

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Dauylbayev, Muratkhan, Marat Akhmet, and Aviltay Nauryzbay. "ASYMPTOTIC EXPANSION OF THE SOLUTION FOR SINGULARPERTURBED LINEAR IMPULSIVE SYSTEMS." Journal of Mathematics, Mechanics and Computer Science 122, no. 2 (2024): 14–26. http://dx.doi.org/10.26577/jmmcs2024-122-02-b2.

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In this study, a singularly perturbed linear impulsive system with singularly perturbed impulses is considered. Many books discuss different types of singular perturbation problems. In the present work, an impulse system is considered in which a small parameter is introduced into the impulse equation. This is the main novelty of our study, since other works [25] have only considered a small parameter in the differential equation. A necessary condition is also established to prevent the impulse function from bloating as the parameter approaches zero. As a result, the notion of singularity for discontinuous dynamics is greatly extended. An asymptotic expansion of the solution of a singularly perturbed initial problem with an arbitrary degree of accuracy for a small parameter is constructed. A theorem for estimating the residual term of the asymptotic expansion is formulated, which estimates the difference between the exact solution and its approximation. The results extend those of [32], which formulates an analogue of Tikhonov’s limit transition theorem. The theoretical results are confirmed by a modelling example.
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Sharip, B., and А. Т. Yessimova. "ESTIMATION OF A BOUNDARY VALUE PROBLEM SOLUTION WITH INITIAL JUMP FOR LINEAR DIFFERENTIAL EQUATION." BULLETIN Series of Physics & Mathematical Sciences 69, no. 1 (2020): 168–73. http://dx.doi.org/10.51889/2020-1.1728-7901.28.

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The paper considers a boundary value problem for a singularly perturbed linear differential equation with constant third-order coefficients. In this problem, a small parameter is indicated before the highest derivatives that are part of the differential equation and the boundary condition at t = 0.The fundamental system of solutions of a homogeneous singularly perturbed differential equation is constructed on the basis of asymptotic representations obtained for the roots of the corresponding characteristic equation. This system was used to construct the Cauchy function, special functions of boundary value problems, and also the Green function. With the help of these functions, an analytical formula is obtained for solving a singularly perturbed boundary value problem and it turns out that this solution has an initial zero-order jump at t = 0. It is proved that the solution to the considered singularly perturbed boundary value problem tends to the corresponding unperturbed problem obtained from it under .
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PERJAN, ANDREI, and GALINA RUSU. "Abstract linear second order differential equations with two small parameters and depending on time operators." Carpathian Journal of Mathematics 33, no. 2 (2017): 233–46. http://dx.doi.org/10.37193/cjm.2017.02.10.

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In a real Hilbert space H consider the following singularly perturbed Cauchy problem. We study the behavior of solutions uεδ to this problem in two different cases: ε → 0 and δ ≥ δ0 > 0; ε → 0 and δ → 0, relative to solution to the corresponding unperturbed problem. We obtain some a priori estimates of solutions to the perturbed problem, which are uniform with respect to parameters, and a relationship between solutions to both problems. We establish that the solution to the perturbed problem has a singular behavior, relative to the parameters, in the neighbourhood of t = 0. We show the boundary layer and boundary layer function in both cases.
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Zhumanazarova, Assiya, and Young Im Cho. "Asymptotic Convergence of the Solution of a Singularly Perturbed Integro-Differential Boundary Value Problem." Mathematics 8, no. 2 (2020): 213. http://dx.doi.org/10.3390/math8020213.

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In this study, the asymptotic behavior of the solutions to a boundary value problem for a third-order linear integro-differential equation with a small parameter at the two higher derivatives has been examined, under the condition that the roots of the additional characteristic equation are negative. Via the scheme of methods and algorithms pertaining to the qualitative study of singularly perturbed problems with initial jumps, a fundamental system of solutions, the Cauchy function, and the boundary functions of a homogeneous singularly perturbed differential equation are constructed. Analytical formulae for the solutions and asymptotic estimates of the singularly perturbed problem are obtained. Furthermore, a modified degenerate boundary value problem has been constructed, and it was stated that the solution of the original singularly perturbed boundary value problem tends to this modified problem’s solution.
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Mirzakulova, A. E., and K. T. Konisbayeva. "Uniform asymptotic expansion of the solution for the initial value problem with a piecewise constant argument." BULLETIN OF THE KARAGANDA UNIVERSITY-MATHEMATICS 116, no. 4 (2024): 138–48. https://doi.org/10.31489/2024m4/138-148.

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The article is devoted to the study of a singularly perturbed initial problem for a linear differential equation with a piecewise constant argument second-order for a small parameter. This paper is considered the asymptotic expansion of the solution to the Cauchy problem for singularly perturbed differential equations with piecewise-constant argument. The initial value problem for first order linear differential equations with piecewise-constant argument was obtained that determined the regular members. The Cauchy problems for linear nonhomogeneous differential equations with a constant coefficient were obtained, which determined the boundary layer terms. An asymptotic estimate for the remainder term of the solution of the Cauchy problem was obtained. Using the remainder term, we construct a uniform asymptotic solution with accuracy O(εN+1) on the θi ≤ t ≤ θi+1, i = 0, p segment of the singularly perturbed Cauchy problem with a piecewise constant argument.
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Vaculíková, Ľudmila, and Vladimír Liška. "Singularly Perturbed Linear Neumann Problem with the Characteristic Roots on the Imaginary Axis." Research Papers Faculty of Materials Science and Technology Slovak University of Technology 18, no. 28 (2010): 163–68. http://dx.doi.org/10.2478/v10186-010-0020-4.

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Singularly Perturbed Linear Neumann Problem with the Characteristic Roots on the Imaginary Axis We investigate the problem of existence and asymptotic behavior of solutions for the singularly perturbed linear Neumann problem <img src="/fulltext-image.asp?format=htmlnonpaginated&src=C6551P41673P4147_html\Journal10186_Volume18_Issue28_20_paper.gif" alt=""/> Our approach relies on the analysis of integral equation equivalent to the problem above.
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Mane, Shilpkala T., and Ram Kishun Lodhi. "Quartic B-Spline Technique for Third-Order Linear Singularly Perturbed Boundary Value Problem with Discontinuous Source Term." International Journal of Mathematical, Engineering and Management Sciences 10, no. 4 (2024): 1178–91. https://doi.org/10.33889/ijmems.2025.10.4.056.

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In this paper, we developed an effective computational technique for addressing third-order linear singularly perturbed problems having the source term discontinuous. Boundary or interior layers are frequently present in singular perturbation issues, making traditional numerical techniques more challenging. Here, we present a quartic B-spline method (QBSM) for the approximate solution of the third-order singularly perturbed boundary value problem, improving both the accuracy and efficiency of the solutions. In addition, the proposed method's convergence and error are investigated. The performance of the current technique is demonstrated through numerous numerical tests. The numerical findings are compared to other approaches reported in the literature.
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Tsekhan, Olga. "Complete Controllability Conditions for Linear Singularly Perturbed Time-Invariant Systems with Multiple Delays via Chang-Type Transformation." Axioms 8, no. 2 (2019): 71. http://dx.doi.org/10.3390/axioms8020071.

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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.
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Grossmann, Christian, Lars Ludwig, and Hans-Görg Roos. "Layer-adapted methods for a singularly perturbed singular problem." Computational Methods in Applied Mathematics 11, no. 2 (2011): 192–205. http://dx.doi.org/10.2478/cmam-2011-0010.

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Abstract In the present paper we analyze linear finite elements on a layer adapted mesh for a boundary value problem characterized by the overlapping of a boundary layer with a singularity. Moreover, we compare this approach numerically with the use of adapted basis functions, in our case modified Bessel functions. It turns out that as well adapted meshes as adapted basis functions are suitable where for our one-dimensional problem adapted bases work slightly better.
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Akmatov, A. "Investigation of Solutions to a System of Singularly Perturbed Differential Equations." Bulletin of Science and Practice 8, no. 5 (2022): 15–23. http://dx.doi.org/10.33619/2414-2948/78/01.

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Solutions of linear systems of singularly perturbed differential equations are investigated in the work, in the case when the matrix function had multiple eigenvalues. And also in the study of solutions to a system of singularly perturbed differential equations, we apply the level line method. We define a stable and unstable interval. We take the starting point in stable intervals. Passing to the complex domain, we define the domain that we study for solutions of the problem under consideration. We divide the defined areas near the singular point into several areas. In each area, we estimate the solutions of the problem. To do this, we choose the integration path and prove the lemma and theorem. As a result, we will prove the asymptotic proximity of the solutions of the perturbed and unperturbed problems.
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Dissertations / Theses on the topic "Linear singularly perturbed problem"

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Howe, Sei. "Upper and lower bounds for singularly perturbed linear quadratic optimal control problems." Thesis, Imperial College London, 2017. http://hdl.handle.net/10044/1/54758.

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The question of how to optimally control a large scale system is widely considered to be difficult to solve due to the size of the problem. This difficulty is further compounded when a system exhibits a two time-scale structure where some components evolve slowly and others evolve quickly. When this occurs, the optimal control problem is regarded as singularly perturbed with a perturbation parameter epsilon representing the ratio of the slow time-scale to the fast time-scale. As epsilon goes to zero, the system becomes stiff resulting in a computationally intractable problem. In this thesis, we propose an analytic method for constructing bounds on the minimum cost of a singularly perturbed, linear-quadratic optimal control problem that hold for any arbitrary value of epsilon.
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Tang, Ying. "Stability analysis and Tikhonov approximation for linear singularly perturbed hyperbolic systems." Thesis, Université Grenoble Alpes (ComUE), 2015. http://www.theses.fr/2015GREAT054/document.

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Les dynamiques des systèmes modélisés par des équations aux dérivées partielles (EDPs) en dimension infinie sont largement liées aux réseaux physiques. La synthèse de la commande et l'analyse de la stabilité de ces systèmes sont étudiées dans cette thèse. Les systèmes singulièrement perturbés, contenant des échelles de temps multiples sont naturels dans les systèmes physiques avec des petits paramètres parasitaires, généralement de petites constantes de temps, les masses, les inductances, les moments d'inertie. La théorie des perturbations singulières a été introduite pour le contrôle à la fin des années $1960$, son assimilation dans la théorie du contrôle s'est rapidement développée et est devenue un outil majeur pour l'analyse et la synthèse de la commande des systèmes. Les perturbations singulières sont une façon de négliger la transition rapide, en la considérant dans une échelle de temps rapide séparée. Ce travail de thèse se concentre sur les systèmes hyperboliques linéaires avec des échelles de temps multiples modélisées par un petit paramètre de perturbation. Tout d'abord, nous étudions une classe de systèmes hyperboliques linéaires singulièrement perturbés. Comme le système contient deux échelles de temps, en mettant le paramètre de la perturbation à zéro, deux sous-systèmes, le système réduit et la couche limite, sont formellement calculés. La stabilité du système complet de lois de conservation implique la stabilité des deux sous-systèmes. En revanche un contre-exemple est utilisé pour illustrer que la stabilité des deux sous-systèmes ne suffit pas à garantir la stabilité du système complet. Cela montre une grande différence avec ce qui est bien connu pour les systèmes linéaires en dimension finie modélisés par des équations aux dérivées ordinaires (EDO). De plus, sous certaines conditions, l'approximation de Tikhonov est obtenue pour tels systèmes par la méthode de Lyapunov. Plus précisément, la solution de la dynamique lente du système complet est approchée par la solution du système réduit lorsque le paramètre de la perturbation est suffisamment petit. Deuxièmement, le théorème de Tikhonov est établi pour les systèmes hyperboliques linéaires singulièrement perturbés de lois d'équilibre où les vitesses de transport et les termes sources sont à la fois dépendant du paramètre de la perturbation ainsi que les conditions aux bords. Sous des hypothèses sur la continuité de ces termes et sous la condition de la stabilité, l'estimation de l'erreur entre la dynamique lente du système complet et le système réduit est obtenue en fonction de l'ordre du paramètre de la perturbation. Troisièmement, nous considérons des systèmes EDO-EDP couplés singulièrement perturbés. La stabilité des deux sous-systèmes implique la stabilité du système complet où le paramètre de la perturbation est introduit dans la dynamique de l'EDP. D'autre part, cela n'est pas valable pour le système où le paramètre de la perturbation est présent dans l'EDO. Le théorème Tikhonov pour ces systèmes EDO-EDP couplés est prouvé par la technique de Lyapunov. Enfin, la synthèse de la commande aux bords est abordée en exploitant la méthode des perturbations singulières. Le système réduit converge en temps fini. La synthèse du contrôle aux bords est mise en œuvre pour deux applications différentes afin d'illustrer les résultats principaux de ce travail<br>Systems modeled by partial differential equations (PDEs) with infinite dimensional dynamics are relevant for a wide range of physical networks. The control and stability analysis of such systems become a challenge area. Singularly perturbed systems, containing multiple time scales, often occur naturally in physical systems due to the presence of small parasitic parameters, typically small time constants, masses, inductances, moments of inertia. Singular perturbation was introduced in control engineering in late $1960$s, its assimilation in control theory has rapidly developed and has become a tool for analysis and design of control systems. Singular perturbation is a way of neglecting the fast transition and considering them in a separate fast time scale. The present thesis is concerned with a class of linear hyperbolic systems with multiple time scales modeled by a small perturbation parameter. Firstly we study a class of singularly perturbed linear hyperbolic systems of conservation laws. Since the system contains two time scales, by setting the perturbation parameter to zero, the two subsystems, namely the reduced subsystem and the boundary-layer subsystem, are formally computed. The stability of the full system implies the stability of both subsystems. However a counterexample is used to illustrate that the stability of the two subsystems is not enough to guarantee the full system's stability. This shows a major difference with what is well known for linear finite dimensional systems. Moreover, under certain conditions, the Tikhonov approximation for such system is achieved by Lyapunov method. Precisely, the solution of the slow dynamics of the full system is approximated by the solution of the reduced subsystem for sufficiently small perturbation parameter. Secondly the Tikhonov theorem is established for singularly perturbed linear hyperbolic systems of balance laws where the transport velocities and source terms are both dependent on the perturbation parameter as well as the boundary conditions. Under the assumptions on the continuity for such terms and under the stability condition, the estimate of the error between the slow dynamics of the full system and the reduced subsystem is the order of the perturbation parameter. Thirdly, we consider singularly perturbed coupled ordinary differential equation ODE-PDE systems. The stability of both subsystems implies that of the full system where the perturbation parameter is introduced into the dynamics of the PDE system. On the other hand, this is not true for system where the perturbation parameter is presented to the ODE. The Tikhonov theorem for such coupled ODE-PDE systems is proved by Lyapunov technique. Finally, the boundary control synthesis is achieved based on singular perturbation method. The reduced subsystem is convergent in finite time. Boundary control design to different applications are used to illustrate the main results of this work
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Kunert, Gerd. "A note on the energy norm for a singularly perturbed model problem." Universitätsbibliothek Chemnitz, 2001. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-200100062.

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A singularly perturbed reaction-diffusion model problem is considered, and the choice of an appropriate norm is discussed. Particular emphasis is given to the energy norm. Certain prejudices against this norm are investigated and disproved. Moreover, an adaptive finite element algorithm is presented which exhibits an optimal error decrease in the energy norm in some simple numerical experiments. This underlines the suitability of the energy norm.
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Adkins, Jacob. "A Robust Numerical Method for a Singularly Perturbed Nonlinear Initial Value Problem." Kent State University Honors College / OhioLINK, 2017. http://rave.ohiolink.edu/etdc/view?acc_num=ksuhonors1513331499579714.

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Kunert, Gerd. "Robust local problem error estimation for a singularly perturbed problem on anisotropic finite element meshes." Universitätsbibliothek Chemnitz, 2001. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-200100011.

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Singularly perturbed problems often yield solutions ith strong directional features, e.g. with boundary layers. Such anisotropic solutions lend themselves to adapted, anisotropic discretizations. The quality of the corresponding numerical solution is a key issue in any computational simulation. To this end we present a new robust error estimator for a singularly perturbed reaction-diffusion problem. In contrast to conventional estimators, our proposal is suitable for anisotropic finite element meshes. The estimator is based on the solution of a local problem, and yields error bounds uniformly in the small perturbation parameter. The error estimation is efficient, i.e. a lower error bound holds. The error estimator is also reliable, i.e. an upper error bound holds, provided that the anisotropic mesh discretizes the problem sufficiently well. A numerical example supports the analysis of our anisotropic error estimator.
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Grosman, Serguei. "Robust local problem error estimation for a singularly perturbed reaction-diffusion problem on anisotropic finite element meshes." Universitätsbibliothek Chemnitz, 2006. http://nbn-resolving.de/urn:nbn:de:swb:ch1-200600475.

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Singularly perturbed reaction-diffusion problems exhibit in general solutions with anisotropic features, e.g. strong boundary and/or interior layers. This anisotropy is reflected in the discretization by using meshes with anisotropic elements. The quality of the numerical solution rests on the robustness of the a posteriori error estimator with respect to both the perturbation parameters of the problem and the anisotropy of the mesh. An estimator that has shown to be one of the most reliable for reaction-diffusion problem is the <i>equilibrated residual method</i> and its modification done by Ainsworth and Babuška for singularly perturbed problem. However, even the modified method is not robust in the case of anisotropic meshes. The present work modifies the equilibrated residual method for anisotropic meshes. The resulting error estimator is equivalent to the equilibrated residual method in the case of isotropic meshes and is proved to be robust on anisotropic meshes as well. A numerical example confirms the theory.
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Blomberg, Magnus. "High Bandwidth Control of a Small Aerial Vehicle." Thesis, Linköpings universitet, Reglerteknik, 2015. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-119622.

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Small aerial vehicles such as quad-rotors have been widely used commercially, for research and for hobby for the last decade with use still growing. The high interest is mainly due to the vehicles being small, simple, cheap and versatile. Among rigid body dynamics fast dynamics exist cohering to motors and other fast actuators. A linear quadratic control design technique is here investigated. The design technique suggests that the linear quadratic controller can be designed with penalties on the slow states only. The fast dynamics are modeled but the states are not penalised in the linear quadratic design. The design technique is here applied and evaluated. The results show that this in several cases is a suitable design technique for linear quadratic control design. MATLAB and Simulink have been widely used for design and implementation of control systems. With additional toolboxes these control systems can be compiled to and run on remote computers. Small, lightweight computers with high computational capacity are now easily accessible. In this thesis an avionics solution based on a small, powerful computer is presented. Simulink models can be compiled and transferred to the computer from the Simulink environment. The result is a user friendly way of rapid prototyping and evaluation of control systems.
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Dalla, Riva Matteo. "Potential theoretic methods for the analysis of singularly perturbed problems in linearized elasticity." Doctoral thesis, Università degli studi di Padova, 2008. http://hdl.handle.net/11577/3426270.

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The dissertation is made of two chapters. The first chapter is dedicated to the investigation of some properties of the layer potentials of a constant coefficient elliptic partial differential operator. In the second chapter, we focus our attention to the Lamè equations, which are related to the physic of an isotropic homogeneous elastic body. In particular, in the first chapter, we investigate the dependence of the single layer potential upon perturbation of the density, the support and the coefficients of the corresponding operator. Under some more restrictive assumptions on the operator, we prove a real analyticity theorem for the single layer potential and its derivatives. As a first step, we introduce a particular fundamental solution of a given constant coefficient partial differential operator. For this purpose, we exploite the construction of a fundamental solution given by John (1955). We have verified that, if the coefficients of the operator are constrained to a bounded set, then there exist a particular fundamental solution which is a sum of functions which depend real analytically on the coefficients of the operator. Such a result resembles the results of Mantlik (1991, 1992) (see also Tréves (1962)), where more general assumptions on the operator are considered. We observe that it is not a corollary. Indeed, we need a suitably detailed expression for the fundamental solution, which cannot be deduced by Mantlik's results. The next step is to introduce the support of our single layer potentials. It will be a compact sub-manifold of the the n-dimensional euclidean space parametrized by a suitable diffeomorphism defined on the boundary of a fixed domain. Then, we will be ready to state in Theorem 1.7 the main result of this chapter, which is a real analyticity result in the frame of Shauder spaces. The main idea of the proof stems from the papers of Lanza de Cristoforis & Preciso (1999) and by Lanza de Cristoforis & Rossi (2004, 2005) and exploits the Implicit Mapping Theorem for real analytic functions. Indeed, our main Theorem 1.7 is in some sense a natural extension of theorems obtained by Lanza de Cristoforis & Preciso (1999) and by Lanza de Cristoforis & Rossi (2004, 2005), for the Cauchy integral and for the Laplace and Helmholtz operators, respectively. Here we confine our attention to elliptic operators which can be factorized with operators of order 2. In the last section of the first chapter, we consider some applications of Theorem 1.7. In particular, we deduce a real analyticity theorem for the single and double layer potential which arise in the analysis of the boundary value problems for the Lamè equations and for the Stokes system. In the second chapter, we focus our attention to the Lamè equations. We consider some boundary value problems defined in a domain with a small hole. For each of them, we investigate the behavior of the solution and of the corresponding energy integral as the hole shrinks to a point. This kind of problem is not new at all and has been long investigated by the techniques of asymptotic analysis. It is perhaps difficult to give a complete list of contributions. Here we mention the work of Keller, Kozlov, Movchan, Maz'ya, Nazarov, Plamenewskii, Ozawa and Ward. The results that we present are in accordance with the behavior one would expect by looking at the above mentioned literature, but we adopt a different approach proposed by Lanza de Cristoforis (2001, 2002, 2005, 2007.) To do so, we exploit the real analyticity results for the elastic layer potentials obtained in the first chapter. We now briefly outline the main difference between our approach and the one of asymptotic analysis. Let d>0 be a parameter which is proportional to the diameter of the hole, so that the singularity of the domain appears when d=0. By the approach of the asymptotic analysis, we can expect to obtain results which are expressed by means of known functions of d plus an unknown term which is smaller than a positive power of d. Whereas, our results are expressed by means of real analytic functions of d defined in a whole open neighborhood of d=0 and by, possibly singular, but completely known functions of d, such as d^(2-n) or log d. Moreover, not only we can consider the dependence upon d, we can also investigate the dependence of the solution and the corresponding energy integral upon perturbations of the coefficients of the operator, and of the point where the hole is situated, and of the shape of the hole, and of the shape of the outer domain, and of the boundary data on the boundary of the hole, and of the boundary data on the boundary of the outer domain, and of the interior data. Also in this case we obtain results expressed by means of real analytic functions and completely known functions such as d^(2-n) and log d. The first boundary value problem we have studied is a Dirichlet boundary value problem with homogeneous data in the interior. Then, we turned to investigate a Robin boundary value problem with homogeneous data in the interior. In this case we have also described the behavior of the solution and the corresponding energy integral when both the domain and the boundary data display a singularity for d=0. Finally, we have studied a Dirichlet boundary value problem with non-homogeneous data in the interior.
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Kunert, Gerd. "A posteriori H^1 error estimation for a singularly perturbed reaction diffusion problem on anisotropic meshes." Universitätsbibliothek Chemnitz, 2001. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-200100730.

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The paper deals with a singularly perturbed reaction diffusion model problem. The focus is on reliable a posteriori error estimators for the H^1 seminorm that can be applied to anisotropic finite element meshes. A residual error estimator and a local problem error estimator are proposed and rigorously analysed. They are locally equivalent, and both bound the error reliably. Furthermore three modifications of these estimators are introduced and discussed. Numerical experiments for all estimators complement and confirm the theoretical results.
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Maddah, Sumayya Suzy. "Formal reduction of differential systems : Singularly-perturbed linear differential systems and completely integrable Pfaffian systems with normal crossings." Thesis, Limoges, 2015. http://www.theses.fr/2015LIMO0065/document.

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Dans cette thèse, nous nous sommes intéressés à l'analyse locale de systèmes différentiels linéaires singulièrement perturbés et de systèmes de Pfaff complètement intégrables et multivariés à croisements normaux. De tels systèmes ont une vaste littérature et se retrouvent dans de nombreuses applications. Cependant, leur résolution symbolique est toujours à l'étude. Nos approches reposent sur l'état de l'art de la réduction formelle des systèmes linéaires singuliers d'équations différentielles ordinaires univariées (ODS). Dans le cas des systèmes différentiels linéaires singulièrement perturbés, les complications surviennent essentiellement à cause du phénomène des points tournants. Nous généralisons les notions et les algorithmes introduits pour le traitement des ODS afin de construire des solutions formelles. Les algorithmes sous-jacents sont également autonomes (par exemple la réduction de rang, la classification de la singularité, le calcul de l'indice de restriction). Dans le cas des systèmes de Pfaff, les complications proviennent de l'interdépendance des multiples sous-systèmes et de leur nature multivariée. Néanmoins, nous montrons que les invariants formels de ces systèmes peuvent être récupérés à partir d'un ODS associé, ce qui limite donc le calcul à des corps univariés. De plus, nous donnons un algorithme de réduction de rang et nous discutons des obstacles rencontrés. Outre ces deux systèmes, nous parlons des singularités apparentes des systèmes différentiels univariés dont les coefficients sont des fonctions rationnelles et du problème des valeurs propres perturbées. Les techniques développées au sein de cette thèse facilitent les généralisations d'autres algorithmes disponibles pour les systèmes différentiels univariés aux cas des systèmes bivariés ou multivariés, et aussi aux systèmes d''equations fonctionnelles<br>In this thesis, we are interested in the local analysis of singularly-perturbed linear differential systems and completely integrable Pfaffian systems in several variables. Such systems have a vast literature and arise profoundly in applications. However, their symbolic resolution is still open to investigation. Our approaches rely on the state of art of formal reduction of singular linear systems of ordinary differential equations (ODS) over univariate fields. In the case of singularly-perturbed linear differential systems, the complications arise mainly from the phenomenon of turning points. We extend notions introduced for the treatment of ODS to such systems and generalize corresponding algorithms to construct formal solutions in a neighborhood of a singularity. The underlying components of the formal reduction proposed are stand-alone algorithms as well and serve different purposes (e.g. rank reduction, classification of singularities, computing restraining index). In the case of Pfaffian systems, the complications arise from the interdependence of the multiple components which constitute the former and the multivariate nature of the field within which reduction occurs. However, we show that the formal invariants of such systems can be retrieved from an associated ODS, which limits computations to univariate fields. Furthermore, we complement our work with a rank reduction algorithm and discuss the obstacles encountered. The techniques developed herein paves the way for further generalizations of algorithms available for univariate differential systems to bivariate and multivariate ones, for different types of systems of functional equations
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Books on the topic "Linear singularly perturbed problem"

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Glizer, Valery Y. Controllability of Singularly Perturbed Linear Time Delay Systems. Springer International Publishing, 2021. http://dx.doi.org/10.1007/978-3-030-65951-6.

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Gajić, Zoran, Djordjija Petkovski, and Xuemin Shen, eds. Singularly Perturbed and Weakly Coupled Linear Control Systems. Springer-Verlag, 1990. http://dx.doi.org/10.1007/bfb0005209.

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Gajić, Zoran. Singularly perturbed and weakly coupled linear control systems: A recursive approach. Springer-Verlag, 1990.

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Myo-Taeg, Lim, ed. Optimal control of singularly perturbed linear systems and applications: High-accuracy techniques. Marcel Dekker, 2001.

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Boglaev, Igor. Domain decomposition in boundary layers for a singularly perturbed parabolic problem. Faculty of Information and Mathematical Sciences, Massey University, 1997.

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Glizer, Valery Y. Controllability of Singularly Perturbed Linear Time Delay Systems. Springer International Publishing AG, 2022.

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Glizer, Valery Y. Controllability of Singularly Perturbed Linear Time Delay Systems. Springer International Publishing AG, 2021.

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Convection Diffusion Problems: An Introduction to Their Analysis and Numerical Solution. American Mathematical Society, 2018.

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Gajic, Zoran. Optimal Control of Singularly Perturbed Linear Systems and Applications. Taylor & Francis Group, 2001.

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Gajic, Zoran. Optimal Control of Singularly Perturbed Linear Systems and Applications. Taylor & Francis Group, 2001.

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Book chapters on the topic "Linear singularly perturbed problem"

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Wasow, Wolfgang. "A Singularly Perturbed Turning Point Problem." In Linear Turning Point Theory. Springer New York, 1985. http://dx.doi.org/10.1007/978-1-4612-1090-0_11.

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Dalla Riva, Matteo, Massimo Lanza de Cristoforis, and Paolo Musolino. "Other Problems with Linear Boundary Conditions in a Domain with a Small Hole." In Singularly Perturbed Boundary Value Problems. Springer International Publishing, 2021. http://dx.doi.org/10.1007/978-3-030-76259-9_9.

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Mahendran, R., and V. Subburayan. "Fitted Numerical Method with Linear Interpolation for Third-Order Singularly Perturbed Delay Problems." In Springer Proceedings in Mathematics & Statistics. Springer Singapore, 2021. http://dx.doi.org/10.1007/978-981-16-7546-1_6.

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Nhan, Thái Anh, and Niall Madden. "Cholesky Factorisation of Linear Systems Coming from Finite Difference Approximations of Singularly Perturbed Problems." In Lecture Notes in Computational Science and Engineering. Springer International Publishing, 2015. http://dx.doi.org/10.1007/978-3-319-25727-3_16.

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Swaminathan, Parthiban, Valarmathi Sigamani, and Franklin Victor. "Numerical Method for a Singularly Perturbed Boundary Value Problem for a Linear Parabolic Second Order Delay Differential Equation." In Springer Proceedings in Mathematics & Statistics. Springer India, 2016. http://dx.doi.org/10.1007/978-81-322-3598-9_7.

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Saminathan, Parthiban, and Franklin Victor. "Numerical Method for a Boundary Value Problem for a Linear System of Partially Singularly Perturbed Parabolic Delay Differential Equations of Reaction-Diffusion Type." In Springer Proceedings in Mathematics & Statistics. Springer Singapore, 2021. http://dx.doi.org/10.1007/978-981-16-7546-1_4.

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Glizer, Valery Y. "Singularly Perturbed Linear Time Delay Systems." In Systems & Control: Foundations & Applications. Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-65951-6_2.

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Bauer, S. M., S. B. Filippov, A. L. Smirnov, P. E. Tovstik, and R. Vaillancourt. "Singularly Perturbed Linear Ordinary Differential Equations." In Asymptotic methods in mechanics of solids. Springer International Publishing, 2015. http://dx.doi.org/10.1007/978-3-319-18311-4_4.

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Gajić, Zoran, and Xuemin Shen. "Singularly Perturbed Weakly Coupled Linear Control Systems." In Parallel Algorithms for Optimal Control of Large Scale Linear Systems. Springer London, 1993. http://dx.doi.org/10.1007/978-1-4471-3219-6_10.

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Dalla Riva, Matteo, Massimo Lanza de Cristoforis, and Paolo Musolino. "A Dirichlet Problem in a Domain with Two Small Holes." In Singularly Perturbed Boundary Value Problems. Springer International Publishing, 2021. http://dx.doi.org/10.1007/978-3-030-76259-9_10.

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Conference papers on the topic "Linear singularly perturbed problem"

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Chitour, Yacine, Jamal Daafouz, Ihab Haidar, Paolo Mason, and Mario Sigalotti. "Necessary conditions for the stability of singularly perturbed linear systems with switching slow-fast behaviors." In 2024 IEEE 63rd Conference on Decision and Control (CDC). IEEE, 2024. https://doi.org/10.1109/cdc56724.2024.10886229.

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Myshkov, Stanislav K., and Vladimir V. Karelin. "Minimax control in the singularly perturbed linear-quadratic stabilization problem." In 2015 International Conference "Stability and Control Processes" in Memory of V.I. Zubov (SCP). IEEE, 2015. http://dx.doi.org/10.1109/scp.2015.7342130.

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Sagara, Muneomi, Hiroaki Mukaidani, and Toru Yamamoto. "Numerical computation of linear quadratic control problem for singularly perturbed stochastic systems." In 2009 International Conference on Networking, Sensing and Control (ICNSC). IEEE, 2009. http://dx.doi.org/10.1109/icnsc.2009.4919367.

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Kodra, Kliti, and Zoran Gajic. "Linear-quadratic-Gaussian problem for a new class of singularly perturbed stochastic systems." In 2016 IEEE 55th Conference on Decision and Control (CDC). IEEE, 2016. http://dx.doi.org/10.1109/cdc.2016.7799407.

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Gajic, Z., Dj Petkovski, and N. Harkara. "The Recursive Algorithm for the Optimal Static Output Feedback Control Problem of Linear Singularly Perturbed Systems." In 1988 American Control Conference. IEEE, 1988. http://dx.doi.org/10.23919/acc.1988.4789818.

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Radisavljevic-Gajic, Verica. "A Simplified Two-Stage Design of Linear Discrete-Time Feedback Controllers With Applications to Systems With Slow and Fast Modes." In ASME 2014 Dynamic Systems and Control Conference. American Society of Mechanical Engineers, 2014. http://dx.doi.org/10.1115/dscc2014-6278.

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Abstract:
In this paper we have shown how to simplify an algorithm for the two stage design of linear feedback controllers by reducing computational requirements. The algorithm is further simplified for linear discrete-time systems with slow and fast modes (multi-time scale systems or singularly perturbed systems) providing independent and accurate designs in slow and fast time scales. The simplified design procedure and its very high accuracy are demonstrated on the eigenvalue assignment problem of a steam power system.
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Edwards Jr., David. "A Numerical Solution of the Semi Linear Singularly Perturbed Boundary Value Problem Using Multi Region Finite Difference Method." In 2009 11th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC). IEEE, 2009. http://dx.doi.org/10.1109/synasc.2009.35.

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Kurina, Galina Alekseevna, and Thi Hoai Nguyen. "On zero order asymptotic solution of singularly perturbed linear - quadratic problems in a critical case." In International Conference "Optimal Control and Differential Games" dedicated to the 110th anniversary of L. S. Pontryagin. Steklov Mathematical Institute, 2018. http://dx.doi.org/10.4213/proc23000.

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Shaldanbayev, Amir, Manat Shomanbayeva, and Asylzat Kopzhassarova. "Solution of a singularly perturbed Cauchy problem for linear systems of ordinary differential equations by the method of spectral decomposition." In INTERNATIONAL CONFERENCE ON ANALYSIS AND APPLIED MATHEMATICS (ICAAM 2016). Author(s), 2016. http://dx.doi.org/10.1063/1.4959704.

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Kurina, Galina A., and Nguyen Thi Hoai. "Projector approach for constructing the zero order asymptotic solution for the singularly perturbed linear-quadratic control problem in a critical case." In INTERNATIONAL CONFERENCE ON ANALYSIS AND APPLIED MATHEMATICS (ICAAM 2018). Author(s), 2018. http://dx.doi.org/10.1063/1.5049067.

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Reports on the topic "Linear singularly perturbed problem"

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Ferguson, Warren E., and Jr. Analysis of a Singularly-Perturbed Linear Two-Point Boundary-Value Problem. Defense Technical Information Center, 1986. http://dx.doi.org/10.21236/ada172582.

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Lou, Xi-Cheng, Alan S. Willsky, and George C. Verghese. An Algebraic Approach to Time Scale Analysis of Singularly Perturbed Linear Systems,. Defense Technical Information Center, 1986. http://dx.doi.org/10.21236/ada186040.

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