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Linear Singularly Perturbed Systems
Ukrainian Mathematical Journal, 2002 This paper deals with singularly perturbed linear systems of the form \[ \epsilon B(t)\dot x=(A_0(t)+\epsilon A_1(t))x. \] The case is studied where the matrix pencil \(A_0(t)-\lambda B(t)\) is singular so that \(\det B(t)\equiv0\) and \(\det(A_0(t)-\lambda B(t))\equiv0\).
Starun, I. I., Shkil', M. I.
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On the Exponential Stability of Singularly Perturbed Systems
SIAM Journal on Control and Optimization, 1992For singularly perturbed systems \(\dot x=f(t,x,z,\mu)\), \(\mu\dot z=g(t,x,z,\mu)\), \textit{A. Saberi} and \textit{H. Khalil} [IEEE Trans. Autom. Control AC--29, 542-550 (1984; Zbl 0538.93049)] have shown that if both the reduced-order system \((\mu=0)\) and the boundary-layer system are exponentially stable, then also the full-order system is stable
Martin Corless, Luigi Glielmo
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ON THE CONTROL OF SINGULARLY PERTURBED NONLINEAR SYSTEMS
IFAC Proceedings Volumes, 1992Abstract Using the integral manifold concept, a recent systematic method to control e-perturbed nonlinear systems is here extended to a rather wide class of singularly perturbed MIMO systems. This class includes singularly perturbed systems exibiting an affine structure in the control after order reduction.
Barbot J. P. +3 more
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Singularly Perturbed Evolution Inclusions
SIAM Journal on Control and Optimization, 2010Consider the control system \[ \dot{x}(t)+ A_{1}x \in F(x,y,u(t)),\quad x(0) = x^{0}\in H_{1},\quad u(t)\in U, \tag{1} \] \[ \varepsilon\dot{y}(t)+ A_{2}y \in G(x,y,u(t)),\quad y(0) = y^{0}\in H_2,\quad t\in [0,1], \tag{2} \] where \(F:H_{1}\times H_{2}\times U \rightrightarrows H_{1}\), \(G:H_{1}\times H_{2}\times U \rightrightarrows H_{2}\) are ...
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Applicable Analysis, 1987
This paper discusses existence and nonexistence of C1 quasi-steady-states to singularly perturbed problems near a singular point. In contrast to the existence and uniqueness result well known for the same problem near a regular point, the answer depends on generic conditions involving both the differential and the transcendental equations of the system.
Patrick J. Rabier, Shiva Shankar
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This paper discusses existence and nonexistence of C1 quasi-steady-states to singularly perturbed problems near a singular point. In contrast to the existence and uniqueness result well known for the same problem near a regular point, the answer depends on generic conditions involving both the differential and the transcendental equations of the system.
Patrick J. Rabier, Shiva Shankar
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Singularly Perturbed Eigenvalue Problems
SIAM Journal on Applied Mathematics, 1987This paper is concerned with eigenvalue problems of singularly perturbed linear ordinary differential equations. A common way to treat such problems is to derive an approximating eigenvalue problem by the use of matched asymptotic expansions. It is shown that under appropriate assumptions a domain in the complex plane can be identified, in which the ...
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Singularly Perturbed Semilinear Systems
Studies in Applied Mathematics, 1979Solutions of a singularly perturbed vector boundary‐value problem are studied under the principal assumption that the trivial solution of the unperturbed equation is stable in certain senses. This is accomplished by constructing special invariant regions in which solutions display the kind of nonuniformity known as boundary‐layer behavior.
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Quasilinear singularly perturbed problem with boundary perturbation
Journal of Zhejiang University-SCIENCE A, 2004A class of quasilinear singularly perturbed problems with boundary perturbation is considered. Under suitable conditions, using theory of differential inequalities we studied the asymptotic behavior of the solution for the boundary value problem.
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Unfolding Singularly Perturbed Bogdanov Points
SIAM Journal on Mathematical Analysis, 2000Summary: Bogdanov points that occur in the fast dynamics of singular perturbation problems are often encountered in applications; e.g., in the van der Pol-Duffing oscillator [\textit{M. Koper}, Physica D 80, No. 1-2, 72--94 (1995; Zbl 0889.34034)] or in the FitzHugh-Nagumo equation [\textit{W.-J. Beyn} and \textit{M. Stiefenhofer}, J. Dyn.
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