• Korean Journal of Mathematics
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• v.21 no.3
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• pp.345-350
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• 2013

Some results on the sensitivity of the solution of the generalized Lyapunov equation $$A^{n-1}X+A^{n-2}XA^*+{\cdots}+X(A^*)^{n-1}=B$$ are shown follow easily from well-known theorems in functional analysis.

• Journal of the Institute of Electronics Engineers of Korea SC
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• v.46 no.4
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• pp.49-57
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• 2009

In this paper we consider the specific types of the generalized continuous-time Lyapunov equation and the existence of solution. This is motivated to analyze the system stability in situations where descriptor system has infinite eigenvalue. As main results, firstly the necessary and sufficient condition for stability of the descriptor system with index one or two will be proposed. Secondly, for the general case of any index, the similar condition for stability of descriptor system will be proposed with the specific type of the generalized Lyapunov equation. Finally some examples are used to show the validity of proposed methods.

• Bulletin of the Korean Mathematical Society
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• v.60 no.1
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• pp.257-279
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• 2023

Applying the Lyapunov-Schmidt reduction, we consider spectral stability of small amplitude stationary periodic solutions bifurcating from an equilibrium of the generalized Swift-Hohenberg equation. We follow the mathematical framework developed in [15, 16, 19, 23] to construct such periodic solutions and to determine regions in the parameter space for which they are stable by investigating the movement of the spectrum near zero as parameters vary.

• International Journal of Control, Automation, and Systems
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• v.1 no.4
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• pp.459-463
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• 2003

In this paper, new results using bounds for the solution of the discrete Lyapunov matrix equation are proposed, and some of the existing works are generalized. The bounds obtained are advantageous in that they provide nontrivial upper bounds even when some existing results yield trivial ones.

• Journal of Institute of Control, Robotics and Systems
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• v.18 no.6
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• pp.513-518
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• 2012

In this paper, we propose a generalized index independent stability condition for a descriptor systemwithout any transformations of system matrices. First, the generalized Lyapunov equation with a specific right-handed matrix form is considered. Furthermore, the existence theorem and the necessary and sufficient conditions for asymptotically stable descriptor systems are presented. Finally, some suitable examples are used to show the validity of the proposed method.

• International Journal of Control, Automation, and Systems
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• v.2 no.4
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• pp.516-522
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• 2004

In this paper we investigate the properties of a generalized impulse response Gramian. The recursive relationship satisfied by the family of Gramians is established. It is shown that the generalized impulse response Gramian contains information on the characteristic polynomial of a linear time-invariant continuous system. The results are applied to model reduction problem.

• Journal of the Institute of Electronics Engineers of Korea SC
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• v.46 no.1
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• pp.34-42
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• 2009

In this paper, new bounds for the solution of the unified Lyapunov matrix equation for decentralized singularly perturbed systemare obtained, and some of the existing works using deficient assumptions are also generalized.

• Proceedings of the Korean Society for Noise and Vibration Engineering Conference
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• 1997.10a
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• pp.145-153
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• 1997

Dynamic characteristics are investigated when a nonlinear system showing periodic and chaotic responses under harmonic excitation is exposed to random perturbation. About two well potential problem, probability of homoclinic bifurcation is estimated using stochastic generalized Meinikov process and quantitive characteristics are investigated by calculation of Lyapunov exponent. Critical excitaion is calculated by various assumptions about Gaussian Melnikov process. To verify the phenomenon graphically Fokker-Planck equation is solved numerically and the original nonlinear equation is numerically simulated. Numerical solution of Fokker-Planck equation is calculated on Poincare section and noise induced chaos is studied by solving the eigenvalue problem of discretized probability density function.

• Journal of the Korean Institute of Telematics and Electronics
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• v.25 no.12
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• pp.1625-1632
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• 1988

This paper aims not only to introduce the concept of linear singular systems, geometric structure, and feedback but also to provide applications of the multivariable linear singular system theories to electric circuits which may appear in some electronic equipments. The impulsive or discontinuous behavior which is not desirable can be removed by the set of admissible initial conditions. The output-nulling supremal (A,E,B) invariant subspace and the singular system structure algorithm are applied to this double-input double-output electric circuit. The Weierstrass form of the pencil (s E-A) is related to the output-nulling supremal (A,E,B) invariant subspace from which the time domain solutions of the finite and the infinite subsystems are found. The generalized Lyapunov equation for this application with feedback is studied and finally, the use of orthogonal functions in singular systems is discussed.