• Communications of the Korean Mathematical Society
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• v.21 no.2
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• pp.293-320
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• 2006

Using calculus inequalities and embedding theorems in $R^1$, we establish $W^1_2$-estimates for the solutions of prey-predator population model with cross-diffusion and self-diffusion terms. Two cases are considered; (i) $d_1\;=\;d_2,\;{\alpha}_{12}\;=\;{\alpha}_{21}\;=\;0$, and (ii) $0\;<\;{\alpha}_{21}\;<\;8_{\alpha}_{11},\;0\;<\;{\alpha}_{12}\;<\;8_{\alpha}_{22}$. It is proved that solutions are bounded uniformly pointwise, and that the uniform bounds remain independent of the growth of the diffusion coefficient in the system. Also, convergence results are obtained when $t\;{\to}\;{\infty}$ via suitable Liapunov functionals.

• Journal of the Korean Mathematical Society
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• v.31 no.4
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• pp.669-678
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• 1994

The subject of Liapunov functions constitutes a central theme in the theory of differential equations. It provides powerful tools that can be used to study the behavior of the solutions.

• Journal of the Chungcheong Mathematical Society
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• v.1 no.1
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• pp.11-17
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• 1988

For a given autonomous system x' = f(x), we obtain some properties on the location of positive limit sets and investigate some stability concepts which are based upon the existence of Liapunov functions.

• Journal of the Chungcheong Mathematical Society
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• v.3 no.1
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• pp.69-81
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• 1990

In this paper we investigate the problem of exponential asymptotic stability (EAS) in perturbed nonlinear systems of the differential system x' = f(t, x). Also, a simple method for constructing Liapunov functions is used to prove a kind of Massera type converse theorem.

• Journal of the Chungcheong Mathematical Society
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• v.3 no.1
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• pp.83-88
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• 1990

It is well-known that under suitable conditions, uniform asymptotic stability implies total stability. We prove this theorem of Malkin by using Liapunov-like functions and so our proof is a detailed version of Yoshizawa's proof.

• Proceedings of the KIEE Conference
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• 1991.07a
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• pp.809-812
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• 1991

Two control strategies are proposed for two arm robots; i.e. position-position control and position-force control. For the proof of these control strategies, the stability analysis is conducted with robot dynamics included. First, the closed form dynamic equation of the robot is derived, then it is transformed into the operational space for further analysis. Finally, Liapunov method is applied to the dynamic equation in operational space.

• Journal of the Korean Institute of Telematics and Electronics
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• v.13 no.4
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• pp.18-23
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• 1976

We have obtained the solution of van der Pol's equation characterized by an arctangent nonlinearity, using the perturbation method by writing periodicity conditions: $$X^{(n)}(2{\pi})-X^{(n)}(0)=0$$ $$X^{(n)'}(2{\pi})-X^{(n)'}(0)=0 (n=0,1,2......)$$ together with the starting condition: $$X^{(n)}(\frac{\pi}{2})=0,\;X^{(n)}'(\frac{\pi}{2})=-R^{(n)}$$. Our results agree with Liapunov's theorem and our calculated value is more similar to Murata's measured value than Murata's calculated value.

• Journal of Astronomy and Space Sciences
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• v.15 no.1
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• pp.209-220
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• 1998

The stability problem of steady motion of a rigid body with multi-elastic appendages and multi-liquid-filled cavities, in the presence of no external forces or torque, is considered in this paper. The flexible appendages are modeled as the clamped -free-free-free rectangular plates, or/and as the discrete mass- spring sub-system. The motion of liquid in every single ellipsoidal cavity is modeled as the uniform vortex motion with a finite number of degrees of freedom. Assuming that stationary holonomic constraints imposed on the body allow its rotation about a spatially fixed axis, the equation of motion for such a systematic configuration can be very complex. It consists of a set of ordinary differential equations for the motion of the rigid body, the uniform rotation of the contained liquids, the motion of discrete elastic parts, and a set of partial differential equations for the elastic appendages supplemented by appropriate initial and boundary conditions. In addition, for such a hybrid system, under suitable assumptions, their equations of motion have four types of first integrals, i.e., energy and area, Helmholtz' constancy of liquid - vortexes, and the constant of the Poisson equation of motion. Chetaev's effective method for constructing Liapunov functions in the form of a set of first integrals of the equations of the perturbed motion is employed to investigate the sufficient stability conditions of steady motions of the complete system in the sense of Liapunov, i.e., with respect to the variables determining the motion of the solid body and to some quantities which define integrally the motion of flexible appendages. These sufficient conditions take into account the vortexes of the contained liquids, the vibration of the flexible components, and coupling among the liquid-elasticity solid.

• Journal of Power Electronics
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• v.6 no.4
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• pp.347-355
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• 2006

A new control method for precision robust position control of a permanent magnet synchronous motor (PMSM) is presented. In direct drive motor systems, a load torque disturbance directly affects the motor shaft. The application of the load torque observer is published in using a fixed gain to solve this problem. However, the motor flux linkage cannot be determined precisely for a load torque observer. Therefore, an asymptotically stable adaptive observer base on a deadbeat observer is considered to overcome the problems of unknown parameters, torque disturbance and a small chattering effect. To find the critical parameters the system stability analysis is carried out using the Liapunov stability theorem.

• Proceedings of the KIPE Conference
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• 1997.07a
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• pp.97-100
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• 1997

A new control method for the robust position control of a brushless DC(BLDC) motor using the asymptotically stable adaptive load torque observer is presented. A precision position control is obtained for the BLDC motor system approximately linearized using the field-orientation method. And the application of the load torque observer is published in [1] using fixed gain. However, the flux linkage is not exactly known for a load torque observer. Therefore, a model reference adaptive observer is considered to overcome the problem of the unknown parameter in this paper. And stability analysis is carried out using Liapunov stability theorem. As a result, asymptotically stable observer gain can be obtained without affecting the overall system response. The load disturbance detected by the asymptotically stable adaptive observer is compensated by feedforwarding the equivalent current having the fast response.