• Nonlinear Functional Analysis and Applications
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• v.26 no.1
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• pp.35-64
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• 2021

In this paper, we investigate an initial boundary value problem for a nonlinear pseudoparabolic equation. At first, by applying the Faedo-Galerkin, we prove local existence and uniqueness results. Next, by constructing Lyapunov functional, we establish a sufficient condition to obtain the global existence and exponential decay of weak solutions.

• Bulletin of the Korean Mathematical Society
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• v.58 no.3
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• pp.669-688
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• 2021

The aim of this paper is to prove the existence of an admissible inertial manifold for mild solutions to infinite delay evolution equation of the form $$\{{\frac{du}{dt}}+Au=F(t,\;u_t),\;t{\geq}s,\\\;u_s({\theta})={\phi}({\theta}),\;{\forall}{\theta}{\in}(-{{\infty}},\;0],\;s{\in}{\mathbb{R}},$$ where A is positive definite and self-adjoint with a discrete spectrum, the Lipschitz coefficient of the nonlinear part F may depend on time and belongs to some admissible function space defined on the whole line. The proof is based on the Lyapunov-Perron equation in combination with admissibility and duality estimates.

• Structural Engineering and Mechanics
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• v.58 no.6
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• pp.1127-1143
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• 2016

The pounding phenomenon in adjacent structures happens in severing earthquakes that can cause great damages. Connecting neighboring structures with active and semi-active control devices is an effective method to avoid mutual colliding between neighboring buildings. One of the most important issues in control systems is applying online control force. There will be a time delay if the prose of producing control force does not perform on time. This paper proposed a time-delay compensation method in coupled structures control, with semi-active Magnetorheological (MR) damper. This method based on Newmark's integration is adopted to mitigate the time-delay effect. In this study, Lyapunov's direct approach is employed to compute demanded voltage for MR dampers. Using Lyapunov's direct algorithm guarantees the system stability to design a controller based on feedback. Because of the strong nonlinearity of MR dampers, the equation of motion of coupled structures becomes an involved equation, and it is impossible to solve it with the common time step methods. In present paper modified Newmark-Beta integration based on the instantaneous optimal control algorithm, used to solve the involved equation. In this method, the response of a coupled system estimated base on optimal control force. Two MDOF structures with different degrees of freedom are finally considered as a numeric example. The numerical results show, the Newmark compensation is an efficient method to decrease the negative effect of time delay in coupled systems; furthermore, instantaneous optimal control algorithm can estimate the response of structures suitable.

• Bulletin of the Korean Mathematical Society
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• v.57 no.2
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• pp.281-294
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• 2020

The initial-boundary value problem for a class of semilinear Klein-Gordon equation with logarithmic nonlinearity in bounded domain is studied. The existence of global solution for this problem is proved by using potential well method, and obtain the exponential decay of global solution through introducing an appropriate Lyapunov function. Meanwhile, the blow-up of solution in the unstable set is also obtained.

• Journal of Mechanical Science and Technology
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• v.17 no.11
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• pp.1714-1724
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• 2003

Feedback strategies of a qualitative competitive game between two players can be designed such as to influence parameters of a mechanical system to induce chaotic behavior. The purpose is to reduce the options and effects of the opponent's strategy. We show it on a case with dynamics specified by a nonautonomous Duffing equation with the players represented by damping and external forcing, respectively. It seems however that the conclusions can be made valid generally.

• Journal of the Korean Mathematical Society
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• v.58 no.5
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• pp.1279-1298
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• 2021

The aim of this paper is to prove the existence of the global attractor of the Cauchy problem for a semilinear degenerate hyperbolic equation involving strongly degenerate elliptic differential operators. The attractor is characterized as the unstable manifold of the set of stationary points, due to the existence of a Lyapunov functional.

• Journal of Electrical Engineering and Technology
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• v.6 no.5
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• pp.697-705
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• 2011

In this paper, a closed-loop system constructed with a linear plant and nonlinearity in the feedback connection is considered to argue against its planar orbital stability. Through a state space approach, a main result that presents a sufficient stability criterion of the limit cycle predicted by solving the harmonic balance equation is given. Preliminarily, the harmonic balance of the nonlinear feedback loop is assumed to have a solution that determines the characteristics of the limit cycle. Using a state-space approach, the nonlinear loop equation is reformulated into a linear perturbed model through the introduction of a residual operator. By considering a series of transformations, such as a modified eigenstructure decomposition, periodic averaging, change of variables, and coordinate transformation, the stability of the limit cycle can be simply tested via a scalar function and matrix. Finally, the stability criterion is addressed by constructing a composite Lyapunov function of the transformed system.

• 제어로봇시스템학회:학술대회논문집
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• 1988.10b
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• pp.749-752
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• 1988

In this paper, the delay margins of the LQ and the LQG regulators are obtained in the time domain. These margins are represented in terms of the singular values of system matrices and the solutions of a Riccati equation and a Lyapunov one. And their asymptotical properties when gains tend to infinity are investigated.

• 제어로봇시스템학회:학술대회논문집
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• 1988.10b
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• pp.1051-1054
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• 1988

In this paper we present initial convergence properties of the Kalman filtering algorithm, we put an arbitrary small positive correlation matrix as an initial condition in the recursive algorithm. This arbitrary small initial condition perturbs the Kalman filtering algorithm and may lead to initial instability. We derive a condition which insures the stable operation of the Kalman filtering algorithm from the stochastic Lyapunov difference equation.

• Journal of the Korean Institute of Intelligent Systems
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• v.13 no.2
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• pp.175-179
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• 2003

Convergence condition determines performance of iterative learning control (ILC), for example, convergence speed, remaining error, etc. Hence, the performance can be elevated and a feasible set of learning controllers grows if a less conservative condition is obtained. In the frequency domain, the $H_{\infty}$ norm of the transfer function between consecutive errors has been currently used to test convergence of a learning system. However, even if the convergence condition based on the $H_{\infty}$ norm has a clear property about monotonic convergence, it has a few drawbacks, especially in MIMO plants. In this paper, the relation between the condition and the monotonicity of convergence is clarified and a modified convergence condition is found out using a frequency domain Lyapunov equation, which supersedes the conventional one in the frequency domain.