• Journal of the Korean Mathematical Society
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• v.56 no.5
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• pp.1419-1439
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• 2019

In this paper, we consider a class of noncooperative fourth-order elliptic systems involving nonlocal terms and critical growth in a bounded domain. With the help of Limit Index Theory due to Li [32] combined with the concentration compactness principle, we establish the existence of infinitely many solutions for the problem under the suitable conditions on the nonlinearity. Our results significantly complement and improve some recent results on the existence of solutions for fourth-order elliptic equations and Kirchhoff type problems with critical growth.

• Journal of the Korean Mathematical Society
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• v.60 no.3
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• pp.565-586
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• 2023

We study the Dirichlet problem for the degenerate nonlocal parabolic equation u_t - a(||∇u||²_L²(Ω))∆u = C_b ||u||^β_L²(Ω) |u|^q(x,t)-2 u log |u| + f in Q_T, where Q_T := Ω × (0, T), T > 0, Ω ⊂ ℝ^N, N ≥ 2, is a bounded domain with a sufficiently smooth boundary, q(x, t) is a measurable function in Q_T with values in an interval [q^-, q⁺] ⊂ (1, ∞) and the diffusion coefficient a(·) is a continuous function defined on ℝ₊. It is assumed that a(s) → 0 or a(s) → ∞ as s → 0⁺, therefore the equation degenerates or becomes singular as ||∇u(t)||2 → 0. For both cases, we show that under appropriate conditions on a, β, q, f the problem has a global in time strong solution which possesses the following global regularity property: ∆u ∈ L²(Q_T) and a(||∇u||²_L²(Ω))∆u ∈ L²(Q_T ).

• Journal of the Computational Structural Engineering Institute of Korea
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• v.32 no.1
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• pp.55-63
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• 2019

In engineering problems, many random variables have correlation, and the correlation of input random variables has a great influence on reliability analysis results of the mechanical systems. However, correlated variables are often treated as independent variables or modeled by specific parametric joint distributions due to difficulty in modeling joint distributions. Especially, when there are insufficient correlated data, it becomes more difficult to correctly model the joint distribution. In this study, multivariate kernel density estimation with bounded data is proposed to estimate various types of joint distributions with highly nonlinearity. Since it combines given data with bounded data, which are generated from confidence intervals of uniform distribution parameters for given data, it is less sensitive to data quality and number of data. Thus, it yields conservative statistical modeling and reliability analysis results, and its performance is verified through statistical simulation and engineering examples.

• 제어로봇시스템학회:학술대회논문집
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• 1997.10a
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• pp.732-735
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• 1997

Variable Structure Controller(VSC) is usually known to have robustness to bounded exogenous disturbances. The robustness is attributed to the discontinuous term in the control input. However, this discontinuous term also causes an undesirable effect called chattering. To alleviate chattering, a hybrid controller consisting of VSC and Fuzzy Logic Controller(FLC) is proposed, which belongs to the category of Fuzzy Sliding Mode Controller(FSMC). The role of FLC in FSMC is to replace a fixed gain of a discontinuous term with a time-varying one based on a specified rule base. The characteristics of proposed controller are shown to be similar to those of VSC with a saturation function instead of sign function. The only remarkable difference is the nonlinearity whose form can be adjusted by free parameters, normalize gain, denormalize gain, and membership functions. Applied to AC servo motor, the proposed controller is compared with VSC in a regulation problem as well as a speed tracking problem. The simulation results show a substantial chatter reduction.

• International Journal of Control, Automation, and Systems
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• v.1 no.1
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• pp.19-27
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• 2003

It is well known that one can easily design a high-gain adaptive output feedback control for a class of nonlinear systems which satisfy a certain condition called output feedback exponential passivity (OFEP). The designed high-gain adaptive controller has simple structure and high robustness with regard to bounded disturbances and unknown order of the controlled system. However, from the viewpoint of practical application, it is important to consider a robust control scheme for controlled systems for which some of the assumptions of output feedback stabilization are not valid. In this paper, we design a robust high-gain adaptive output feedback control for the OFEP nonlinear systems with uncertain nonlinearities and/or disturbances. The effectiveness of the proposed method is shown by numerical simulations.

• Korean Journal of Mathematics
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• v.17 no.3
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• pp.283-291
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• 2009

We investigate the existence of multiple solutions (${\xi},{\eta}$) for perturbations of the parabolic system with Dirichlet boundary condition $$(0.1)\;\begin{array}{lcr}{\xi}_t=-L{\xi}+g_1(3{\xi}+{\eta})-s{\phi}_1-h_1(x,t)\;in\;{\Omega}{\times}(0,\;2{\pi}),\\{\eta}_t=-L{\eta}+g_2(3{\xi}+{\eta})-s{\phi}_1-h_2(x,t)\;in\;{\Omega}{\times}(0,\;2{\pi}).\end{array}$$ We show the existence of multiple solutions (${\xi},{\eta}$) for perturbations of the parabolic system when the nonlinearity $g^{\prime}_1,\;g^{\prime}_2$ are bounded and $3g^{\prime}_1(-{\infty})+g^{\prime}_2(-{\infty})<{\lambda}_1,\;{\lambda}_n<3g^{\prime}_1(+{\infty})+g^{\prime}_2(+{\infty})<{\lambda}_{n+1}$.

• Journal of the Korean Mathematical Society
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• v.55 no.3
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• pp.531-551
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• 2018

In this paper we consider a class of nonlocal parabolic equations in bounded domains with Dirichlet boundary conditions and a new class of nonlinearities. We first prove the existence and uniqueness of weak solutions by using the compactness method. Then we study the existence and fractal dimension estimates of the global attractor for the continuous semigroup generated by the problem. We also prove the existence of stationary solutions and give a sufficient condition for the uniqueness and global exponential stability of the stationary solution. The main novelty of the obtained results is that no restriction is imposed on the upper growth of the nonlinearities.

• Bulletin of the Korean Mathematical Society
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• v.50 no.5
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• pp.1693-1710
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• 2013

In this paper, we consider the biharmonic elliptic systems of the form $$\{{\Delta}^2u=F_u(u,v)+{\lambda}{\mid}u{\mid}^{q-2}u,\;x{\in}{\Omega},\\{\Delta}^2v=F_v(u,v)+{\delta}{\mid}v{\mid}^{q-2}v,\;x{\in}{\Omega},\\u=\frac{{\partial}u}{{\partial}n}=0,\; v=\frac{{\partial}v}{{\partial}n}=0,\;x{\in}{\partial}{\Omega},$$, where ${\Omega}{\subset}\mathbb{R}^N$ is a bounded domain with smooth boundary ${\partial}{\Omega}$, ${\Delta}^2$ is the biharmonic operator, $N{\geq}5$, $2{\leq}q$ < $2^*$, $2^*=\frac{2N}{N-4}$ denotes the critical Sobolev exponent, $F{\in}C^1(\mathbb{R}^2,\mathbb{R}^+)$ is homogeneous function of degree $2^*$. By using the variational methods and the Ljusternik-Schnirelmann theory, we obtain multiplicity result of nontrivial solutions under certain hypotheses on ${\lambda}$ and ${\delta}$.

• 제어로봇시스템학회:학술대회논문집
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• 2002.10a
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• pp.102.4-102
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• 2002

A deadzone compensator is designed for a positioning system using fuzzy logic. The classification property of fuzzy logic systems make them a natural candidate for the rejection of errors induced by the deadzone, which has regions in which it behaves differently. A tuning algorithm is given for the fuzzy logic parameters, so that the deadzone compensation scheme becomes adaptive, guaranteeing small tracking errors and bounded parameter estimates, formal nonlinear stability proofs are given to show that the tracking error is small. The fuzzy logic deadzone compensator is implemented on a positioning system to show its efficacy. 1. Deadzone Compansation 2. XY positioning table 3. Fuzzy Logic 4. Actuator nonlinearity

• Journal of the Institute of Electronics Engineers of Korea SC
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• v.41 no.2
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• pp.17-28
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• 2004

A deadzone compensator is designed for a XY positioning table using fuzzy logic. The classification property of fuzzy logic systems makes them a natural candidate for the rejection of errors induced by the deadzone, which has regions in which it behaves differently. A tuning algorithm is given for the fuzzy logic parameters, so that the deadzone compensation scheme becomes adaptive, guaranteeing small tracking errors and bounded parameter estimates. Formal nonlinear stability proofs are given to show that the tracking error is small. The fuzzy logic deadzone compensator is implemented on a XY positioning table to show its efficacy.