• Journal of Institute of Control, Robotics and Systems
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• v.19 no.4
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• pp.281-284
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• 2013

This paper describes a reduced observer design problem for one-sided Lipschitz nonlinear systems which are considered as a generalization of Lipschitz systems. The sufficient conditions to ensure the existence of reduced order observer are provided by using linear matrix inequalities. Moreover, it is shown that existence conditions of reduced order observer can be obtained from sufficient conditions on the existence of full order observer. As a result, this fact implies that the existence of full order observer for one-sided Lipschitz systems guarantees that of reduced order observer. Finally, a simulation example is given to verify the validness of the proposed design.

• Journal of the Korean Mathematical Society
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• v.34 no.4
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• pp.959-971
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• 1997

In the paper we introduce Hausdorff measures which are suitable or the study of Lipschitz regularity of M-harmonic function in the unit ball B in $C^n$. For an M-harmonic function h which satisfies certain integrability conditions, we show that there is an open set $\Omega$, whose Hausdorff content is arbitrarily small, such that h is Lipschitz smooth on $B \backslash \Omega$.

• Journal of applied mathematics & informatics
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• v.16 no.1_2
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• pp.1-17
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• 2004

The famous Newton-Kantorovich hypothesis has been used for a long time as a sufficient condition for the convergence of Newton method to a solution of an equation in connection with the Lipschitz continuity of the Frechet-derivative of the operator involved. Using Lipschitz and center-Lipschitz conditions we show that the Newton-Kantorovich hypothesis is weakened. The error bounds obtained under our semilocal convergence result are finer and the information on the location of the solution more precise than the corresponding ones given by the dominating Newton-Kantorovich theorem, and under the same hypotheses/computational cost, since the evaluation of the Lipschitz also requires the evaluation of the center-Lipschitz constant. In the case of local convergence we obtain a larger convergence radius than before. This observation is important in computational mathematics and can be used in connection to projection methods and in the construction of optimum mesh independence refinement strategies.

• Journal of the Korean Mathematical Society
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• v.49 no.6
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• pp.1301-1321
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• 2012

In this paper we study the unique solvability of backward stochastic Volterra integral equations (BSVIEs in short), in terms of both the adapted M-solutions introduced in [19] and the adapted solutions via a new method. A general existence and uniqueness of adapted M-solutions is proved under non-Lipschitz conditions by virtue of a briefer argument than the ones in [13] and [19], which modifies and extends the results in [13] and [19] respectively. For the adapted solutions, the unique solvability of BSVIEs under more general stochastic non-Lipschitz conditions is shown, which improves and generalizes the results in [7], [14] and [15].

• Journal of applied mathematics & informatics
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• v.10 no.1_2
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• pp.75-81
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• 2002

Lipschitz stability and Lipschitz ø$_{o}$ - equistability of the functional differential equation x'= B(x)f(t, x, $x_{t}$), $x_{to}$ =$\theta$$_{o}$ are discussed. Sufficient conditions are given using the comparison with the corresponding scalar equation.ion.n.

• Korean Journal of Mathematics
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• v.24 no.1
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• pp.1-13
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• 2016

This paper shows that the solutions to the perturbed functional dierential system $$y^{\prime}=f(t,y)+{\int_{t_0}^{t}}g(s,y(s),Ty(s))ds$$ have uniformly Lipschitz stability and asymptotic property. To sRhow these properties, we impose conditions on the perturbed part ${\int_{t_0}^{t}}g(s,y(s),Ty(s))ds$ and the fundamental matrix of the unperturbed system $y^{\prime}=f(t,y)$.

• Journal of Institute of Control, Robotics and Systems
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• v.14 no.8
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• pp.837-841
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• 2008

This paper presents a robust reduced order state observer for a class of Lipschitz nonlinear systems with external disturbance. Sufficient conditions on the existence of the proposed observer are characterized by linear matrix inequalities. It is also shown that the proposed observer design can reduce the effect on the estimation error of external disturbance up to the prescribed level. Finally, a numerical example is provided to verify the proposed design method.

• Journal of the Chungcheong Mathematical Society
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• v.29 no.3
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• pp.429-442
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• 2016

In this paper we show that the solutions to the perturbed differential system $$y^{\prime}=f(t,y)+{\int}_{to}^{t}g(s,y(s),Ty(s))ds$$ have uniformly Lipschitz stability and asymptotic behavior by imposing conditions on the perturbed part $\int_{to}^{t}g(s,y(s),Ty(s))ds$ and the fundamental matrix of the unperturbed system y' = f(t, y).

• Journal of the Korean Mathematical Society
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• v.43 no.2
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• pp.241-258
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• 2006

This paper gives conditions under which necessary optimality conditions in a locally Lipschitz program can be expressed as the invexity of the active constraint functions or the type I invexity of the objective function and the constraint functions on the feasible set of the program. The results are nonsmooth extensions of those of Hanson and Mond obtained earlier in differentiable case.

• Journal of the Korean Mathematical Society
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• v.44 no.4
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• pp.873-887
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• 2007

In this paper cellular neutral networks with time-varying delays and continuously distributed delays are considered. Without assuming the global Lipschitz conditions of activation functions, some sufficient conditions for the existence and exponential stability of the almost periodic solutions are established by using the fixed point theorem and differential inequality techniques. The results of this paper are new and complement previously known results.