• Journal of applied mathematics & informatics
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• 제16권1_2호
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• pp.265-277
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• 2004

This article deals with the number of periodic solutions of the second order polynomial differential equation using the Riccati equation, and applies the property of the solutions of the Riccati equation to study the property of the solutions of the more complicated differential equations. Many valuable criterions are obtained to determine the number of the periodic solutions of these complex differential equations.

• 한국경영과학회지
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• 제9권2호
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• pp.28-33
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• 1984

In this paper some procedures are given whereby an analytic solution may be found for the Riccati differential equation and algebraic Riccati equation in optimal control theory. Some iterative techniques for solving these equations are presented. Rate of convergence and initialization of the iterative processes are discussed.

• 대한전자공학회논문지
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• 제16권5호
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• pp.18-26
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• 1979

본 논문에서는 algebraic matrix Lyapunov equations과 a1gebraic matrix Riccati equations에 관하여 잘 알려져 있는 중요한 결과를 확장한다. 본 연구는 Matrix 미분 방정식에서 양단 경계조건이 존재하는 문제를 다루며 여기에서 얻어지는 결과는 기존하고 있는 결과를 포함하게 된다. 특히 선형 시스템이 periodic feedback gain control로 안정화되는 필요충분조건을 구하며, two-point boundary Riccati equations의 해를 쉽게 구하는 반복 계산방법을 제시한다. 또한 interalwise reeceding horizon을 이용한 새로운 periodic feedback gain control이 시스템을 안전화시켜줌을 보여준다.

• 대한수학회지
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• 제50권1호
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• pp.173-187
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• 2013

This paper studies the properties of solutions of the Riccati equation arising from the quadratic optimal control problem of the general damped second order system. Using the semigroup theory, we establish the weak differential characterization of the Riccati equation for a general class of the second order distributed systems with arbitrary damping terms.

• Coupled systems mechanics
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• 제2권3호
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• pp.255-269
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• 2013

This paper targets to investigate the solution of fuzzy quadratic Riccati differential equations with various types of fuzzy environment using Homotopy Perturbation Method (HPM). Fuzzy convex normalized sets are used for the fuzzy parameter and variables. Obtained results are depicted in term of plots to show the efficiency of the proposed method.

• 제어로봇시스템학회:학술대회논문집
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• 제어로봇시스템학회 1995년도 Proceedings of the Korea Automation Control Conference, 10th (KACC); Seoul, Korea; 23-25 Oct. 1995
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• pp.10-13
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• 1995

This paper considers the linear-quadratic optimal regulator problem for nonstandard singularly perturbed systems making use of the recursive technique. We first derive a generalized Riccati differential equation by the Hamilton-Jacobi equation. In order to obtain the feedback gain, we must solve the generalized algebraic Riccati equation. Using the recursive technique, we show that the solution of the generalized algebraic Riccati equation converges with the rate of convergence of O(.epsilon.). The existence of a bounded solution of error term can be proved by the implicit function theorem. It is enough to show that the corresponding Jacobian matrix is nonsingular at .epsilon. = 0. As a result, the solution of optimal regulator problem for nonstandard singularly perturbed systems can be obtained with an accuracy of O(.epsilon.$^{k}$ ). The proposed technique represents a significant improvement since the existing method for the standard singularly perturbed systems can not be applied to the nonstandard singularly perturbed systems.

• Korean Journal of Mathematics
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• 제23권1호
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• pp.11-27
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• 2015

Nonlinear partial differential equations are more suitable to model many physical phenomena in science and engineering. In this paper, we consider three nonlinear partial differential equations such as Novikov equation, an equation for surface water waves and the Geng-Xue coupled equation which serves as a model for the unidirectional propagation of the shallow water waves over a at bottom. The main objective in this paper is to apply the generalized Riccati equation mapping method for obtaining more exact traveling wave solutions of Novikov equation, an equation for surface water waves and the Geng-Xue coupled equation. More precisely, the obtained solutions are expressed in terms of the hyperbolic, the trigonometric and the rational functional form. Solutions obtained are potentially significant for the explanation of better insight of physical aspects of the considered nonlinear physical models.

• 대한수학회지
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• 제44권6호
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• pp.1453-1467
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• 2007

By means of a Riccati transform some oscillation or nonoscillation criteria are established for nonlinear differential equations of second order $$(E_1)\;[p(t)|x#(t)|^{\alpha}sgn\;x#(t)]#+q(t)|x(\tau(t)|^{\alpha}sgn\;x(\tau(t))=0$$. $$(E_2),\;(E_3)\;and\;(E_4)\;where\;0<{\alpha}$$ and $${\tau}(t){\leq}t,\;{\tau}#(t)>0,\;{\tau}(t){\rightarrow}{\infty}\;as\;t{\rightarrow}{\infty}$$. In this paper we improve some previous results.

• 대한수학회보
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• 제45권2호
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• pp.241-252
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• 2008

By means of a Riccati transform and averaging technique some oscillation criteria are established for perturbed nonlinear differential equations of second order $(P_1)\;(p(t)x'(t))'+q(t)|x({\phi}(t)|^{{\alpha}+1}sgnx({\phi}(t))+g(t,\;x(t))=0$ $(P_2)$ and $(P_3)$ satisfying the condition (H). A comparison theorem and examples are given.

• Journal of applied mathematics & informatics
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• 제27권5_6호
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• pp.1165-1176
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• 2009

This paper is concerned with the interval oscillation of the second order nonlinear ordinary differential equation (r(t)|y'(t)|$^{{\alpha}-1}$ y'(t))'+p(t)|y'(t)|$^{{\alpha}-1}$ y'(t)+q(t)f(y(t))g(y'(t))=0. By constructing ageneralized Riccati transformation and using the method of averaging techniques, we establish some interval oscillation criteria when f(y) is not differetiable but satisfies the condition $\frac{f(y)}{|y|^{{\alpha}-1}y}$ ${\geq}{\mu}_0$ > 0 for $y{\neq}0$.