Global asymptotic stability of nonautonomous systems of Liénard type

Abstract

This paper deals with nonautonomous Liénard-type systems. Sufficient conditions are given for the zero solution of the systems to be globally asymptotically stable. The main result is proved by means of phase plane analysis with a Liapunov function. Examples are included to contrast our theorem with results which were presented by Hatvani and Cantarelli. Some global phase portraits are also attached.

An Integrated Approach to Hypersonic Entry Attitude Control

“…However, additionally considering dynamic response features, we choose α0i = 0.85 for the final controller. These two cases show that, as already discussed for (26), proper NF can feasibly improve control performance over LF. The basic reason lies in that NF provides a nonlinear mechanism that completely agrees with the intuition obtained from practical engineering experiences.…”

“…For this well-known system, mathematicians and physicists have obtained many interesting results on its boundedness, stability, and existence of periodic solutions and almost periodic solutions. The interested reader is advised to refer to [25][26][27] and the references quoted therein. Next, we utilize these qualitative properties to obtain some more comprehensive results on nonlinear ESO stability and estimation error than those in [28][29][30], particularly on the asymptotic behavior of solutions with disturbances that are absolutely integrable.…”

Necessary and sufficient conditions for exponential stability of a family of generalized Liénard differential equations

“…The Lie´nard equation has been studied in a great number of papers and it is included in most textbooks and monographs on nonlinear dynamical systems and qualitative theory of differential equations (see, e.g. [5][6][7][8][9][10][11][12][13]). Equation (1) is equivalent to the system…”

“…Since it belongs to class (4), (5), inequality (11) may serve as a sufficient stability condition for system (2), (3) (note that sufficient stability conditions for this system obtained by the Lyapunov function method can be found, in particular, in [9] and [13]). However, as is mentioned above, the necessity of condition (11) for system (4), (5) does not imply its necessity for (2), (3).…”

On the asymptotic behavior of solutions of certain second-order differential equations

“…(1) or different forms of differential equations of Lienard type. The interested reader is advised to look up the references cited in the listed publications [1][2][3][4][5][6][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][25][26][27][28] and the references quoted therein for some works. In [8], Hatvani established sufficient conditions for stability of the zero solution of the differential equation:…”