TY - JOUR
T1 - A remark about asymptotic stability in Duffing equations
T2 - lateral stability in Comb-drive finger MEMS
AU - Núñez, D.
AU - Murcia, L.
N1 - Publisher Copyright:
© 2023, The Author(s).
PY - 2023
Y1 - 2023
N2 - In this short paper we tackle two subjects. First, we provide a lower bound for the first eigenvalue of the antiperiodic problem for a Hill’s equation based on Lp -conditions, and as a consequence, we introduce an adjusted statement of the main result about the asymptotic stability of periodic solutions for the general Duffing equation in (Torres in Mediterr. J. Math. 1(4):479–486, 2004) (Theorem 4). This appropriate version of the result arises because of one subtlety in the proof provided in (Torres in Mediterr. J. Math. 1(4):479–486, 2004). More precisely, the lower bound of the first antiperiodic eigenvalue associated with Hill’s equations of potential a(t) employed there may be negative, thus the conclusion is not completely attained. Hence, the adjustments considered here provide a mathematically correct result. On the other hand, we apply this result to obtain a lateral asymptotic stable periodic oscillation in the Comb-drive finger MEMS model with a cubic nonlinear stiffness term and linear damping. This fact is not typical in Comb-drive finger devices, thus our results could provide a new possibility; a new design principle for stabilization in Comb-drive finger MEMS.
AB - In this short paper we tackle two subjects. First, we provide a lower bound for the first eigenvalue of the antiperiodic problem for a Hill’s equation based on Lp -conditions, and as a consequence, we introduce an adjusted statement of the main result about the asymptotic stability of periodic solutions for the general Duffing equation in (Torres in Mediterr. J. Math. 1(4):479–486, 2004) (Theorem 4). This appropriate version of the result arises because of one subtlety in the proof provided in (Torres in Mediterr. J. Math. 1(4):479–486, 2004). More precisely, the lower bound of the first antiperiodic eigenvalue associated with Hill’s equations of potential a(t) employed there may be negative, thus the conclusion is not completely attained. Hence, the adjustments considered here provide a mathematically correct result. On the other hand, we apply this result to obtain a lateral asymptotic stable periodic oscillation in the Comb-drive finger MEMS model with a cubic nonlinear stiffness term and linear damping. This fact is not typical in Comb-drive finger devices, thus our results could provide a new possibility; a new design principle for stabilization in Comb-drive finger MEMS.
KW - Antiperiodic eigenvalues
KW - Asymptotic stability
KW - Comb-drive
KW - Lower and upper solutions method
KW - MEMS
KW - Periodic solution
UR - http://www.scopus.com/inward/record.url?scp=85175704058&partnerID=8YFLogxK
U2 - 10.1186/s13660-023-03050-9
DO - 10.1186/s13660-023-03050-9
M3 - Article
AN - SCOPUS:85175704058
SN - 1025-5834
VL - 2023
JO - Journal of Inequalities and Applications
JF - Journal of Inequalities and Applications
IS - 1
M1 - 142
ER -