In this paper we investigate in detail the conditions for the stability of the zero solution of a system of ordinary differential equations of the form (∇pG(u,u′))′−∇uG(u,u′)+f(t,u)=Q(t,u,u′) on (0,∞) where u=(u1,⋯,uN) and f(t,u)=∇uF(t,u). Among other conditions it is assumed that (f(t,u),u)>0 if u≠0, −(Q(t,u,p),p)≥φ(u,p)>0 if u≠0, p≠0 and (Q(t,u,p),u)≤δ(t)ρ(p) in a neighbourhood of u=0, p=0. Then, a precise condition on δ(t) for any bounded solution u(t) to decay to 0 as t→∞ is given. The derived condition allows δ(t) to be unbounded as t→∞ in a certain sense, which is a complementary result we proved in Part I [Acta Math. 170 (1993), 275-307], where the case δ(t)→0 as t→∞ is mainly discussed.
Precise damping conditions for global asymptotic stability for nonlinear second order systems, II
PUCCI, Patrizia;
1994
Abstract
In this paper we investigate in detail the conditions for the stability of the zero solution of a system of ordinary differential equations of the form (∇pG(u,u′))′−∇uG(u,u′)+f(t,u)=Q(t,u,u′) on (0,∞) where u=(u1,⋯,uN) and f(t,u)=∇uF(t,u). Among other conditions it is assumed that (f(t,u),u)>0 if u≠0, −(Q(t,u,p),p)≥φ(u,p)>0 if u≠0, p≠0 and (Q(t,u,p),u)≤δ(t)ρ(p) in a neighbourhood of u=0, p=0. Then, a precise condition on δ(t) for any bounded solution u(t) to decay to 0 as t→∞ is given. The derived condition allows δ(t) to be unbounded as t→∞ in a certain sense, which is a complementary result we proved in Part I [Acta Math. 170 (1993), 275-307], where the case δ(t)→0 as t→∞ is mainly discussed.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
