Asymptotic stability for intermittently controlled nonlinear oscillators

In this paper we obtain sharp conditions for the uniform asymptotic stability of u=0 with respect to an equation having the form (ΔL(t,u,u′))′−ΔuL(t,u,u′)=Q(t,u,u′) with L=G(u,u′)−F(t,u). This form includes a variety of oscillatory-type equations with damping. The sought for conditions are assumed to hold on time intervals In, n=1,2,⋯, which are arbitrarily spaced in [0,∞); the coefficients may be arbitrary otherwise, e.g., no damping at all or unbounded damping which amounts to freezing the state, etc. Thus the damping on the intervals should imply the asymptotic stability. Several more concrete examples illustrate and motivate the general results, which are at times surprising. For instance, in the particular case u′′+A(t,u,u′)u′+f(u)=0, under the standard conditions and with A(t,u,u′) continuous and uniformly positive definite on the union of In, the asymptotic stability is guaranteed by ∑|In|3=∞, and the exponent 3 is the best possible. Other interesting examples are provided.