A sharp stability criterion for single well Duffing and Duffing-like equations. (January 2020) - Haraux, Alain
Abstract: We refine some previous sufficient conditions for exponential stability of the linear ODE u ′ ′ + c u ′ + ( b + a ( t ) ) u = 0 where b, c > 0 and a is a bounded nonnegative time dependent coefficient. This allows to improve some results on uniqueness and asymptotic stability of periodic or almost periodic solutions of the equation u ′ ′ + c u ′ + g ( u ) = f ( t ) where c > 0, f ∈ L ∞ ( R ) and g ∈ C 1 ( R ) satisfies some sign hypotheses. The typical case is g ( u ) = b u + a | u | p u with a ≥ 0, b > 0 . Similar properties are valid for evolution equations of the form u ′ ′ + c u ′ + ( B + A ( t ) ) u = 0 where A ( t ) and B are self-adjoint operators on a real Hilbert space H with B coercive and A ( t ) bounded in L ( H ) with a sufficiently small bound of its norm in L ∞ ( R +, L ( H ) ) .