Vol. 16, No. 3, 1966

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Stability of linear differential equations with periodic coefficients in Hilbert space

Gert Einar Torsten Almkvist

Vol. 16 (1966), No. 3, 383–391

In this paper we study the stability of the solutions of the differential equation

u′(t) = A(t)⋅u(t)

for t 0 in a separable Hilbert space. It is assumed that A(t) is periodic with period one and satisfies the following symmetry condition: There exists a continuous constant invertible operator Q such that

A(t)∗ = − Q ⋅A(t)⋅Q− 1 for all t ≧ 0.

We use a perturbation technique. Let A(t) = A0(t) + B(t) where A0(t) is compact and antihermitian for all t. We denote by U0(t) the solution operator of u(t) = A0(t)u(t). It is shown that (1) is stable if B(t) satisfies a certain smallness condition involving the distribution of the eigenvalues of U0(1) and the action of B(t) on the eigenvectors of U0(1). The results can be applied to the second order equation

y  + C(t)y = 0

where C(t) is selfadjoint for all t.

Mathematical Subject Classification
Primary: 34.95
Secondary: 34.51
Received: 20 June 1964
Published: 1 March 1966
Gert Einar Torsten Almkvist