Stabilization by delay distributed feedback control

Abstract

In this paper, a new approach to stability of integro-differential equations x′′t+β 1 ∫ t-τ1tte-α1t-sxsds+β 2 ∫ t-τ2tte-α2t-sxsds=0 and x′′t+β 1 ∫ 0t-τ1te-α1t-sxsds +β 2 ∫ 0t-τ2te-α2t-sxsds=0 is proposed. Under corresponding conditions on the coefficients α 1 , α 2 , β 1 and β 2 the first equation is exponentially stable if the delays τ 1 (t ) and τ 2 (t ) are large enough and the second equation is exponentially stable if these delays are small enough. On the basis of these results, assertions on stabilization by distributed input control are proven. It should be stressed that stabilization of this sort, according to common belief, requires a damping term in the second order differential equation. Results obtained in this paper demonstrate that this is not the case.