State space representation of flexible structure dynamics in air flow

Recently [1] (Balakrishnan, 2000) it was shown that the linearized dynamics of the response of a flexible structure in inviscid incompressible airflow can be modeled as a convolution/evolution equation in a separable Hilbert space H(1)Y˙(t)=AY(t)+∫0tL(t-s)Y˙(s)dswhere Y(·) is the structure state, A is the infinitesimal generator of a c0 semigroup, with a compact resolvent, and L(t) is linear bounded and strongly continuous in t≥0. The convolution is characteristic of the aerodynamic force and moment but H is no longer an adequate state space, essential before we can consider controllability. Indeed currently there is no design theory for stability enhancement controllers. In this paper we develop a state space representation in the form(2)Z˙(t)=A˚Z(t)where the state space Z now a Banach space in which H is imbedded and A˚ is the infinitesimal generator of a c0 semigroup. A control term is readily added. The structure state is given by(3)Y(t)=PZ(t)where P is a projection operator onto H. The aeroelastic modes are now identified as eigenvalues (point spectrum) of A˚.