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 ) = A Y ( t ) + ∫ 0 t L ( t - s ) Y ˙ ( s ) ds where Y ( · ) is the structure state, A is the infinitesimal generator of a c 0 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 c 0 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 ˚ .