Equivalent conditions for exponential stability for a special class of conservative linear systems

Let A0 be a possibly unbounded positive operator on the Hilbert space H, which is boundedly invertible. Let C0 be a bounded operator from D(A^1/20) (with the norm ||z||²1/2 = 〈A0z, z〉) to another Hilbert space U. It is known that the system of equations z(t) + A0 z(t) + 1/2 C^∗0 C0 Z(t) = C^∗0u(t), y(t) = − C0 z(t) + u(t), determines a well-posed linear system Σ with input u and output y, input and output space U and state space X = D(A^1/20) × H. Moreover, Σ is conservative, which means that a certain energy balance equation is satisfied both by the trajectories of Σ and by those of its dual system. In this paper we present various conditions which are equivalent to the exponential stability of such a systems. Among the equivalent conditions are exact controllability and exact observability. Denoting V(s) = (s² I + s/2 C^∗0 C0 + A0)⁻¹, we also obtain that the system is exponentially stable if and only if s → A^1/20V(s) is a bounded L(H)-valued function on the imaginary axis. This is also equivalent to the condition that s → sV(s) is a bounded L(H)-valued function on the imaginary axis (or equivalently, on the open right half-plane).