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

Let A
0
be a possibly unbounded positive operator on the Hilbert space H, which is boundedly invertible. Let C
0
be a bounded operator from D(A
^1/2
0
) (with the norm ||z||
²
1/2
= 〈A
0
z, z〉) to another Hilbert space U. It is known that the system of equations z(t) + A
0
z(t) + 1/2 C
^∗
0
C
0
Z(t) = C
^∗
0
u(t), y(t) = − C
0
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
^1/2
0
) × 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) = (s
²
I + s/2 C
^∗
0
C
0
+ A
0
)
⁻¹
, we also obtain that the system is exponentially stable if and only if s → A
^1/2
0
V(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).