How to get a conservative well-posed linear system out of thin air. Part II. Controllability and stability

Let A ₀ be a possibly unbounded positive operator on the Hilbert space H, which is boundedly mvertible. Let C _o be a bounded operator from D(A ₀ ^1/2) (with the norm ||z|| _1/2 ² = (A ₀z, z)) to another Hilbert space U. In Part I of this work we have proved that the system of equations z̈(t) + A ₀z(f) + 1/2C ₀*C ₀z̈(t) = C ₀*u(t), y(t) = -C ₀ż(t) + u(t) determines a well-posed linear system E with input u and output y, input and output space U, and state space X = D(A ₀ ^1/2) × 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 show that Σ is exactly controllable if and only if it is exactly observable, if and only if it is exponentially stable. Moreover, if we denote by A the generator of the contraction semigroup associated with Σ (which acts on X), then Σ is exponentially stable if and only if one of the entries in the second column of (iωI-A) ^-1 is uniformly bounded as a function of ω ∈ R. We also show that, under a mild assumption, Σ is approximately controllable if and only if it is approximately observable, if and only if it is strongly stable, if and only if the dual system is strongly stable. We prove many related results and we give examples based on wave and beam equations.