Spectral Assignability for Distributed Parameter Systems with Unbounded Scalar Control

This article studies a class of control systems in a Hilbert space H given x(t) = Ax(t) + bu(t), where A generates a holomorphic semigroup on H, $u(t)$ is a scalar control, and the control input b is possibly unbounded. Many systems with boundary or point control can be represented in this form. The author considers the question of what eigenvalues \ α ₖ \ k ∈ I, the closed-loop system can have when $u(t)$ is a feedback control. Shun-Hua Sun’s condition on \ α ₖ \ k ∈ I [SIAM J. Control Optim., 19 (1981), pp. 730–743] is generalized to the case where b is unbounded but satisfies an admissibility criterion; this condition is generalized further when unbounded feedback elements are allowed. These results are applied to a structurally damped elastic beam with a single point actuator. Similar techniques also prove a spectral assignability result for a damped elastic beam with a moment control force at one end, even though the associated input element is not admissible in the appropriate sense.