Zero dynamics and root locus for a boundary controlled heat equation

We consider a single-input-single output systems whose internal dynamics are described by the heat equation on some domain Ω ⊂Rd with sufficiently smooth boundary ∂Ω . The input is formed by the Neumann boundary values; the output consists of the integral over the Dirichlet boundary values. We show that the transfer function admits some partial fraction expansion with positive residuals. The location of transmission and invariant zeros of this system is furthermore investigated. We prove that the transmission zeros have an interlacing property in the sense that there is exactly one transmission zero between two poles of the transfer function. The set of transmission zeros is shown to be a subset of the invariant zeros. Thereafter we consider the zero dynamics of this system. We prove that these are fully described by a self-adjoint and exponentially stable semigroup. The eigenvalues of the generator of this semigroup are proven to coincide with the set of invariant zeros. Finally, we consider proportional output feedback. We show that any positive proportional gain results in an exponentially stable system. We further prove the root locus property: If the proportional gain tends to infinity, then the eigenvalues of the generator of the closed loop system will converge to the invariant zeros.