Zero dynamics and root locus for a boundary controlled heat equation

Abstract

We consider single-input single-output systems whose internal dynamics are described by the heat equation on some domain $\varOmega \subset {\mathbb {R}}^d$ with sufficiently smooth boundary $\partial \varOmega$. The scalar input is formed by the Neumann boundary values which are forced to be constant in space; the output consists of the integral over the Dirichlet boundary values. We show that the transfer function admits some partial fraction expansion with positive residues. The location of the transmission zeros and invariant zeros is further analyzed. Thereafter we show that the zero dynamics are fully described by a self-adjoint and exponentially stable semigroup. The spectrum of its generator is proven to be the set of invariant zeros. Finally, we show that any positive proportional output feedback results in an exponentially stable system. We further analyze the root loci: as the proportional gain tends to infinity, the eigenvalues of the generator of the closed-loop system converge to the invariant zeros.