Approximating parabolic boundary control problems with delayed actuator dynamics

In this paper we consider a control problem for the convection diffusion equation and investigate the impact of including actuator dynamics with delays. The problem is motivated by applications to control of energy efficient buildings where actuation is provided by a HVAC system. The basic model is governed by a parabolic partial differential equation (PDE) with boundary inputs. The boundary inputs are assumed to be the output of an actuator governed by a delay differential equation. Thus, one augments the PDE with a delay equation model of an actuator with delays. The combined system is described by a coupled delay partial differential equation. We show that under suitable conditions, the coupled delay PDE system is well posed in a standard Hilbert space and we use this corresponding abstract formulation to construct numerical methods for control design. We apply these results to a simple 1D boundary control system to illustrate the ideas and numerical methods.