On the transfer function of the zero dynamics for a boundary controlled heat problem in two dimensions

We recently introduced a new approach, zero dynamics inverse (ZDI) design, for designing a feedback compensation scheme achieving asymptotic regulation, i.e., asymptotic tracking and/or disturbance rejection, for a linear or nonlinear distributed parameter system (DPS) in the case when only the value of the signal w(t) to be tracked or rejected are known are known at any instant of time. In analogy with the non-equilibrium formulation of output regulation the control objective is to achieve zero steady-state error together with ultimate boundedness of the state of the system and the controller(s), with a bound determined by bounds on the norms of the initial data and w. In particular, a controller solving this problem depends only on a bound on the norm of w not on the particular choice of w, which is used only as an input to the controller. ZDI design consists of the interconnection of a stabilizing feedback compensator and a cascade controller, designed in a universal way from the zero dynamics of the closed-loop feedback system. This methodology has evolved over a series of papers for asymptotic regulation for specific linear boundary control systems and for set-point control of linear and nonlinear boundary control systems in one spatial dimension, in which case the input and output spaces for the transfer functions were finite dimensional. In this paper, we formulate the main ingredients of the zero dynamics inverse design methodology for a class of abstract linear boundary control systems for problems with infinite dimensional input and output space. We conclude by illustrating the design for a boundary controlled heat equation on a two dimensional rectangle.