Stabilization of cascade interconnection of an 1D heat equation in polar coordinates and an ODE using LQR design

In this paper, we consider ODE-PDE cascade systems in which the input is applied to the ODE system whose output drives the PDE system and PDE-ODE cascade systems in which the input is applied to the PDE system whose output drives the ODE system. In both the cascade systems we take the PDE system to be an 1D heat equation in polar coordinates containing a singular term. We address the problem of designing state-feedback control laws for stabilizing these cascade systems via finite-dimensional approximation and LQR design. Using the Galerkin technique we first obtain an N
^th
-order ODE approximation for the cascade system. Next we fix a quadratic cost function for the cascade system and consider the corresponding operator algebraic Riccati equation (ARE). Then we associate an appropriate matrix ARE with the ODE approximation. Under some natural assumptions on the cascade system, we verify that certain well-known hypothesis in the literature hold, which imply that the solution Π
N
of the matrix ARE converges to the solution Π of the operator ARE strongly as N→∞. This implies that the stabilizing gain for the ODE approximation determined by Π
N
converges to a stabilizing gain for the cascade system. We illustrate this in simulations.