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In this paper, we study the finite-dimensional stabilization problem of the cascade of the one-dimensional transport-diffusion process with an unstable ODE plant, where it is assumed that the ODE plant is controllable and observable and the plant is connected with the transport-diffusion process through a filter. The input to the whole system is only Neumann boundary input to the transport-diffusion process, and the outputs are the data at the boundary of process domain and the output of the ODE plant. We use the previous result that the one-dimensional transport-diffusion process can be formulated as a system with A-bounded output operator. In this paper, it is proved that the finite-dimensional model of the whole system is controllable and observable when the filter is a Residual Mode Filter (RMF). This fact enables us to construct a finite-dimensional stabilizing controller by using an RMF approach. The point that the output from the same RMF is used in the two parts features this paper.

Stabilization of Unstable Mixed Parameter Systems by Using a Finite-dimensional Controller