Bilateral Boundary Control Design for a Cascaded Diffusion-ODE System Coupled at an Arbitrary Interior Point

We present a methodology for designing bilateral boundary controllers for a class of systems consisting of a coupled diffusion equation with an unstable ODE at an arbitrary interior point. A folding transformation is applied about the coupling point, transforming the system into an ODE with an input channel consisting of two coupled diffusive actuation paths. A target system with an exponentially stable trivial solution in the sense of L^2 X R^n is proposed, and the stability property is shown via the Lyapunov method. The stabilizing control laws are formulated via tiered Volterra transformations of the second kind, establishing an equivalence relation between the stable target system and the original plant under boundary feedback. Stability properties of the plant under feedback is inferred from the equivalence relation. The well-posedness of the backstepping transformations involved are studied, and the existence of bounded Volterra kernels is shown, constituting a sufficient condition for the invertibility of the Volterra transformations.