Backstepping-forwarding control of parabolic PDEs with partially separable kernels

This paper considers a boundary control problem of parabolic partial differential equations (PDEs) with an integral of the state variable over the whole spatial domain. To handle this non-strict feedback term, we use a Fredholm-type state transformation that involves two integral terms with kernels defined on the lower and the upper triangular domains. It is shown that the transformation converts the original system with an associated state feedback law into an exponentially stable target system, if the kernels are solutions to a boundary value problem of coupled second-order hyperbolic PDEs. Moreover, the upper kernel is separable if a coefficient of the original system is characterized by a certain parameterized ordinary integro-differential equation. Our approach also provides an explicit feedback gain in a special case. The effectiveness of the proposed controller is confirmed by numerical simulation.