On the ISS properties of a class of parabolic DPS’ with discontinuous control using sampled-in-space sensing and actuation

A reaction–diffusion–advection equation with uncertain parameters, and additionally subject to disturbances of both matched and mismatched nature, is considered. It is assumed that only a finite number of point-wise sensing and actuation devices, suitably located in an equi-spaced manner along the 1-dimensional spatial domain of interest, is available. The variable structure control approach is adopted to design the underlying discontinuous feedback control laws. The existence of the resulting closed-loop trajectories is addressed in depth. The state of the closed-loop system is shown to feature an exponential ISS property with respect to mismatched disturbances thereby constituting a new result in the DPS’ setting, capturing point-wise sensing and actuation. Apart from this, it is established that an arbitrary level of the mismatched disturbance attenuation can be achieved by employing sufficiently many sensing and actuation devices. It is also established that the matched disturbances entering the control channels can be fully rejected by the proposed design. Tuning rules of the controller parameters are constructively derived by means of the Lyapunov approach, and simulation results are brought into play to support the theoretical development.