Boundary prescribed–time stabilization of a pair of coupled reaction–diffusion equations

We study boundary feedback stabilization of a pair of coupled reaction-diffusion equations with distinct diffusivities, where attractivity to the origin is required to occur in a finite time which is prescribed independently of initial conditions. Our approach is twofold: we first develop control laws which render the system to a cascade form; we then utilize two different time-varying control laws which ensure prescribed-time stabilization of each equation. To achieve the desired stabilization while ensuring that the resulting boundary feedback controllers remain bounded, it is necessary to prescribe two different rates of attractivity to the origin. We achieve this by using distinct time-varying gains with square and cubic "blow-up" functions, which diverge at the prescribed terminal time. The ensuing control laws achieve stabilization of the plant in a prescribed terminal time.