Observer-based dynamic local piecewise control of a linear parabolic PDE using non-collocated local piecewise observation

This paper employs the observer-based output feedback control technique to study the problem of local piecewise control for a linear parabolic partial differential equation(PDE) using non-collocated local piecewise observation. By the non-collocated local piecewise observation, a Luenberger-type PDE observer is first constructed to exponentially track the state of the PDE in the sense of both L₂ norm and H¹ norm. Based on the estimated state, a collocated local piecewise state feedback controller is then proposed for exponential stabilisation of the PDE. Sufficient conditions on exponential stabilisation of the PDE by the suggested feedback controller in the sense of both L₂ norm and H¹ norm are developed by utilising Lyapunov direct method, integration by parts, Wirtinger's inequality, and first mean value theorem for definite integrals, and presented in terms of standard linear matrix inequalities. The well-posedness is also analysed for both open-loop and resulting closed-loop PDE by using C₀-semigroup approach. Finally, numerical simulation results are presented to show the effectiveness of the proposed design method.