Spatiotemporal adaptive state feedback control of a linear parabolic partial differential equation

This article deals with the issue of asymptotic stabilization for a linear parabolic partial differential equation (PDE) with an unknown space-varying reaction coefficient and multiple local piecewise uniform control. Clearly, the unknown reaction coefficient belongs to a function space. Hence, the fundamental difficulty for such issue lies in the lack of a conceptually simple but effective parameter identification technique in a function space. By the Lyapunov technique combined with a variant of Poincaré-Wirtinger inequality, an update law is derived for estimate of the unknown reaction coefficient in a function space. Then a spatiotemporal adaptive state feedback control law is constructed such that the estimate of the unknown coefficient is bounded and the closed-loop PDE is asymptotically stable in the sense of spatial (Formula presented.) norm if a sufficient condition given in terms of space-time varying linear matrix inequalities (LMIs) is fulfilled for the estimated coefficient and the control gains. Both analytical and numerical approaches are proposed to construct a feasible solution to the space-time varying LMI problem. With the aid of the semigroup theory, the well-posedness and regularity of the closed-loop PDE is also analyzed. Moreover, two extensions of the proposed adaptive control scheme are discussed: the PDE in (Formula presented.) -D space and the PDE with unknown diffusion and reaction coefficients. Finally, numerical simulation results are presented to support the proposed spatiotemporal adaptive control design.