Nonlinear feedback control of a class of semilinear parabolic PDEs

In the present paper a nonlinear feedback control design for a class of coupled 1-D semilinear partial differential equations with in-domain point measurements and in-domain averaged or homogeneously distributed actuator configurations is addressed. The objective is to stabilize a desired (potentially open-loop unstable) profile with a nonlinear controller that mitigates the effect of the nonlinearity at the most sensitive location, for this purpose the actuator and sensor characteristics are key design degrees of freedom. Stability of the zero profile for the closed-loop system is ensured by studying it as a Lur'e system and by applying modal techniques while its uniqueness is studied using a bifurcation analysis and aims to an auxiliary sensor location criterion. The performance of the proposed controller is illustrated using numerical simulations.