A Semiglobal Observer for the Korteweg–de Vries–Burgers Equation

In the work, the problem of state estimation of the generalized Korteweg–de Vries–Burgers equation is considered under the assumption that a finite number of spatially localized (in-domain) measurements are available. A nonlinear observer is proposed, constructed on the basis of recent results on the stabilization of the system described by the Korteweg–de Vries–Burgers equation using a finite number of spatially localized sensors and actuators. Sufficient conditions are obtained for the system parameters, the number of sensors, and the observer gain coefficient, ensuring exponential convergence of the observer error to zero. A qualitative analysis of these conditions shows that, in contrast to all existing results on state estimation of the Korteweg–de Vries–Burgers equation, the proposed observer is semi-global in the sense that convergence of the estimation error to zero can be proved for arbitrary initial conditions, provided that the number of available measurements is sufficiently large.