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This article is intended to present a qualitative and numerical analysis ofwell-posedness and boundary stabilization problems of the well-knownKorteweg-de Vries-Burgers equation. Assuming that the boundary control is ofmemory type, the history approach is adopted in order to deal with the memoryterm. Under sufficient conditions on the physical parameters of the system andthe memory kernel of the control, the system is shown to be well-posed bycombining the semigroups approach of linear operators and the fixed pointtheory. Then, energy decay estimates are provided by applying the multipliermethod. An application to the Kuramoto-Sivashinsky equation will be also given.Moreover, we present a numerical analysis based on a finite differences methodand provide numerical examples illustrating our theoretical results.

Boundary stabilization of the Korteweg-de Vries-Burgers equation with an  infinite memory-type control and applications: a qualitative and numerical  analysis