Controllability aspects of the Korteweg–de Vries Burgers equation on unbounded domains

Abstract

The aim of this work is to consider the controllability problem of the linear system associated to Korteweg-de Vries Burgers equation posed in the whole real line. We obtain a sort of exact controllability for solutions in L 2 loc (R 2 ) by deriving an internal observability inequality and a Global Carlemann estimate. Following the ideas contained in [26], the problem is reduced to prove an approximate theorem.2010 Mathematics Subject Classification. Primary: 35Q53, Secondary: 37K10, 93B05, 93D15.

Observer‐based H∞ control of a stochastic Korteweg–de Vries–Burgers equation

“…It shows that system (37) achieves the mean-square H ∞ performance under observer-based controller (6). Figure 5(B) demonstrates the mean-square H ∞ performance of the closed-loop system.…”

“…To prove the stability of the closed-loop system (8) or ( 9), it is sufficient to show the stability of the system (4), ( 5) subject to (6) or (7). Thus, we start with the stability analysis of the system (4), ( 5) subject to (6) or (7). The Lyapunov-Krasovskii functional (LKF) is selected for (4) and ( 5) as follows:…”

“…It usually appears in the study of the weak effects of nonlinearity, dissipation, and dispersion in waves propagating in a liquid-filled elastic tube (see References 2,3). In recent years, many fruitful works on stabilization of KdVB equation have been studied widely (see References 1,[3][4][5][6][7]. The controllability problem of the KdVB equation on bounded domains (see Reference 4) and unbounded domains (see Reference 6).…”

Local exact controllability to the trajectories of the Korteweg–de Vries–Burgers equation on a bounded domain with mixed boundary conditions

“…That is the reason that the Laplacian is added. From a mathematical point of view, there exist several results for the KdVB equation in both bounded and unbounded domains, concerning the global and local well-posedness problem [8], [26], [14] and [3]; the optimal control problem [4], [11]; the internal controllability problem on unbounded domain [17]; and the boundary feedback stabilization problem [23]. As far as we know, the internal controllability problem for (1) has not been studied and thus, our paper will fill this gap.…”