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

Abstract

This paper studies the internal control of the Korteweg-de Vries-Burgers (KdVB) equation on a bounded domain. The diffusion coefficient is time-dependent and the boundary conditions are mixed in the sense that homogeneous Dirichlet and periodic Neumann boundary conditions are considered. The exact controllability to the trajectories is proven for a linearized system by using duality and getting a new Carleman estimate. Then, using an inversion theorem we deduce the local exact controllability to the trajectories for the original KdVB equation, which is nonlinear.

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

“…Theorem 1. Consider the closed-loop system (8) or (9), whereẑ is governed by (4). Given positive scalars , Δ and a tuning parameter C 1 > 0.…”

“…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).…”

On the Stability and Null-Controllability of an Infinite System of Linear Differential Equations

“…Control problems in Banach or Hilbert spaces arise naturally in processes described by partial differential equations (see for example [ 1 , 3 , 7 , 8 , 11 , 13 , 15 , 16 , 19 , 22 ] and references therein). Sometimes it is useful to reduce the control problem for partial differential equations to infinite systems of ODEs [ 4 , 5 , 9 , 10 ].…”

On the stability and null-controllability of an infinite system of linear differential equations

“…Control problems in Banach or Hilbert spaces arise naturally in processes described by partial differential equations (see for example [2,6,7,10,12,14,15,19,22] and references therein). Sometimes it is useful to reduce the control problem for partial differential equations to infinite systems of ODEs [3,4,8,9].…”