Nonlinear Adaptive Boundary Control of the Modified Generalized Korteweg–de Vries–Burgers Equation

Abstract

In this paper, we study the nonlinear adaptive boundary control problem of the modified generalized Korteweg–de Vries–Burgers equation (MGKdVB) when the spatial domain is 0,1. Four different nonlinear adaptive control laws are designed for the MGKdVB equation without assuming the nullity of the physical parameters ν, μ, γ1, and γ2 and depending whether these parameters are known or unknown. Then, using Lyapunov theory, the L2-global exponential stability of the solution is proven in each case. Finally, numerical simulations are presented to illustrate the developed control schemes.

Stable and functional solutions of the Klein-Fock-Gordon equation with nonlinear physical phenomena

“…¢ G G -expansion method Now considering the homogeneous balance between ¶ Y ¶G 2 2 and Ψ 3 in equation ( 13), then we find N = 1. Therefore, we take: (14) into equation (13), then collecting the coefficients of F i ( ± 1, ± 2, K...., ± N), and calculating the resulting system with the help of Maple, we find: By using the case 1 into equation ( 14), we find:…”

“…-expansion method [9,10], novel (G′/G)-expansion method [11,12], nonlinear adaptive control scheme [13], the unified method and its generalized form [14][15][16][17], Linear non-adaptive boundary control law [18,19], the improved Bernoulli sub-equation function method [20], tanh-coth expansion method [21], rational sine-cosine method [22], the sine-Gordon expansion method [23,24], the enhanced modified simple equation method [25], the exponential rational function method [26], the simplified Hirotas method [27], Darboux transformation method [28], N-fold Darboux transformation [29], Lie symmetry approach [30], the power index method [31], generalized Kudryashov method [32], novel operational method [33], extended simple equation [34], (G′/G)-expansion method [35], the extended sinh-Gordon equation expansion method [36,37], the Hirota bilinear approach [38,39], the extended rational sinecosine and rational sinh-cosh methods [40], new extended rational SGEE method [41], the extended unified method [42,43] and many other approaches.…”

Qualitative and numerical study of the stability of a nonlinear time-delayed dispersive equation

“…has also been extensively investigated by many researchers in finite and infinite domains (see for example [14,15,16,17,18,19,27,51,52,53,54,55,57]).…”