Abstract
The linear stabilization problem of the modified generalized Korteweg–de Vries–Burgers equation (MGKdVB) is considered when the spatial variable lies in [0,1]. First, the existence and uniqueness of global solutions are proved. Next, the exponential stability of the equation is established in L^{2} (0,1). Then, a linear adaptive boundary control is put forward. Finally, numerical simulations for both non-adaptive and adaptive problems are provided to illustrate the analytical outcomes.
Highlights
1 Introduction This paper deals with the well-posedness and exponential stability of the modified generalized Korteweg–de Vries–Burgers (MGKdVB) equation ut + γ1uαux – σ uxx + μuxxx + γ2uxxxx = 0, x ∈ (0, 1), t > 0, (1.1)
The MGKdVB equation has been extensively studied in literature but in some very special cases of the physical parameters
5 Linear adaptive boundary control law for MGKdVB equation (1.1) we present a linear adaptive boundary control law for MGKdVB equation (1.1), subject to the same boundary conditions as in (1.2), that is, u(1, t) = u(0, t) = uxx(0, t) = 0, (5.1)
Summary
In [24], the author considered the generalized KuramotoSivashinsky equation (i.e., when α = γ1 = 1, μ > 0, and γ2 = –σ > 0 in (1.1)) He proved the well-posedness of the problem and the exponential decay of the solution provided that σ and the norm of the initial conditions are sufficiently small. K uk(x, t) = fjk(t)φj(x), j=1 where φj(x) are given in Lemma 1 and fjk(t) are solutions to the initial-value problem, which consists of the system of k ordinary differential equations: Puk, φj (t) = ukt , φj (t) + γ1 uk αukx, φj (t) + μ ∂x3uk, φj (t) – σ ukxx, φj (t) + γ2 ∂x4uk, φj (t) = 0, fjk(0) = u0, φj , j = 1, . Inserting (3.25) in (3.22), we obtain d w(x, t) 2 ≤ γ1K w(t) 2 + γ1K δ – 2σ dt (α + 1)δ α+1 wx(t) 2
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