Abstract
In this paper, we discuss two main problems. In the first section, we establish a new global distributed exact controllability of the periodic two-component $\mu\rho$-Hunter-Saxton system on the circle by means of a distributed control. And in the second section, we present corresponding result of the asymptotic stabilization problem about the periodic two-component $\mu\rho$-Hunter-Saxton system. By presenting concrete form of the feedback law, an equivalent system is got.
Highlights
The Hunter- Saxton (HS) equation mt + 2mux + mxu = 0, m = −∂x2u, (1.1)where u(x, t) is the function of spatial variable x and t being a slow time variable [14, 30], which is originally derived and studied in [21] as a model for propagation of orientation waves in a massive nematic liquid crystal director field
The HS equation can be regarded of the Camassa-Holm (CH) equation as α −→ ∞
Which was introduced to model the unidirectional propagation of shallow water waves over a flat bottom [4]
Summary
Where u(x, t) is the function of spatial variable x and t being a slow time variable [14, 30], which is originally derived and studied in [21] as a model for propagation of orientation waves in a massive nematic liquid crystal director field. Periodic two-component μρ-Hunter-Saxton system, global distributed exact controllability, asymptotic stabilization. For the exact controllability and asymptotic stabilization: Glass in [34] investigated the problem of exact controllability of the Camassa-Holm equation in the circle S := R/Z, by means of a distributed and compactly supported type feedback control. As for the problems of exact controllability and asymptotic stabilization of two-component Camassa-Holm equation and generalized two-component μ-Hunter-Saxton equation need us to investigate. In the paper, we will investigate the exact controllability and asymptotic stabilization problems of the wider range of the equations. We call equation (1.6) that the generalized periodic two-component μρ-Hunter-Saxton equation. In order to solve the above global distributed exact controllability problem and asymptotic stabilization problem, we give the following Theorems and Propositions. (Global distributed controllability of the generalized periodic twocomponent μρ-Hunter-Saxton equation).
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