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Stabilization of the Marine Riser model by controllers depending on finitely many parameters

V. K. Kalantarov, A. A. Namazov, E. S. Titi

Abstract

We prove global stabilization of the marine riser models using a feedback controller that depend on finitely many finite-volume elements and finitely many nodal observables. Our approach is based on a feedback control design for dissipative nonlinear partial differential equations, inspired by the methodology introduced in [Evol. Equ. Control Theory, Vol. 3 (2014), 579-594]. The proposed control strategy ensures asymptotic stabilization while maintaining computational feasibility, making it suitable for practical applications.

Stabilization of the Marine Riser model by controllers depending on finitely many parameters

Abstract

We prove global stabilization of the marine riser models using a feedback controller that depend on finitely many finite-volume elements and finitely many nodal observables. Our approach is based on a feedback control design for dissipative nonlinear partial differential equations, inspired by the methodology introduced in [Evol. Equ. Control Theory, Vol. 3 (2014), 579-594]. The proposed control strategy ensures asymptotic stabilization while maintaining computational feasibility, making it suitable for practical applications.
Paper Structure (4 sections, 4 theorems, 88 equations)

This paper contains 4 sections, 4 theorems, 88 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Stabilization of the equation with nonlinear damping term
  4. Stabilization of the linear equation .

Key Result

Lemma 2.1

(see AzTi) Let $\phi\in H^1(0,L)$. Then, for any positive integer $N$, with $h:=\frac{L}{N}$, one has and where $J_k:=\left[(k-1)\frac{L}{N}, k\frac{L}{N}\right),$ for $k=1,2,\cdots N-1$ and $J_N=[\frac{N-1}{N}L, L]$, and $\chi_{J_k}(x)$ is the characteristic function of the interval $J_k$.

Theorems & Definitions (6)

  • Lemma 2.1
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 4.1
  • Remark 4.2
  • Remark 4.3