Three-dimensional real-valued Maxwell-Bloch equations with controls
type: Item
instance of: scholarly article
title: Three-dimensional real-valued Maxwell-Bloch equations with controls
zbMATH Open document ID: 0963.93069
DOI: 10.1016/0034-4877(96)84072-9
published in: Reports on Mathematical Physics
publication date: 2001-06-28T00:00:00Z
review text: The author presents some geometrical and dynamical properties of the following Maxwell-Bloch equations: \[ \dot x_1= x_2,\quad \dot x_2= x_1 x_3+ u,\quad \dot x_3= -x_1 x_2\tag{1} \] with particular classes of controls about the \(x_2\)-axis, namely, the linear feedback \[ u= -kx_1\tag{2} \] respectively the quadratic control \[ u= kx_1 x_3,\tag{3} \] where \(k\in\mathbb{R}\) is the feedback gain parameter. The controlled systems (1), (2) and (1), (3) are Hamilton-Poisson mechanical systems with the phase space \(\mathbb{R}^3\), the Hamiltonian \(H= {1\over 2}(x^2_2+ x^2_3)+{1\over 2} kx^2_1\) respectively \(H={1\over 2}(x^2_2+ x^2_3)+{1\over 2} kx^2_3\), and with respect to the Poisson structure \[ \{f, g\}= \text{det} \left[\begin{matrix} x_1 & 0 & 1\\ {\partial f\over\partial x_1} & {\partial f\over\partial x_2} & {\partial f\over\partial x_3}\\ {\partial g\over\partial x_1} & {\partial g\over\partial x_2} & {\partial g\over\partial x_3}\end{matrix}\right],\;f,g\in C^\infty(\mathbb{R}^3, \mathbb{R}) \] respectively \[ \{f, g\}= \text{det} \left[\begin{matrix} (d+ ck)x_1 & cx_2 & cx_3+d\\ {\partial f\over\partial x_1} & {\partial f\over\partial x_2} & {\partial f\over\partial x_3}\\ {\partial g\over\partial x_1} & {\partial g\over\partial x_2} & {\partial g\over\partial x_3}\end{matrix}\right],\;f,g\in C^\infty(\mathbb{R}^3, \mathbb{R}), \] where \(a,b\in \mathbb{R}\), \(ad-bc= 1\). It is shown that each of the above systems may be nonlinearly stabilized about the equilibrium \((x_1, x_2,x_3)= (0,0,M)\) for \(k> M\), respectively about the equilibrium state \((x_1, x_2, x_3)= (M,0,0)\) for \(k> -1\). It is also shown that each of the controlled system (1), (2) and (1), (3) may be realized as a Hamilton-Poisson mechanical system in an infinite number of different ways, with the phase space \(P= (\mathbb{R}^3_{cd})^*\simeq \mathbb{R}^3\) and a suitable Hamiltonian. On the other hand, the dynamics of the controlled system (1), (2) reduces to the Duffing oscillator dy…
reviewed by: Marian Ivanovici
Mathematics Subject Classification ID: 93D15
Mathematics Subject Classification ID: 70H05
Mathematics Subject Classification ID: 93C10
zbMATH DE Number: 1090421
zbMATH Keywords: stabilization
zbMATH Keywords: Hamilton-Poisson mechanical systems
zbMATH Keywords: Hamiltonian
zbMATH Keywords: Duffing oscillator
zbMATH Keywords: pendulum dynamics
zbMATH Keywords: homoclinic orbit
zbMATH Keywords: Maxwell-Bloch equations
author: Mircea Puta
MaRDI profile type: Publication
full work available at URL: 0034-4877(96)84072-9
OpenAlex ID: W2016495931
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Wikidata QID: Q127586805
description: scientific article; zbMATH DE number 1090421