Three-dimensional real-valued Maxwell-Bloch equations with controls
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 dynamics with one linear control. The same property holds for the system (1), (3) with one control on the level surface of the function \(C''= d(x_3+ {1\over 2} x^2_1)\) and \(k>-1\). Moreover, the dynamics of (1), (3) reduces to the pendulum dynamics. Finally, one discusses the problem of controlling the homoclinic orbit of the Maxwell-Bloch equations with three independent controls of the form: \[ \dot x_1= x_2+ u_1,\quad \dot x_2= x_1 x_3+ u_2,\quad \dot x_3= -x_1 x_2+ u_3. \]
scientific article; zbMATH DE number 1090421