Stabilization of linear systems by rotation
The problem addressed in this paper can be seen as a stabilization problem. Given a linear system \(\dot x= Ax+ u\), \(x\in\mathbb{R}^n\), the authors seek a stabilizing law of the form \(u = k(t)\Sigma x\), where \(\Sigma\) is skew-symmetric and \(k(t)\) is real valued, under the standing assumption that the trace of \(A\) is negative. The first result states the existence of a \(k^*\geq 0\) such that the system is stabilizable with \(|k|\geq k^*\) and \(\Sigma= \Sigma_A= U\Sigma_n U^T\). Here, \(U\) is an orthogonal matrix whose columns are eigenvectors of \(A+ A^T\), and the elements of \(\Sigma_n\) are \[ \sigma= \begin{cases} 0 &\text{if }i= j,\\ -1 &\text{if }i< j,\\ 1 &\text{if }i> j.\end{cases} \] The authors give an example which shows that the conclusion is not true if \(k\) is replaced by an arbitrary \(k(t)\geq k^*\). However, they prove that stabilization is achieved if either \(k(t)\) is nondecreasing and \(k\geq k^*_1\) or \(k(t)= kp(t)\) where \(p(t)\) is periodic and \(k\geq k^*_2\) (\(k^*_1\) and \(k^*_2\) are in general greater than \(k^*\)). The latter statement is compared with the classical principle of vibrational control. In fact the principle of vibrational control is improved under some respects by the present result. It is also proved that a stabilizing gain \(k(t)\) can be obtained in a adaptive and robust way, as a solution of \(\dot k= \| x\|^p\). Numerical examples are given. The proofs require an accurate analysis of properties of eigenvalues of skew-symmetric matrices and perturbation matrix theory.
scientific article; zbMATH DE number 5137563
scientific article