Lyapunov Transformations Research Articles

Stabilization of the stationary motions of non-holonomic mechanical systems

The possible stabilization of the unstable stationary motions of a non-holonomic system is studied from the standpoint of general control theory /1, 2/. As distinct from the case previously considered /3/, when forces of a certain structure are applied with respect to both positional and cyclical coordinates, the stabilization is obtained here by applying control forces only with respect to the cyclical coordinates /4/; the control forces may be applied with respect to some or all of the cyclical coordinates, and depend on the positional coordinates, the velocities, and the corresponding cyclical momenta. It is shown that, just as in the case of holonomic systems /5, 6/, depending on the control properties of the corresponding linear subsystem, the stationary motions, whether stable or unstable, can be stabilized, up to asymptotic stability with respect to all the phase variables, or asymptotic stability with respect to some of the phase variables and stability with respect to the remaining variables. The type of stabilization with respect to the given phase variables depends on the Lyapunov transformations which are needed in order to reduce the critical cases obtained to singular cases /7, 8/.

Ranges of Lyapunov Transformations for operator algebras

In this paper we shall extend results obtained in [5] to the W*-algebra setting.Let be a C*-algebra and let + denote the set of positive elements in . Given a fixed element A in , the Lyapunov transformation LA corresponding to A is the mapping of into itself which sends X to AX+XA*. We are interested in characterizing those Bin for which

Ranges of Lyapunov transformations for Hilbert space

Interest in the ranges of Lyapunov transformations began with Taussky in [5]. Recently in a series of papers, Loewy has studied the ranges of Lyapunov transformations on matrices. In particular in [2] and [3], the following result was obtained.

On ranges of Lyapunov transformations IV

Let ℂn,n denote the space of n × n matrices with complex entries and let ℋn denote the set of n × n hermitian matrices. Given any matrix A∊ℂn,n, the Lyapunov transformation corresponding to A is defined by ℐA(H) = AH+HA*, where H∊ℋn. Let PSD(n) be the set of all n × n hermitian positive semidefinite matrices. Taussky [8, 9] raised the problems of determiningandBoth of these problems seem to be difficult.

On Ranges of Lyapunov Transformations. III

The Lyapunov transformation $\mathcal{L}_A $ corresponding to the matrix $A \in \mathbb{C}^{n,n} $ is a linear transformation on the space $\mathcal{H}_n $ of Hermitian matrices $H \in \mathbb{C}^{n,n} $, of the form $\mathcal{L}_A (H) = AH + HA^ * $. Let \[ C_1 (A) = \{ {AH + HA^ * :H\,{\text{is Hermitian and positive semidefinite}}} \}\]. Given $A \in \mathbb{C}^{n,n} $ such that $\mathcal{L}_A $ is invertible, we characterize the matrices $B \in \mathbb{C}^{n,n} $ such that $C_1 (A) = C_1 (B)$.

On ranges of real Lyapunov transformations

Let S be the space of n× n real symmetric matrices and let S + n be the cone of n× n positive semidefinite matrices in S n . If A∈ R n,n, then the real Lyapunov transformation R A : S n → S n corresponding to A is defined by R A(S)=AS + SA t . We prove that given A,B∈ R n,n such that R A is invertible, we have R A( S + n) = R B( S + n if and only if B= μA or B= μA −1 for some μ > 0, and that this condition is also necessary and sufficient for R −1 A( S + n) = R −1 B( S + n .

On ranges of Lyapunov transformations

A Lyapunov transformation is a linear transformation on the set H n of hermitian matrices H ϵ C n,n of the form L A ( H) = A ∗ H + HA, where A ϵ C n,n . Given a positive stable A ϵ C n,n , the Stein-Pfeffer Theorem characterizes those K ϵ H n for which K = L B ( H), where B is similar to A and H is positive definite. We give a new proof of this result, and extend it in several directions. The proofs involve the idea of a controllability subspace, employed previously in this context by Snyders and Zakai.