Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
4628675 | Applied Mathematics and Computation | 2013 | 24 Pages |
Abstract
In an earlier paper, the author formed a positive definite matrix R as the sum of positive semi-definite matrices that are eigenmatrices of a matrix eigenproblem associated with the Lyapunov matrix equation. This positive definite matrix R was then used to define the weighted norm â·âR in Cn which led to new two-sided bounds on the solution of the initial value problem xÌ=Ax,x(t0)=x0, in any vector norm â·â. In the present paper, the quantity âx(t)âR itself is analyzed showing that the solution x(t) in the weighted norm âx(t)âR suppresses the vibration behavior inherent to x(t). This new sight at the norm âx(t)âR may be of considerable practical use in technical dynamical systems. As a direct important application in engineering, the quantity âx(t)âR may be used as a measure to assess the damping behavior of the studied free dynamical system since the vibratory part is absent. Moreover, it is shown that âx(t)âR is monotonically decreasing for sufficiently large t if matrix A is asymptotically stable. Thus, as a further application, a two-sided estimate of the form c0|D+âx(t)âR|⩽âxÌ(t)âR⩽c1|D+âx(t)âR| can be derived, which is shown to be invalid for the norm â·â2. This result is also of interest on its own. The obtained findings are underpinned by two numerical examples and illustrated by graphs.
Related Topics
Physical Sciences and Engineering
Mathematics
Applied Mathematics
Authors
L. Kohaupt,