On the vibration-suppression property and monotonicity behavior of a special weighted norm for dynamical systems ẋ=Ax,x(t0)=x0

摘要

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 ‖·‖R in Cn which led to new two-sided bounds on the solution of the initial value problem ẋ=Ax,x(t0)=x0, in any vector norm ‖·‖. In the present paper, the quantity ‖x(t)‖R itself is analyzed showing that the solution x(t) in the weighted norm ‖x(t)‖R suppresses the vibration behavior inherent to x(t). This new sight at the norm ‖x(t)‖R may be of considerable practical use in technical dynamical systems. As a direct important application in engineering, the quantity ‖x(t)‖R 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 ‖x(t)‖R 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+‖x(t)‖R|⩽‖ẋ(t)‖R⩽c1|D+‖x(t)‖R| can be derived, which is shown to be invalid for the norm ‖·‖2. This result is also of interest on its own. The obtained findings are underpinned by two numerical examples and illustrated by graphs.