Two-sided bounds on the mean vector and covariance matrix in linear stochastically excited vibration systems with application of the differential calculus of norms

• Main Author: Ludwig Kohaupt
• Format: Article
• Language: English
• Published: Taylor & Francis Group 2015-12-01
• Series: Cogent Mathematics
• Subjects: linear stochastic vibration system excited by white noise; mean vector; covariance matrix; two-sided bounds; differential calculus of norms
• Online Access: http://dx.doi.org/10.1080/23311835.2015.1021603

For a linear stochastic vibration model in state-space form, $ \dot{x}(t) = A x(t)+b(t), \, x(0)=x_0, $ with system matrix A and white noise excitation $ b(t) $, under certain conditions, the solution $ x(t) $ is a random vector that can be completely described by its mean vector, $ m_x(t):=m_{x(t)} $, and its covariance matrix, $ P_x(t):=P_{x(t)} $. If matrix $ A $ is asymptotically stable, then $ m_x(t) \rightarrow 0 \, (t \rightarrow \infty ) $ and $ P_x(t) \rightarrow P \, (t \rightarrow \infty ) $, where $ P $ is a positive (semi-)definite matrix. As the main new points, in this paper, we derive two-sided bounds on $ \Vert m_x(t)\Vert _2 $ and $ \Vert P_x(t)- P\Vert _2 $ as well as formulas for the right norm derivatives $ D_+^k \Vert P_x(t)- P\Vert _2, \, k=0,1,2 $, and apply these results to the computation of the best constants in the two-sided bounds. The obtained results are of special interest to applied mathematicians and engineers.