Two-sided bounds on the mean vector and covariance matrix in linear stochastically excited vibration systems with application of the differential calculus of norms
Ludwig Kohaupt · Cogent Mathematics · 2015
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.