Analysis of the Dynamical System x˙(t) = A x(t) +h(t, x(t)), x(t0) = x0 in a Special Time-Dependent Norm

Yıl 2019, Cilt: 2 Sayı: 1, 27 - 47, 22.03.2019

Ludwig Kohaupt

https://doi.org/10.33434/cams.460724

Öz

As the main new result, we show that one can construct a time-dependent positive definite matrix $R(t,t_0)$ such that the solution $x(t)$ of the initial value problem $\dot{x}(t)=A\,x(t)+h(t,x(t)), \; x(t_0)=x_0,$ under certain conditions satisfies the equation $\|x(t)\|_{R(t,t_0)} = \|x_A(t)\|_R$ where $x_A(t)$ is the solution of the above IVP when $h \equiv 0$ and $R$ is a constant positive definite matrix constructed from the eigenvectors and principal vectors of $A$ and $A^{\ast}$ and where $\|\cdot\|_{R(t,t_0)}$ and $\|\cdot\|_R$ are weighted norms. Applications are made to dynamical systems, and numerical examples underpin the theoretical findings.

Anahtar Kelimeler

Nonlinear initial value problem with linear principal part, Vibration suppression, Monotonicity behavior, Two-sided bounds, Weighted norm, Weighted norm

Kaynakça

• [1] L. Kohaupt, Computation of optimal two-sided bounds for the asymptotic behavior of free lineardynamical systems with application of the differential calculus of norms, Journal of ComputationalMathematics and Optimization 2(3)(2006)127-173.
• [2] L. Kohaupt, Construction of a biorthogonal system of principal vectors of the matrices A andA* with applications to the initial value problem dx/dt = Ax; x(t0) = x0, Journal of ComputationalMathematics and Optimization 3(3)(2007)163-192.
• [3] L. Kohaupt, Solution of the matrix eigenvalue problem V A + A*V = \mu V with applications tothe study of free linear systems, J. Comp. Appl. Math. 213(1)(2008)142-165.
• [4] L. Kohaupt, Biorthogonalization of the principal vectors for the matrices A and A* with applicationto the computation of the explicit representation of the solution x(t) of dx/dt = Ax; x(t_0) = x_0,Applied Mathematical Sciences 2(20)(2008)961-974.
• [5] L. Kohaupt, Two-sided bounds for the asymptotic behavior of free nonlinear vibration systemswith application of the differential calculus of norms, International Journal of Computer Mathematics87(3) (2010) 653-667.
• [6] L. Kohaupt, On the vibration-suppression property and monotonicity behavior of a specialweighted norm for dynamical dx/dt = Ax; x(t_0) = x_0, Applied Mathematics and Computation222(2013)307-330.