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

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.

Keywords:

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

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