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

Authors : Ludwig KOHAUPT

Pages : 27-47

Doi:10.33434/cams.460724

View : 42 | Download : 10

Publication Date : 2019-03-22

Article Type : Research Paper

Abstract :As the main new result, we show that one can construct a time-dependent positive definite matrix $Rinsert ignore into journalissuearticles values(t,t_0);$ such that the solution $xinsert ignore into journalissuearticles values(t);$ of the initial value problem $\dot{x}insert ignore into journalissuearticles values(t);=A\,xinsert ignore into journalissuearticles values(t);+hinsert ignore into journalissuearticles values(t,xinsert ignore into journalissuearticles values(t););, \; xinsert ignore into journalissuearticles values(t_0);=x_0,$ under certain conditions satisfies the equation $\|xinsert ignore into journalissuearticles values(t);\|_{Rinsert ignore into journalissuearticles values(t,t_0);} = \|x_Ainsert ignore into journalissuearticles values(t);\|_R$ where $x_Ainsert ignore into journalissuearticles values(t);$ 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 $\|\cdot\|_{Rinsert ignore into journalissuearticles values(t,t_0);}$ 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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