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Thread: Proving Stability and Asymptotic Stability of Homogeneous Equations

  1. #1
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    Proving Stability and Asymptotic Stability of Homogeneous Equations

    Hi,
    Using these definitions:

    A homogeneous equation with constant coefficients is said to be stable if all solutions remain bounded as t-->infinity

    A homogeneous equation with constant coefficients is said to be asymptotically stable if all solutions converge to the zero solution as t-->infinity

    I'm trying to prove these two statements:

    1) The system is stable if and only if for every root $\displaystyle \lambda = \alpha + i\beta$ of the characteristic polynomial, $\displaystyle 0 \geq \alpha $ and $\displaystyle \alpha < 0 $ if the multiplicity of lambda is bigger than one.

    2) The system is aymptotically stable if and only if for every root $\displaystyle \lambda = \alpha + i\beta$ of the characteristic polynomial, $\displaystyle \alpha < 0 $.

    Thanks a lot!
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  2. #2
    Super Member Rebesques's Avatar
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    ...When is, for fixed n, the limit $\displaystyle \lim_{x\rightarrow+\infty}x^n\exp(\alpha x)\cos(\beta x)$ zero? And when is it infinite?
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