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Thread: Assymptotic stable DE and Jacoby matrix

  1. #1 Assymptotic stable DE and Jacoby matrix 
    srm is offline
    Forum Freshman
    Join Date
    Mar 2011
    Hi. Sorry for my English.

    I have the following system of differential equations:
    , (1)
    where , , , , .
    is a matrix with unknown components, but I know that this matrix is symmetric and positive defined.

    I need to find function such that the system (1) has asymptotically stable equilibrium at zero.

    My ideas. I can choose function such that its Jacoby matrix is
    . (2)
    In this case the system (1) will be asymptotically stable, since the direvative of the Lapunov function is and the matrix is positive defined.

    The problem is the equation (2) is not solved, i.e. the function such that the (2) hold is not exists.

    The else way is choose as , then . In this case dynamic of the system (1) will be stable only locally since the derivative of the Lapunov function will contain 2 terms. The first is and the second is .

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