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Math Help - Maximization problem

  1. #1
    Member garymarkhov's Avatar
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    Maximization problem

    Find the functions x(t) and y(t) that maximize \int_0^1(4x-y^2)dt subject to \dot x = y, x(0)=1, x(1)=2

    So I formed the Hamiltonian:

    H=4x-y^2 + \lambda y

    And took the partials:

    \frac{\partial H}{\partial x} = 4 = -\dot \lambda

    I've assumed that x must be the state variable, but can someone describe why it must be so?

    \frac{\partial H}{\partial y} = -2y + \lambda = 0

    Now what? Please explain as if you are talking to an idiot child, which I am.
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  2. #2
    Super Member Rebesques's Avatar
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    Please explain as if you are talking to an idiot child, which I am.
    Relax. We 're all dumb kids in this grand game called life


    Now, for the functional I=\int_0^1(4x-y^2)dt, the argument is x=x(t), so maybe it makes more sence to write I=\int_0^1(4x-x'^2)dt. Now, the Euler-Lagrange equation (the "derivative" of I) gives us that, at a stationary point x,

    \frac{\partial}{\partial t}\frac{\partial}{\partial x'}(4x-x'^2)=\frac{\partial}{\partial x}(4x-x'^2)

    or x''(t)=-2. Solve and use the boundary conditions.
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