簡單混合型端點變化變分問題的橫截條件

2023-07-29 11:36 作者:艾琳娜的糖果屋 0人讀過 | 我要投稿

我們考慮泛函 $%0AJ%5Cleft%5B%20x%2Cy%20%5Cright%5D%20%3D%5Cvarphi%20%5Cleft(%20t_2%2Cx_2%2Cy_2%20%5Cright)%20%2B%5Cint_%7Bt_1%7D%5E%7Bt_2%7D%7BF%5Cleft(%20t%2Cx%2Cy%2Cx%5Cprime%2Cy%5Cprime%20%5Cright)%20%5Cmathrm%7Bd%7Dt%7D%0A%0A$ ，其中 $%0A%5C%2C%5C%2Cx%5Cleft(%20t_1%20%5Cright)%20%3Dx_1%2Cy%5Cleft(%20t_1%20%5Cright)%20%3Dy_1%5C%2C%5C%2Ct_2%0A%0A$ 待定，并且滿足不完全約束 $%0A%5Cphi%20%5Cleft(%20t%2Cx%2Cy%2Cx%5Cprime%2Cy%5Cprime%20%5Cright)%20%3D0%0A%0A$

引入拉格朗日乘子 $%0A%5Clambda%20%5Cleft(%20t%20%5Cright)%20%0A%0A$ ,記

$%0A%5Coverline%7BJ%5Cleft%5B%20x%2Cy%20%5Cright%5D%20%7D%3D%5Cvarphi%20%5Cleft(%20t_2%2Cx_2%2Cy_2%20%5Cright)%20%2B%5Cint_%7Bt_1%7D%5E%7Bt_2%7D%7BF%5Cleft(%20t%2Cx%2Cy%2Cx%5Cprime%2Cy%5Cprime%20%5Cright)%20%2B%5Clambda%20%5Cleft(%20t%20%5Cright)%20%5Cphi%20%5Cleft(%20t%2Cx%2Cy%2Cx%5Cprime%2Cy%5Cprime%20%5Cright)%20%5Cmathrm%7Bd%7Dt%7D%5C%5C%3D%5Cvarphi%20%5Cleft(%20t_2%2Cx_2%2Cy_2%20%5Cright)%20%2B%5Cint_%7Bt_1%7D%5E%7Bt_2%7D%7BH%5Cleft(%20t%2Cx%2Cy%2C%5Clambda%20%2Cx%5Cprime%2Cy%5Cprime%20%5Cright)%20%5Cmathrm%7Bd%7Dt%7D%0A%0A$

取變分則有

$%0A%5Cdelta%20%5Coverline%7BJ%7D%3D%5Cvarphi%20_%7Bt_2%7D%5Cdelta%20t_2%2B%5Cvarphi%20_%7Bx_2%7D%5Cdelta%20x_2%2B%5Cvarphi%20_%7By_2%7D%5Cdelta%20y_2%5C%5C%2B%5Cint_%7Bt_1%7D%5E%7Bt_2%7D%7B%5Cleft%5C%7B%20F_x%5Cdelta%20x%2BF_y%5Cdelta%20y%2BF_%7Bx%5Cprime%7D%5Cdelta%20x%5Cprime%2BF_%7By%5Cprime%7D%5Cdelta%20y%5Cprime%2B%5Clambda%20%5Cleft(%20t%20%5Cright)%20%5Cleft(%20%5Cphi%20_x%5Cdelta%20x%2B%5Cphi%20_y%5Cdelta%20y%2B%5Cphi%20_%7Bx%5Cprime%7D%5Cdelta%20x%5Cprime%2B%5Cphi%20_%7By%5Cprime%7D%5Cdelta%20y%5Cprime%20%5Cright)%20%5Cright%5C%7D%20%5Cmathrm%7Bd%7Dt%7D%5C%5C%2B%5Cleft(%20F%2B%5Clambda%20%5Cphi%20%5Cright)%20%5Cmid_%7Bt%3Dt_2%7D%5E%7B%7D%5Cdelta%20t_2%0A%0A$

右側(cè)

$%0A%3D%5Cvarphi%20_%7Bt_2%7D%5Cdelta%20t_2%2B%5Cvarphi%20_%7Bx_2%7D%5Cdelta%20x_2%2B%5Cvarphi%20_%7By_2%7D%5Cdelta%20y_2%2B%5Cleft(%20F_%7Bx%5Cprime%7D%5Cdelta%20x%2BF_%7By%5Cprime%7D%5Cdelta%20y%20%5Cright)%20%5Cmid_%7Bt%3Dt_2%7D%5E%7B%7D%2B%5Cleft(%20F%2B%5Clambda%20%5Cphi%20%5Cright)%20%5Cmid_%7Bt%3Dt_2%7D%5E%7B%7D%5Cdelta%20t_2%2B%5C%5C%5Cint_%7Bt_1%7D%5E%7Bt_2%7D%7B%5Cleft(%20F_x-%5Cfrac%7B%5Cmathrm%7Bd%7DF_%7Bx%5Cprime%7D%7D%7B%5Cmathrm%7Bd%7Dt%7D%2B%5Clambda%20%5Cleft(%20t%20%5Cright)%20%5Cphi%20_x%20%5Cright)%20%5Cdelta%20x%5Cmathrm%7Bd%7Dt%7D%2B%5Cint_%7Bt_1%7D%5E%7Bt_2%7D%7B%5Cleft(%20F_y-%5Cfrac%7B%5Cmathrm%7Bd%7DF_%7By%5Cprime%7D%7D%7B%5Cmathrm%7Bd%7Dt%7D%2B%5Clambda%20%5Cleft(%20t%20%5Cright)%20%5Cphi%20_y%20%5Cright)%20%5Cdelta%20y%5Cmathrm%7Bd%7Dt%7D%2B%5Cint_%7Bt_1%7D%5E%7Bt_2%7D%7B%5Clambda%20%5Cleft(%20t%20%5Cright)%20%5Cleft(%20%5Cphi%20_%7Bx%5Cprime%7D%5Cdelta%20x%5Cprime%2B%5Cphi%20_%7By%5Cprime%7D%5Cdelta%20y%5Cprime%20%5Cright)%20%5Cmathrm%7Bd%7Dt%7D%0A%0A$ $%0A%0A%0A$

利用變分與導數(shù)的關系

$%0A%5Cdelta%20x%5Cmid_%7Bt%3Dt_2%7D%5E%7B%7D%3D%5Cdelta%20x_2-x%5Cprime%5Cleft(%20t_2%20%5Cright)%20%5Cdelta%20t_2%5C%2C%5C%2C%20%5C%5C%5Cdelta%20y%5Cmid_%7Bt%3Dt_2%7D%5E%7B%7D%3D%5Cdelta%20y_2-y%5Cprime%5Cleft(%20t_2%20%5Cright)%20%5Cdelta%20t_2%0A%0A$

并假設 $%5Cdelta%20t_2%2C%5Cdelta%20x_2%2C%5Cdelta%20y_2$ 是相互無關的，則有橫截條件與歐拉-拉格朗日方程

$%0A%5Cvarphi%20_%7Bt_2%7D%3D%5Cleft%5B%20x%5Cprime%5Cleft(%20F_%7Bx%5Cprime%7D%2B%5Clambda%20%5Cphi%20_%7Bx%5Cprime%7D%20%5Cright)%20%2By%5Cprime%5Cleft(%20F_%7By%5Cprime%7D%2B%5Clambda%20%5Cphi%20_%7By%5Cprime%7D%20%5Cright)%20-%5Cleft(%20F%2B%5Clambda%20%5Cphi%20%5Cright)%20%5Cright%5D%20%5Cmid_%7Bt%3Dt_2%7D%5E%7B%7D%5C%2C%5C%2C%20%0A%5C%5C%0A%5Cvarphi%20_%7Bx_2%7D%3D-%5Cleft%5B%20F_%7Bx%5Cprime%7D%2B%5Clambda%20%5Cphi%20_%7Bx%5Cprime%7D%20%5Cright%5D%20%5Cmid_%7Bt%3Dt_2%7D%5E%7B%7D%0A%5C%5C%0A%5Cvarphi%20_%7By_2%7D%3D-%5Cleft%5B%20F_%7By%5Cprime%7D%2B%5Clambda%20%5Cphi%20_%7By%5Cprime%7D%20%5Cright%5D%20%5Cmid_%7Bt%3Dt_2%7D%5E%7B%7D%0A%5C%5C%0AF_x-%5Cfrac%7B%5Cmathrm%7Bd%7DF_%7Bx%5Cprime%7D%7D%7B%5Cmathrm%7Bd%7Dt%7D%2B%5Clambda%20%5Cleft(%20t%20%5Cright)%20%5Cphi%20_x-%5Cfrac%7B%5Cmathrm%7Bd%7D%7D%7B%5Cmathrm%7Bd%7Dt%7D%5Clambda%20%5Cleft(%20t%20%5Cright)%20%5Cphi%20_x%0A%5C%5C%0AF_y-%5Cfrac%7B%5Cmathrm%7Bd%7DF_%7By%5Cprime%7D%7D%7B%5Cmathrm%7Bd%7Dt%7D%2B%5Clambda%20%5Cleft(%20t%20%5Cright)%20%5Cphi%20_y-%5Cfrac%7B%5Cmathrm%7Bd%7D%7D%7B%5Cmathrm%7Bd%7Dt%7D%5Clambda%20%5Cleft(%20t%20%5Cright)%20%5Cphi%20_y%0A%0A$

最優(yōu)控制例子

$%0AJ%3Dt_2%2B%5Cint_0%5E%7Bt_2%7D%7B%5Cfrac%7B1%7D%7B2%7Du%5E2%5Cleft(%20t%20%5Cright)%20%5Cmathrm%7Bd%7Dt%7D%5C%2C%5C%2Cx%5Cprime%5Cleft(%20t%20%5Cright)%20%3Du%5Cleft(%20t%20%5Cright)%20%5C%2C%5C%2Cx%5Cleft(%200%20%5Cright)%20%3D1%20%EF%BC%8Cx%5Cleft(%20t_2%20%5Cright)%20%3D0%0A%0A$

$%0A%5Coverline%7BJ%7D%3Dt_2%2B%5Cint_0%5E%7Bt_2%7D%7B%5Cleft%5C%7B%20%5Cfrac%7B1%7D%7B2%7Du%5E2%5Cleft(%20t%20%5Cright)%20%2B%5Clambda%20%5Cleft(%20t%20%5Cright)%20%5Cleft(%20x%5Cprime%5Cleft(%20t%20%5Cright)%20-u%5Cleft(%20t%20%5Cright)%20%5Cright)%20%5Cright%5C%7D%20%5Cmathrm%7Bd%7Dt%7D%0A%0A$

此問題的 $%0A%5Cvarphi%20%5Cleft(%20t_2%2Cx_2%2Cy_2%20%5Cright)%20%3Dt_2%0A%0A$ ，因此僅有第一個橫截條件成立，剩余兩個自動消失

于是可以寫出方程組

$%0A1%3D%5Clambda%20%5Cleft(%20t%20%5Cright)%20u%5Cleft(%20t%20%5Cright)%20-%5Cfrac%7B1%7D%7B2%7Du%5E2%5Cleft(%20t%20%5Cright)%20%5Cmid_%7Bt%3Dt_2%7D%5E%7B%7D%0A%5C%5C%0Au%5Cleft(%20t%20%5Cright)%20-%5Clambda%20%5Cleft(%20t%20%5Cright)%20%3D0%0A%5C%5C%0A%5Clambda%20%5Cprime%5Cleft(%20t%20%5Cright)%20%3D0%0A%0A$

于是可解出

$%0Au%5Cleft(%20t%20%5Cright)%20%3D-%5Csqrt%7B2%7D%0A%5C%5C%0Ax%5Cleft(%20t%20%5Cright)%20%3D-%5Csqrt%7B2%7Dt%2B1%0A%5C%5C%0At_2%3D%5Cfrac%7B%5Csqrt%7B2%7D%7D%7B2%7D%0A%0A$

標簽：變分法

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簡單混合型端點變化變分問題的橫截條件

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