Q1. Find an extremal for the functional $$ f(x)=\int_{1}^{t_{f}}\left[2 x(t)+\frac{1}{2} \dot{x}^{2}(t)\right] d t $$ The boundary conditions are $$ x(1)=4, x\left(t_{f}\right)=4 $$ and $$ t_{f}1 $$ is free.

Question

Q1. Find an extremal for the functional $$ f(x)=\int_{1}^{t_{f}}\left[2 x(t)+\frac{1}{2} \dot{x}^{2}(t)\right] d t $$ The boundary conditions are $$ x(1)=4, x\left(t_{f}\right)=4 $$ and $$ t_{f}>1 $$ is free.

Wang Wagner · Accepted Answer

The extremal function $$ x(t) $$ is given by $$ x(t) = t^2 + C_1 t + C_2 $$, where $$ C_1 $$ and $$ C_2 $$ are constants determined by the boundary conditions $$ x(1) = 4 $$ and $$ x(t_f) = 4 $$. The specific form of $$ x(t) $$ depends on the choice of $$ t_f $$ and the corresponding values of $$ C_1 $$ and $$ C_2 $$.

Q1. Find an extremal for the functional \[ f(x)=\int_{1}^{t_{f}}\left[2 x(t)+\frac{1}{2} \dot{x}^{2}(t)\right] d t \] The boundary conditions are \( x(1)=4, x\left(t_{f}\right)=4 \) and \( t_{f}>1 \) is free.

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Solución paso a paso

Q1. Find an extremal for the functional \[ f(x)=\int_{1}^{t_{f}}\left[2 x(t)+\frac{1}{2} \dot{x}^{2}(t)\right] d t \] The boundary conditions are \( x(1)=4, x\left(t_{f}\right)=4 \) and \( t_{f}>1 \) is free.

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Respuesta rápida

Solución paso a paso

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