How to minimize the next functional using the Pontryagin Maximum Principle?
$begingroup$
Best regards,
I am asked to minimize the next functional
$T(v)=int_{A}^{B}frac{dx}{v(x)}$,
with $vneq 0text{ and } vin V$, where $V={vin C([A,B]):v(A)=v(B)=0}$.
If we assume that there exists $v^*$ that minimizes the functional, then $v^*$ satisfies the Euler-Lagrange equation.
Euler-Lagrange Equation: $frac{partial L}{partial y}-frac{d}{dx}(frac{partial L}{partial y'})=0$.
If $L=frac{1}{v},vneq 0$
$y=vRightarrow frac{partial L}{partial y}=-frac{1}{v^2},$
$frac{partial L}{partial y'}=0Rightarrowfrac{d}{dx}(frac{partial L}{partial y'})=0,$
then $frac{partial L}{partial y}-frac{d}{dx}(frac{partial L}{partial y'})=-frac{1}{v^2}-0=0Rightarrow frac{1}{v^2}=0,$ which is impossible, then must resort to the Principle of Maximum of Pontryagin.
How can I use the Pontryagin maximum principle in this case and I have also been told that the optimal $v$ is reached at point $frac{A+B}{2}$?
calculus-of-variations optimal-control
asked Jan 12 at 17:05
Alex PozoAlex Pozo
519214
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add a comment |
$begingroup$
Best regards,
I am asked to minimize the next functional
$T(v)=int_{A}^{B}frac{dx}{v(x)}$,
with $vneq 0text{ and } vin V$, where $V={vin C([A,B]):v(A)=v(B)=0}$.
If we assume that there exists $v^*$ that minimizes the functional, then $v^*$ satisfies the Euler-Lagrange equation.
Euler-Lagrange Equation: $frac{partial L}{partial y}-frac{d}{dx}(frac{partial L}{partial y'})=0$.
If $L=frac{1}{v},vneq 0$
$y=vRightarrow frac{partial L}{partial y}=-frac{1}{v^2},$
$frac{partial L}{partial y'}=0Rightarrowfrac{d}{dx}(frac{partial L}{partial y'})=0,$
then $frac{partial L}{partial y}-frac{d}{dx}(frac{partial L}{partial y'})=-frac{1}{v^2}-0=0Rightarrow frac{1}{v^2}=0,$ which is impossible, then must resort to the Principle of Maximum of Pontryagin.
How can I use the Pontryagin maximum principle in this case and I have also been told that the optimal $v$ is reached at point $frac{A+B}{2}$?
calculus-of-variations optimal-control
asked Jan 12 at 17:05
Alex PozoAlex Pozo
519214
$endgroup$
$begingroup$
Your assumption that there is a minimiser (with the constraints you gave) is wrong. It is not difficult to construct a sequence of functions $v_n$ such that $T(v_n)to-infty$. In fact, this is what the Euler-Lagrange result is telling you!
$endgroup$
– bjorne
Jan 14 at 19:35
$begingroup$
@bjrone I very much agree with you. How could you explain this "problem" in a formal way?
$endgroup$
– Alex Pozo
Jan 15 at 17:07
add a comment |
$begingroup$
Best regards,
I am asked to minimize the next functional
$T(v)=int_{A}^{B}frac{dx}{v(x)}$,
with $vneq 0text{ and } vin V$, where $V={vin C([A,B]):v(A)=v(B)=0}$.
If we assume that there exists $v^*$ that minimizes the functional, then $v^*$ satisfies the Euler-Lagrange equation.
Euler-Lagrange Equation: $frac{partial L}{partial y}-frac{d}{dx}(frac{partial L}{partial y'})=0$.
If $L=frac{1}{v},vneq 0$
$y=vRightarrow frac{partial L}{partial y}=-frac{1}{v^2},$
$frac{partial L}{partial y'}=0Rightarrowfrac{d}{dx}(frac{partial L}{partial y'})=0,$
then $frac{partial L}{partial y}-frac{d}{dx}(frac{partial L}{partial y'})=-frac{1}{v^2}-0=0Rightarrow frac{1}{v^2}=0,$ which is impossible, then must resort to the Principle of Maximum of Pontryagin.
How can I use the Pontryagin maximum principle in this case and I have also been told that the optimal $v$ is reached at point $frac{A+B}{2}$?
calculus-of-variations optimal-control
asked Jan 12 at 17:05
Alex PozoAlex Pozo
519214
$endgroup$
Best regards,
I am asked to minimize the next functional
$T(v)=int_{A}^{B}frac{dx}{v(x)}$,
with $vneq 0text{ and } vin V$, where $V={vin C([A,B]):v(A)=v(B)=0}$.
If we assume that there exists $v^*$ that minimizes the functional, then $v^*$ satisfies the Euler-Lagrange equation.
Euler-Lagrange Equation: $frac{partial L}{partial y}-frac{d}{dx}(frac{partial L}{partial y'})=0$.
If $L=frac{1}{v},vneq 0$
$y=vRightarrow frac{partial L}{partial y}=-frac{1}{v^2},$
$frac{partial L}{partial y'}=0Rightarrowfrac{d}{dx}(frac{partial L}{partial y'})=0,$
then $frac{partial L}{partial y}-frac{d}{dx}(frac{partial L}{partial y'})=-frac{1}{v^2}-0=0Rightarrow frac{1}{v^2}=0,$ which is impossible, then must resort to the Principle of Maximum of Pontryagin.
How can I use the Pontryagin maximum principle in this case and I have also been told that the optimal $v$ is reached at point $frac{A+B}{2}$?
calculus-of-variations optimal-control
calculus-of-variations optimal-control
asked Jan 12 at 17:05
Alex PozoAlex Pozo
519214
asked Jan 12 at 17:05
Alex PozoAlex Pozo
519214
asked Jan 12 at 17:05
Alex PozoAlex Pozo
519214
asked Jan 12 at 17:05
Alex PozoAlex Pozo
519214
asked Jan 12 at 17:05
Alex PozoAlex Pozo
519214
519214
$begingroup$
Your assumption that there is a minimiser (with the constraints you gave) is wrong. It is not difficult to construct a sequence of functions $v_n$ such that $T(v_n)to-infty$. In fact, this is what the Euler-Lagrange result is telling you!
$endgroup$
– bjorne
Jan 14 at 19:35
$begingroup$
@bjrone I very much agree with you. How could you explain this "problem" in a formal way?
$endgroup$
– Alex Pozo
Jan 15 at 17:07
add a comment |
$begingroup$
Your assumption that there is a minimiser (with the constraints you gave) is wrong. It is not difficult to construct a sequence of functions $v_n$ such that $T(v_n)to-infty$. In fact, this is what the Euler-Lagrange result is telling you!
$endgroup$
– bjorne
Jan 14 at 19:35
$begingroup$
@bjrone I very much agree with you. How could you explain this "problem" in a formal way?
$endgroup$
– Alex Pozo
Jan 15 at 17:07
$begingroup$
Your assumption that there is a minimiser (with the constraints you gave) is wrong. It is not difficult to construct a sequence of functions $v_n$ such that $T(v_n)to-infty$. In fact, this is what the Euler-Lagrange result is telling you!
$endgroup$
– bjorne
Jan 14 at 19:35
$begingroup$
Your assumption that there is a minimiser (with the constraints you gave) is wrong. It is not difficult to construct a sequence of functions $v_n$ such that $T(v_n)to-infty$. In fact, this is what the Euler-Lagrange result is telling you!
$endgroup$
– bjorne
Jan 14 at 19:35
$begingroup$
@bjrone I very much agree with you. How could you explain this "problem" in a formal way?
$endgroup$
– Alex Pozo
Jan 15 at 17:07
$begingroup$
@bjrone I very much agree with you. How could you explain this "problem" in a formal way?
$endgroup$
– Alex Pozo
Jan 15 at 17:07
add a comment |
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$begingroup$
Your assumption that there is a minimiser (with the constraints you gave) is wrong. It is not difficult to construct a sequence of functions $v_n$ such that $T(v_n)to-infty$. In fact, this is what the Euler-Lagrange result is telling you!
$endgroup$
– bjorne
Jan 14 at 19:35
$begingroup$
@bjrone I very much agree with you. How could you explain this "problem" in a formal way?
$endgroup$
– Alex Pozo
Jan 15 at 17:07