Formulation for calculus of variation with state-space constraint
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I'm stuck on this question, let $B = {xin mathbb{R}^n:|x|leq 1}$ be the unit ball in $mathbb{R}^n$, consider the following minimizing problem
$$ inf_{x(cdot) in mathcal{A}} int_0^infty e^{-s} Big(|x'(s)|^2 - V(x(s))Big);ds$$
where $V:mathbb{R}^nlongrightarrow mathbb{R}$ us of class $C^1$ and is bounded $|V(x)| leq C$, subjected to a somewhat unsual constraint
$$ mathcal{A} = Big{x(cdot):[0,infty)longrightarrow B: x'(cdot)in L^1_{mathrm{loc}}([0,infty)), x(0) = x_0in B Big}.$$
How can I find the correct Euler-Lagrange equation for this problem? The problems appear when I need to find a good test function space $gamma(cdot)$ such that $eta+gamma in mathcal{A}$ for all $eta$ and $gamma$, which is not clear how to make $eta(s)+gamma(s) in B$ for all $s$, and also the boundary for the Euler-Lagrange equation is unclear.
optimization calculus-of-variations optimal-control euler-lagrange-equation
asked Dec 7 '18 at 3:21
SeanSean
527513
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add a comment |
$begingroup$
I'm stuck on this question, let $B = {xin mathbb{R}^n:|x|leq 1}$ be the unit ball in $mathbb{R}^n$, consider the following minimizing problem
$$ inf_{x(cdot) in mathcal{A}} int_0^infty e^{-s} Big(|x'(s)|^2 - V(x(s))Big);ds$$
where $V:mathbb{R}^nlongrightarrow mathbb{R}$ us of class $C^1$ and is bounded $|V(x)| leq C$, subjected to a somewhat unsual constraint
$$ mathcal{A} = Big{x(cdot):[0,infty)longrightarrow B: x'(cdot)in L^1_{mathrm{loc}}([0,infty)), x(0) = x_0in B Big}.$$
How can I find the correct Euler-Lagrange equation for this problem? The problems appear when I need to find a good test function space $gamma(cdot)$ such that $eta+gamma in mathcal{A}$ for all $eta$ and $gamma$, which is not clear how to make $eta(s)+gamma(s) in B$ for all $s$, and also the boundary for the Euler-Lagrange equation is unclear.
optimization calculus-of-variations optimal-control euler-lagrange-equation
asked Dec 7 '18 at 3:21
SeanSean
527513
$endgroup$
$begingroup$
Wouldn't $x'(s)=0$ minimize the problem, so $x(s)=x_0$?
$endgroup$
– Kwin van der Veen
Dec 7 '18 at 11:50
$begingroup$
I just fixed it!
$endgroup$
– Sean
Dec 7 '18 at 22:27
add a comment |
$begingroup$
I'm stuck on this question, let $B = {xin mathbb{R}^n:|x|leq 1}$ be the unit ball in $mathbb{R}^n$, consider the following minimizing problem
$$ inf_{x(cdot) in mathcal{A}} int_0^infty e^{-s} Big(|x'(s)|^2 - V(x(s))Big);ds$$
where $V:mathbb{R}^nlongrightarrow mathbb{R}$ us of class $C^1$ and is bounded $|V(x)| leq C$, subjected to a somewhat unsual constraint
$$ mathcal{A} = Big{x(cdot):[0,infty)longrightarrow B: x'(cdot)in L^1_{mathrm{loc}}([0,infty)), x(0) = x_0in B Big}.$$
How can I find the correct Euler-Lagrange equation for this problem? The problems appear when I need to find a good test function space $gamma(cdot)$ such that $eta+gamma in mathcal{A}$ for all $eta$ and $gamma$, which is not clear how to make $eta(s)+gamma(s) in B$ for all $s$, and also the boundary for the Euler-Lagrange equation is unclear.
optimization calculus-of-variations optimal-control euler-lagrange-equation
asked Dec 7 '18 at 3:21
SeanSean
527513
$endgroup$
I'm stuck on this question, let $B = {xin mathbb{R}^n:|x|leq 1}$ be the unit ball in $mathbb{R}^n$, consider the following minimizing problem
$$ inf_{x(cdot) in mathcal{A}} int_0^infty e^{-s} Big(|x'(s)|^2 - V(x(s))Big);ds$$
where $V:mathbb{R}^nlongrightarrow mathbb{R}$ us of class $C^1$ and is bounded $|V(x)| leq C$, subjected to a somewhat unsual constraint
$$ mathcal{A} = Big{x(cdot):[0,infty)longrightarrow B: x'(cdot)in L^1_{mathrm{loc}}([0,infty)), x(0) = x_0in B Big}.$$
How can I find the correct Euler-Lagrange equation for this problem? The problems appear when I need to find a good test function space $gamma(cdot)$ such that $eta+gamma in mathcal{A}$ for all $eta$ and $gamma$, which is not clear how to make $eta(s)+gamma(s) in B$ for all $s$, and also the boundary for the Euler-Lagrange equation is unclear.
optimization calculus-of-variations optimal-control euler-lagrange-equation
optimization calculus-of-variations optimal-control euler-lagrange-equation
asked Dec 7 '18 at 3:21
SeanSean
527513
asked Dec 7 '18 at 3:21
SeanSean
527513
edited Dec 8 '18 at 3:13
Sean
asked Dec 7 '18 at 3:21
SeanSean
527513
asked Dec 7 '18 at 3:21
SeanSean
527513
asked Dec 7 '18 at 3:21
SeanSean
527513
527513
$begingroup$
Wouldn't $x'(s)=0$ minimize the problem, so $x(s)=x_0$?
$endgroup$
– Kwin van der Veen
Dec 7 '18 at 11:50
$begingroup$
I just fixed it!
$endgroup$
– Sean
Dec 7 '18 at 22:27
add a comment |
$begingroup$
Wouldn't $x'(s)=0$ minimize the problem, so $x(s)=x_0$?
$endgroup$
– Kwin van der Veen
Dec 7 '18 at 11:50
$begingroup$
I just fixed it!
$endgroup$
– Sean
Dec 7 '18 at 22:27
$begingroup$
Wouldn't $x'(s)=0$ minimize the problem, so $x(s)=x_0$?
$endgroup$
– Kwin van der Veen
Dec 7 '18 at 11:50
$begingroup$
Wouldn't $x'(s)=0$ minimize the problem, so $x(s)=x_0$?
$endgroup$
– Kwin van der Veen
Dec 7 '18 at 11:50
$begingroup$
I just fixed it!
$endgroup$
– Sean
Dec 7 '18 at 22:27
$begingroup$
I just fixed it!
$endgroup$
– Sean
Dec 7 '18 at 22:27
add a comment |
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$begingroup$
Wouldn't $x'(s)=0$ minimize the problem, so $x(s)=x_0$?
$endgroup$
– Kwin van der Veen
Dec 7 '18 at 11:50
$begingroup$
I just fixed it!
$endgroup$
– Sean
Dec 7 '18 at 22:27