Formulation for calculus of variation with state-space constraint
$begingroup$
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
$endgroup$
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
$endgroup$
$begingroup$
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
$endgroup$
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
edited Dec 8 '18 at 3:13
Sean
asked Dec 7 '18 at 3:21
SeanSean
532513
532513
$begingroup$
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 |
0
active
oldest
votes
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3029419%2fformulation-for-calculus-of-variation-with-state-space-constraint%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3029419%2fformulation-for-calculus-of-variation-with-state-space-constraint%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
$begingroup$
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