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Finding an optimal path “around†a function
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I would like to know how can I put some conditions on a function $d(x)$ when solving this optimalization problem:
Given a continuous function $p:[a,b]rightarrowmathbbR$ on $[a,b]$ find a function $d:[a,b]rightarrowmathbbR_0^+$ where $d(a)=d(b)=0$ such that the function $d(x)$ minimizes the following functional: $$int_a^b
F(x,d,d')dx=int_a^bsqrt1+[p'(x)+d'(x)]^2dx$$
To find $d(x)$ I tried to apply the Euler-Lagrange equation:
$$fracpartial Fpartial d-fracddxfracpartial Fpartial d'=0$$
Hence
$$0-fracddxfracp'+d'sqrt1+[p'+d']^2=0$$
$$p'+d'=csqrt1+[p'+d']^2$$
$$(p'+d')^2=c^2+c^2(p'+d')^2$$
$$p'+d'=c_1$$
$$d(x)=c_1x+c_0-p(x)$$
Applying our boundary conditions we get:
$$d(x)=left(fracp(a)-p(b)a-bright)(x-a)+p(a)-p(x)$$
But becouse $d:[a,b]rightarrowmathbbR_0^+$ this can only be true if $p(x)leleft(fracp(a)-p(b)a-bright)(x-a)+p(a)$ on $[a,b]$.
My question is that if there is some more general solution with no further restrictions (also gives the correct answer if $p(x)gt c_1x+c_0$)
Thanks for any kind of help with this problem.
optimization calculus-of-variations
asked Jul 31 at 16:07
VojtaKlojta
111
add a comment |Â
up vote
1
down vote
favorite
I would like to know how can I put some conditions on a function $d(x)$ when solving this optimalization problem:
Given a continuous function $p:[a,b]rightarrowmathbbR$ on $[a,b]$ find a function $d:[a,b]rightarrowmathbbR_0^+$ where $d(a)=d(b)=0$ such that the function $d(x)$ minimizes the following functional: $$int_a^b
F(x,d,d')dx=int_a^bsqrt1+[p'(x)+d'(x)]^2dx$$
To find $d(x)$ I tried to apply the Euler-Lagrange equation:
$$fracpartial Fpartial d-fracddxfracpartial Fpartial d'=0$$
Hence
$$0-fracddxfracp'+d'sqrt1+[p'+d']^2=0$$
$$p'+d'=csqrt1+[p'+d']^2$$
$$(p'+d')^2=c^2+c^2(p'+d')^2$$
$$p'+d'=c_1$$
$$d(x)=c_1x+c_0-p(x)$$
Applying our boundary conditions we get:
$$d(x)=left(fracp(a)-p(b)a-bright)(x-a)+p(a)-p(x)$$
But becouse $d:[a,b]rightarrowmathbbR_0^+$ this can only be true if $p(x)leleft(fracp(a)-p(b)a-bright)(x-a)+p(a)$ on $[a,b]$.
My question is that if there is some more general solution with no further restrictions (also gives the correct answer if $p(x)gt c_1x+c_0$)
Thanks for any kind of help with this problem.
optimization calculus-of-variations
asked Jul 31 at 16:07
VojtaKlojta
111
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I would like to know how can I put some conditions on a function $d(x)$ when solving this optimalization problem:
Given a continuous function $p:[a,b]rightarrowmathbbR$ on $[a,b]$ find a function $d:[a,b]rightarrowmathbbR_0^+$ where $d(a)=d(b)=0$ such that the function $d(x)$ minimizes the following functional: $$int_a^b
F(x,d,d')dx=int_a^bsqrt1+[p'(x)+d'(x)]^2dx$$
To find $d(x)$ I tried to apply the Euler-Lagrange equation:
$$fracpartial Fpartial d-fracddxfracpartial Fpartial d'=0$$
Hence
$$0-fracddxfracp'+d'sqrt1+[p'+d']^2=0$$
$$p'+d'=csqrt1+[p'+d']^2$$
$$(p'+d')^2=c^2+c^2(p'+d')^2$$
$$p'+d'=c_1$$
$$d(x)=c_1x+c_0-p(x)$$
Applying our boundary conditions we get:
$$d(x)=left(fracp(a)-p(b)a-bright)(x-a)+p(a)-p(x)$$
But becouse $d:[a,b]rightarrowmathbbR_0^+$ this can only be true if $p(x)leleft(fracp(a)-p(b)a-bright)(x-a)+p(a)$ on $[a,b]$.
My question is that if there is some more general solution with no further restrictions (also gives the correct answer if $p(x)gt c_1x+c_0$)
Thanks for any kind of help with this problem.
optimization calculus-of-variations
asked Jul 31 at 16:07
VojtaKlojta
111
I would like to know how can I put some conditions on a function $d(x)$ when solving this optimalization problem:
Given a continuous function $p:[a,b]rightarrowmathbbR$ on $[a,b]$ find a function $d:[a,b]rightarrowmathbbR_0^+$ where $d(a)=d(b)=0$ such that the function $d(x)$ minimizes the following functional: $$int_a^b
F(x,d,d')dx=int_a^bsqrt1+[p'(x)+d'(x)]^2dx$$
To find $d(x)$ I tried to apply the Euler-Lagrange equation:
$$fracpartial Fpartial d-fracddxfracpartial Fpartial d'=0$$
Hence
$$0-fracddxfracp'+d'sqrt1+[p'+d']^2=0$$
$$p'+d'=csqrt1+[p'+d']^2$$
$$(p'+d')^2=c^2+c^2(p'+d')^2$$
$$p'+d'=c_1$$
$$d(x)=c_1x+c_0-p(x)$$
Applying our boundary conditions we get:
$$d(x)=left(fracp(a)-p(b)a-bright)(x-a)+p(a)-p(x)$$
But becouse $d:[a,b]rightarrowmathbbR_0^+$ this can only be true if $p(x)leleft(fracp(a)-p(b)a-bright)(x-a)+p(a)$ on $[a,b]$.
My question is that if there is some more general solution with no further restrictions (also gives the correct answer if $p(x)gt c_1x+c_0$)
Thanks for any kind of help with this problem.
optimization calculus-of-variations
asked Jul 31 at 16:07
VojtaKlojta
111
asked Jul 31 at 16:07
VojtaKlojta
111
asked Jul 31 at 16:07
VojtaKlojta
111
asked Jul 31 at 16:07
VojtaKlojta
111
111
add a comment |Â
add a comment |Â
1 Answer
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In this answer, we'll give a geometric description of OP's variational problem.
Problem. There is given a continuous function profile
$$[a,b]~ni~ x~~mapsto~~ p(x)~in~mathbbR.$$
We want to minimize the arclength of the continuous$^1$ function profile
$$[a,b]~ni~ x~~mapsto~~ f(x)~in~mathbbR.$$
We are allowed to increase (but not decrease!) the function value
$$ f(a)~=~p(a)~~wedge~~f(b)~=~p(b)~~wedge~~ forall x~in~]a,b[:~~f(x)~geq~p(x) .$$
Solution: The optimal function profile $f_ast$ is then given by (the boundary of) the convex upward hull of the $p$-profile.
Examples:
If $p$ is convex upward, then $f_ast=p$.
If $p$ is concave upward, then $f_ast$ is the straight line through the endpoints $(a,p(a))$ and $(b,p(b))$.
--
$^1$ Even though OP does not explicitly say so we assume that $f$ must be continuous. Else the problem is trivial: Just pick a constant majorant function $f$ on the open interval $]a,b[$.
answered Aug 3 at 9:42
Qmechanic
4,36711746
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1 Answer
1
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oldest
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1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
In this answer, we'll give a geometric description of OP's variational problem.
Problem. There is given a continuous function profile
$$[a,b]~ni~ x~~mapsto~~ p(x)~in~mathbbR.$$
We want to minimize the arclength of the continuous$^1$ function profile
$$[a,b]~ni~ x~~mapsto~~ f(x)~in~mathbbR.$$
We are allowed to increase (but not decrease!) the function value
$$ f(a)~=~p(a)~~wedge~~f(b)~=~p(b)~~wedge~~ forall x~in~]a,b[:~~f(x)~geq~p(x) .$$
Solution: The optimal function profile $f_ast$ is then given by (the boundary of) the convex upward hull of the $p$-profile.
Examples:
If $p$ is convex upward, then $f_ast=p$.
If $p$ is concave upward, then $f_ast$ is the straight line through the endpoints $(a,p(a))$ and $(b,p(b))$.
--
$^1$ Even though OP does not explicitly say so we assume that $f$ must be continuous. Else the problem is trivial: Just pick a constant majorant function $f$ on the open interval $]a,b[$.
answered Aug 3 at 9:42
Qmechanic
4,36711746
add a comment |Â
up vote
0
down vote
In this answer, we'll give a geometric description of OP's variational problem.
Problem. There is given a continuous function profile
$$[a,b]~ni~ x~~mapsto~~ p(x)~in~mathbbR.$$
We want to minimize the arclength of the continuous$^1$ function profile
$$[a,b]~ni~ x~~mapsto~~ f(x)~in~mathbbR.$$
We are allowed to increase (but not decrease!) the function value
$$ f(a)~=~p(a)~~wedge~~f(b)~=~p(b)~~wedge~~ forall x~in~]a,b[:~~f(x)~geq~p(x) .$$
Solution: The optimal function profile $f_ast$ is then given by (the boundary of) the convex upward hull of the $p$-profile.
Examples:
If $p$ is convex upward, then $f_ast=p$.
If $p$ is concave upward, then $f_ast$ is the straight line through the endpoints $(a,p(a))$ and $(b,p(b))$.
--
$^1$ Even though OP does not explicitly say so we assume that $f$ must be continuous. Else the problem is trivial: Just pick a constant majorant function $f$ on the open interval $]a,b[$.
answered Aug 3 at 9:42
Qmechanic
4,36711746
add a comment |Â
up vote
0
down vote
up vote
0
down vote
In this answer, we'll give a geometric description of OP's variational problem.
Problem. There is given a continuous function profile
$$[a,b]~ni~ x~~mapsto~~ p(x)~in~mathbbR.$$
We want to minimize the arclength of the continuous$^1$ function profile
$$[a,b]~ni~ x~~mapsto~~ f(x)~in~mathbbR.$$
We are allowed to increase (but not decrease!) the function value
$$ f(a)~=~p(a)~~wedge~~f(b)~=~p(b)~~wedge~~ forall x~in~]a,b[:~~f(x)~geq~p(x) .$$
Solution: The optimal function profile $f_ast$ is then given by (the boundary of) the convex upward hull of the $p$-profile.
Examples:
If $p$ is convex upward, then $f_ast=p$.
If $p$ is concave upward, then $f_ast$ is the straight line through the endpoints $(a,p(a))$ and $(b,p(b))$.
--
$^1$ Even though OP does not explicitly say so we assume that $f$ must be continuous. Else the problem is trivial: Just pick a constant majorant function $f$ on the open interval $]a,b[$.
answered Aug 3 at 9:42
Qmechanic
4,36711746
In this answer, we'll give a geometric description of OP's variational problem.
Problem. There is given a continuous function profile
$$[a,b]~ni~ x~~mapsto~~ p(x)~in~mathbbR.$$
We want to minimize the arclength of the continuous$^1$ function profile
$$[a,b]~ni~ x~~mapsto~~ f(x)~in~mathbbR.$$
We are allowed to increase (but not decrease!) the function value
$$ f(a)~=~p(a)~~wedge~~f(b)~=~p(b)~~wedge~~ forall x~in~]a,b[:~~f(x)~geq~p(x) .$$
Solution: The optimal function profile $f_ast$ is then given by (the boundary of) the convex upward hull of the $p$-profile.
Examples:
If $p$ is convex upward, then $f_ast=p$.
If $p$ is concave upward, then $f_ast$ is the straight line through the endpoints $(a,p(a))$ and $(b,p(b))$.
--
$^1$ Even though OP does not explicitly say so we assume that $f$ must be continuous. Else the problem is trivial: Just pick a constant majorant function $f$ on the open interval $]a,b[$.
answered Aug 3 at 9:42
Qmechanic
4,36711746
edited Aug 4 at 15:18
answered Aug 3 at 9:42
Qmechanic
4,36711746
answered Aug 3 at 9:42
Qmechanic
4,36711746
answered Aug 3 at 9:42
Qmechanic
4,36711746
4,36711746
add a comment |Â
add a comment |Â
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