Given a system of functions, find a system of differential equations which describe that system
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Inspired by modelling phenomena in biology, I'm wondering whether there has been mathematical study on the following question:
Given some $mathbf{X}(t) in mathbb{R}^n$, find $f = f(X_1, dots, X_n)$, where $f$ is time invariant so that $$X'(t) = f(X_1, dots, X_n)$$
To explain, has there been an exposition into the general problem of finding a time invariant differential equation which "best" describes the time evolution of a given system?
ordinary-differential-equations mathematical-modeling control-theory
edited Dec 1 '18 at 0:42
Robert Lewis
46.3k23066
asked Nov 30 '18 at 23:19
libbylibby
1137
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add a comment |
$begingroup$
Inspired by modelling phenomena in biology, I'm wondering whether there has been mathematical study on the following question:
Given some $mathbf{X}(t) in mathbb{R}^n$, find $f = f(X_1, dots, X_n)$, where $f$ is time invariant so that $$X'(t) = f(X_1, dots, X_n)$$
To explain, has there been an exposition into the general problem of finding a time invariant differential equation which "best" describes the time evolution of a given system?
ordinary-differential-equations mathematical-modeling control-theory
edited Dec 1 '18 at 0:42
Robert Lewis
46.3k23066
asked Nov 30 '18 at 23:19
libbylibby
1137
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$begingroup$
I added the "differential-equations" tag to your post Cheers!
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– Robert Lewis
Dec 1 '18 at 0:42
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Not always possible even for 1 dimension, for example $x(t) = sin(t)$ has $x(0)=x(pi)=0$ but $x'(0) neq x'(pi)$. So there is no function $f$ for which $x'(t)=f(x(t))$.
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– Michael
Dec 1 '18 at 5:00
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@Michael But one can extend the state space to also include $x_2=x'$.
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– Kwin van der Veen
Dec 1 '18 at 5:33
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So you are asking what kind of research has been done on continuous nonlinear time-invariant system identification?
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– Kwin van der Veen
Dec 1 '18 at 5:37
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I suppose. Do we have any tools to "solve" for the nonlinear time-invariant system? Even in any special cases?
$endgroup$
– libby
Dec 1 '18 at 18:52
add a comment |
$begingroup$
Inspired by modelling phenomena in biology, I'm wondering whether there has been mathematical study on the following question:
Given some $mathbf{X}(t) in mathbb{R}^n$, find $f = f(X_1, dots, X_n)$, where $f$ is time invariant so that $$X'(t) = f(X_1, dots, X_n)$$
To explain, has there been an exposition into the general problem of finding a time invariant differential equation which "best" describes the time evolution of a given system?
ordinary-differential-equations mathematical-modeling control-theory
edited Dec 1 '18 at 0:42
Robert Lewis
46.3k23066
asked Nov 30 '18 at 23:19
libbylibby
1137
$endgroup$
Inspired by modelling phenomena in biology, I'm wondering whether there has been mathematical study on the following question:
Given some $mathbf{X}(t) in mathbb{R}^n$, find $f = f(X_1, dots, X_n)$, where $f$ is time invariant so that $$X'(t) = f(X_1, dots, X_n)$$
To explain, has there been an exposition into the general problem of finding a time invariant differential equation which "best" describes the time evolution of a given system?
ordinary-differential-equations mathematical-modeling control-theory
ordinary-differential-equations mathematical-modeling control-theory
edited Dec 1 '18 at 0:42
Robert Lewis
46.3k23066
asked Nov 30 '18 at 23:19
libbylibby
1137
edited Dec 1 '18 at 0:42
Robert Lewis
46.3k23066
asked Nov 30 '18 at 23:19
libbylibby
1137
edited Dec 1 '18 at 0:42
Robert Lewis
46.3k23066
edited Dec 1 '18 at 0:42
Robert Lewis
46.3k23066
edited Dec 1 '18 at 0:42
Robert Lewis
46.3k23066
46.3k23066
asked Nov 30 '18 at 23:19
libbylibby
1137
asked Nov 30 '18 at 23:19
libbylibby
1137
asked Nov 30 '18 at 23:19
libbylibby
1137
1137
$begingroup$
I added the "differential-equations" tag to your post Cheers!
$endgroup$
– Robert Lewis
Dec 1 '18 at 0:42
$begingroup$
Not always possible even for 1 dimension, for example $x(t) = sin(t)$ has $x(0)=x(pi)=0$ but $x'(0) neq x'(pi)$. So there is no function $f$ for which $x'(t)=f(x(t))$.
$endgroup$
– Michael
Dec 1 '18 at 5:00
$begingroup$
@Michael But one can extend the state space to also include $x_2=x'$.
$endgroup$
– Kwin van der Veen
Dec 1 '18 at 5:33
$begingroup$
So you are asking what kind of research has been done on continuous nonlinear time-invariant system identification?
$endgroup$
– Kwin van der Veen
Dec 1 '18 at 5:37
$begingroup$
I suppose. Do we have any tools to "solve" for the nonlinear time-invariant system? Even in any special cases?
$endgroup$
– libby
Dec 1 '18 at 18:52
add a comment |
$begingroup$
I added the "differential-equations" tag to your post Cheers!
$endgroup$
– Robert Lewis
Dec 1 '18 at 0:42
$begingroup$
Not always possible even for 1 dimension, for example $x(t) = sin(t)$ has $x(0)=x(pi)=0$ but $x'(0) neq x'(pi)$. So there is no function $f$ for which $x'(t)=f(x(t))$.
$endgroup$
– Michael
Dec 1 '18 at 5:00
$begingroup$
@Michael But one can extend the state space to also include $x_2=x'$.
$endgroup$
– Kwin van der Veen
Dec 1 '18 at 5:33
$begingroup$
So you are asking what kind of research has been done on continuous nonlinear time-invariant system identification?
$endgroup$
– Kwin van der Veen
Dec 1 '18 at 5:37
$begingroup$
I suppose. Do we have any tools to "solve" for the nonlinear time-invariant system? Even in any special cases?
$endgroup$
– libby
Dec 1 '18 at 18:52
$begingroup$
I added the "differential-equations" tag to your post Cheers!
$endgroup$
– Robert Lewis
Dec 1 '18 at 0:42
$begingroup$
I added the "differential-equations" tag to your post Cheers!
$endgroup$
– Robert Lewis
Dec 1 '18 at 0:42
$begingroup$
Not always possible even for 1 dimension, for example $x(t) = sin(t)$ has $x(0)=x(pi)=0$ but $x'(0) neq x'(pi)$. So there is no function $f$ for which $x'(t)=f(x(t))$.
$endgroup$
– Michael
Dec 1 '18 at 5:00
$begingroup$
Not always possible even for 1 dimension, for example $x(t) = sin(t)$ has $x(0)=x(pi)=0$ but $x'(0) neq x'(pi)$. So there is no function $f$ for which $x'(t)=f(x(t))$.
$endgroup$
– Michael
Dec 1 '18 at 5:00
$begingroup$
@Michael But one can extend the state space to also include $x_2=x'$.
$endgroup$
– Kwin van der Veen
Dec 1 '18 at 5:33
$begingroup$
@Michael But one can extend the state space to also include $x_2=x'$.
$endgroup$
– Kwin van der Veen
Dec 1 '18 at 5:33
$begingroup$
So you are asking what kind of research has been done on continuous nonlinear time-invariant system identification?
$endgroup$
– Kwin van der Veen
Dec 1 '18 at 5:37
$begingroup$
So you are asking what kind of research has been done on continuous nonlinear time-invariant system identification?
$endgroup$
– Kwin van der Veen
Dec 1 '18 at 5:37
$begingroup$
I suppose. Do we have any tools to "solve" for the nonlinear time-invariant system? Even in any special cases?
$endgroup$
– libby
Dec 1 '18 at 18:52
$begingroup$
I suppose. Do we have any tools to "solve" for the nonlinear time-invariant system? Even in any special cases?
$endgroup$
– libby
Dec 1 '18 at 18:52
add a comment |
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$begingroup$
I added the "differential-equations" tag to your post Cheers!
$endgroup$
– Robert Lewis
Dec 1 '18 at 0:42
$begingroup$
Not always possible even for 1 dimension, for example $x(t) = sin(t)$ has $x(0)=x(pi)=0$ but $x'(0) neq x'(pi)$. So there is no function $f$ for which $x'(t)=f(x(t))$.
$endgroup$
– Michael
Dec 1 '18 at 5:00
$begingroup$
@Michael But one can extend the state space to also include $x_2=x'$.
$endgroup$
– Kwin van der Veen
Dec 1 '18 at 5:33
$begingroup$
So you are asking what kind of research has been done on continuous nonlinear time-invariant system identification?
$endgroup$
– Kwin van der Veen
Dec 1 '18 at 5:37
$begingroup$
I suppose. Do we have any tools to "solve" for the nonlinear time-invariant system? Even in any special cases?
$endgroup$
– libby
Dec 1 '18 at 18:52