Design a parameter $k$ for a second order ODE $ddot{x}(t) = f(t,k)$ subjected to a set of initial and...
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I have a nonlinear differential equation as follows
begin{align}
ddot{x}(t) = f(theta(t),k),
end{align}
where $f$ is a function of the time variable $t in mathbb{R}^{+}$, $k in mathbb{R}$ is the designed parameter, and $theta(t)$ is a known function of time. The function $theta(t)$ only depends on time and it is a quite complicated so I cannot integrate to find $x(t)$ and $dot{x}(t)$ analytically.
My task is: compute the parameter $k$ such that following initial/terminal conditions should be met as good as possible:
begin{align}
x(0) = 0, quad dot{x}(0) = 0, quad x(t_{f}) = bar{x},
end{align}
where the terminal time $t_{f}$ and the value of $bar{x}$ are both known.
My question is: is there any elegant way to solve such problem?
My current idea: I will formulate an optimization problem as below:
begin{align}
&min quad J = x^{2}(0) + dot{x}^{2}(0) + [x(t_{f}) - bar{x}]^2 \
& quad {k}
end{align}
Then I will use a nonlinear programmer to solve. But I think it is a bit brute to solve in that way.
I appreciate any suggestion. Thank you!
differential-equations optimization nonlinear-system
asked Nov 22 at 16:10
lyhuehue01
133
|
show 1 more comment
up vote
0
down vote
favorite
I have a nonlinear differential equation as follows
begin{align}
ddot{x}(t) = f(theta(t),k),
end{align}
where $f$ is a function of the time variable $t in mathbb{R}^{+}$, $k in mathbb{R}$ is the designed parameter, and $theta(t)$ is a known function of time. The function $theta(t)$ only depends on time and it is a quite complicated so I cannot integrate to find $x(t)$ and $dot{x}(t)$ analytically.
My task is: compute the parameter $k$ such that following initial/terminal conditions should be met as good as possible:
begin{align}
x(0) = 0, quad dot{x}(0) = 0, quad x(t_{f}) = bar{x},
end{align}
where the terminal time $t_{f}$ and the value of $bar{x}$ are both known.
My question is: is there any elegant way to solve such problem?
My current idea: I will formulate an optimization problem as below:
begin{align}
&min quad J = x^{2}(0) + dot{x}^{2}(0) + [x(t_{f}) - bar{x}]^2 \
& quad {k}
end{align}
Then I will use a nonlinear programmer to solve. But I think it is a bit brute to solve in that way.
I appreciate any suggestion. Thank you!
differential-equations optimization nonlinear-system
asked Nov 22 at 16:10
lyhuehue01
133
This ode is linear but non-homogeneous.
– Jon
Nov 22 at 16:26
Thanks a lot @Jon! I figured out that it is not possible to obtain an analytical solution for $x(t)$ and $dot{x}(t)$, and formulating an optimization problem for this task is not a good idea as well. However, I can separate the parameter $k$ such that $ddot{x}(t) = L(theta(t)) + kR(theta(t))$, then using numerical integration to impose the given initial and terminal conditions.
– lyhuehue01
Nov 23 at 7:06
as you have written the equation, the solution is $x(t)=x(0)+{dot x}(o)t+int_0^tdt'int_0^{t'}dt''f(theta(t''),k)$. This has a closed form or not depending the known functions $f$ and $theta$.
– Jon
Nov 23 at 7:36
Yes, that's right. In my case, $f(theta(t),k)$ is a nested trigonometric function (e.g. $tan(sin(t))$) so as far as I know, it has no closed form of antiderivative. Therefore, I am only able to use numerical integration.
– lyhuehue01
Nov 25 at 3:13
Then, in such a situation, the inescapable approach is the numerical one.
– Jon
Nov 25 at 13:27
|
show 1 more comment
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I have a nonlinear differential equation as follows
begin{align}
ddot{x}(t) = f(theta(t),k),
end{align}
where $f$ is a function of the time variable $t in mathbb{R}^{+}$, $k in mathbb{R}$ is the designed parameter, and $theta(t)$ is a known function of time. The function $theta(t)$ only depends on time and it is a quite complicated so I cannot integrate to find $x(t)$ and $dot{x}(t)$ analytically.
My task is: compute the parameter $k$ such that following initial/terminal conditions should be met as good as possible:
begin{align}
x(0) = 0, quad dot{x}(0) = 0, quad x(t_{f}) = bar{x},
end{align}
where the terminal time $t_{f}$ and the value of $bar{x}$ are both known.
My question is: is there any elegant way to solve such problem?
My current idea: I will formulate an optimization problem as below:
begin{align}
&min quad J = x^{2}(0) + dot{x}^{2}(0) + [x(t_{f}) - bar{x}]^2 \
& quad {k}
end{align}
Then I will use a nonlinear programmer to solve. But I think it is a bit brute to solve in that way.
I appreciate any suggestion. Thank you!
differential-equations optimization nonlinear-system
asked Nov 22 at 16:10
lyhuehue01
133
I have a nonlinear differential equation as follows
begin{align}
ddot{x}(t) = f(theta(t),k),
end{align}
where $f$ is a function of the time variable $t in mathbb{R}^{+}$, $k in mathbb{R}$ is the designed parameter, and $theta(t)$ is a known function of time. The function $theta(t)$ only depends on time and it is a quite complicated so I cannot integrate to find $x(t)$ and $dot{x}(t)$ analytically.
My task is: compute the parameter $k$ such that following initial/terminal conditions should be met as good as possible:
begin{align}
x(0) = 0, quad dot{x}(0) = 0, quad x(t_{f}) = bar{x},
end{align}
where the terminal time $t_{f}$ and the value of $bar{x}$ are both known.
My question is: is there any elegant way to solve such problem?
My current idea: I will formulate an optimization problem as below:
begin{align}
&min quad J = x^{2}(0) + dot{x}^{2}(0) + [x(t_{f}) - bar{x}]^2 \
& quad {k}
end{align}
Then I will use a nonlinear programmer to solve. But I think it is a bit brute to solve in that way.
I appreciate any suggestion. Thank you!
differential-equations optimization nonlinear-system
differential-equations optimization nonlinear-system
asked Nov 22 at 16:10
lyhuehue01
133
asked Nov 22 at 16:10
lyhuehue01
133
asked Nov 22 at 16:10
lyhuehue01
133
asked Nov 22 at 16:10
lyhuehue01
133
asked Nov 22 at 16:10
lyhuehue01
133
133
This ode is linear but non-homogeneous.
– Jon
Nov 22 at 16:26
Thanks a lot @Jon! I figured out that it is not possible to obtain an analytical solution for $x(t)$ and $dot{x}(t)$, and formulating an optimization problem for this task is not a good idea as well. However, I can separate the parameter $k$ such that $ddot{x}(t) = L(theta(t)) + kR(theta(t))$, then using numerical integration to impose the given initial and terminal conditions.
– lyhuehue01
Nov 23 at 7:06
as you have written the equation, the solution is $x(t)=x(0)+{dot x}(o)t+int_0^tdt'int_0^{t'}dt''f(theta(t''),k)$. This has a closed form or not depending the known functions $f$ and $theta$.
– Jon
Nov 23 at 7:36
Yes, that's right. In my case, $f(theta(t),k)$ is a nested trigonometric function (e.g. $tan(sin(t))$) so as far as I know, it has no closed form of antiderivative. Therefore, I am only able to use numerical integration.
– lyhuehue01
Nov 25 at 3:13
Then, in such a situation, the inescapable approach is the numerical one.
– Jon
Nov 25 at 13:27
|
show 1 more comment
This ode is linear but non-homogeneous.
– Jon
Nov 22 at 16:26
Thanks a lot @Jon! I figured out that it is not possible to obtain an analytical solution for $x(t)$ and $dot{x}(t)$, and formulating an optimization problem for this task is not a good idea as well. However, I can separate the parameter $k$ such that $ddot{x}(t) = L(theta(t)) + kR(theta(t))$, then using numerical integration to impose the given initial and terminal conditions.
– lyhuehue01
Nov 23 at 7:06
as you have written the equation, the solution is $x(t)=x(0)+{dot x}(o)t+int_0^tdt'int_0^{t'}dt''f(theta(t''),k)$. This has a closed form or not depending the known functions $f$ and $theta$.
– Jon
Nov 23 at 7:36
Yes, that's right. In my case, $f(theta(t),k)$ is a nested trigonometric function (e.g. $tan(sin(t))$) so as far as I know, it has no closed form of antiderivative. Therefore, I am only able to use numerical integration.
– lyhuehue01
Nov 25 at 3:13
Then, in such a situation, the inescapable approach is the numerical one.
– Jon
Nov 25 at 13:27
This ode is linear but non-homogeneous.
– Jon
Nov 22 at 16:26
This ode is linear but non-homogeneous.
– Jon
Nov 22 at 16:26
Thanks a lot @Jon! I figured out that it is not possible to obtain an analytical solution for $x(t)$ and $dot{x}(t)$, and formulating an optimization problem for this task is not a good idea as well. However, I can separate the parameter $k$ such that $ddot{x}(t) = L(theta(t)) + kR(theta(t))$, then using numerical integration to impose the given initial and terminal conditions.
– lyhuehue01
Nov 23 at 7:06
Thanks a lot @Jon! I figured out that it is not possible to obtain an analytical solution for $x(t)$ and $dot{x}(t)$, and formulating an optimization problem for this task is not a good idea as well. However, I can separate the parameter $k$ such that $ddot{x}(t) = L(theta(t)) + kR(theta(t))$, then using numerical integration to impose the given initial and terminal conditions.
– lyhuehue01
Nov 23 at 7:06
as you have written the equation, the solution is $x(t)=x(0)+{dot x}(o)t+int_0^tdt'int_0^{t'}dt''f(theta(t''),k)$. This has a closed form or not depending the known functions $f$ and $theta$.
– Jon
Nov 23 at 7:36
as you have written the equation, the solution is $x(t)=x(0)+{dot x}(o)t+int_0^tdt'int_0^{t'}dt''f(theta(t''),k)$. This has a closed form or not depending the known functions $f$ and $theta$.
– Jon
Nov 23 at 7:36
Yes, that's right. In my case, $f(theta(t),k)$ is a nested trigonometric function (e.g. $tan(sin(t))$) so as far as I know, it has no closed form of antiderivative. Therefore, I am only able to use numerical integration.
– lyhuehue01
Nov 25 at 3:13
Yes, that's right. In my case, $f(theta(t),k)$ is a nested trigonometric function (e.g. $tan(sin(t))$) so as far as I know, it has no closed form of antiderivative. Therefore, I am only able to use numerical integration.
– lyhuehue01
Nov 25 at 3:13
Then, in such a situation, the inescapable approach is the numerical one.
– Jon
Nov 25 at 13:27
Then, in such a situation, the inescapable approach is the numerical one.
– Jon
Nov 25 at 13:27
|
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This ode is linear but non-homogeneous.
– Jon
Nov 22 at 16:26
Thanks a lot @Jon! I figured out that it is not possible to obtain an analytical solution for $x(t)$ and $dot{x}(t)$, and formulating an optimization problem for this task is not a good idea as well. However, I can separate the parameter $k$ such that $ddot{x}(t) = L(theta(t)) + kR(theta(t))$, then using numerical integration to impose the given initial and terminal conditions.
– lyhuehue01
Nov 23 at 7:06
as you have written the equation, the solution is $x(t)=x(0)+{dot x}(o)t+int_0^tdt'int_0^{t'}dt''f(theta(t''),k)$. This has a closed form or not depending the known functions $f$ and $theta$.
– Jon
Nov 23 at 7:36
Yes, that's right. In my case, $f(theta(t),k)$ is a nested trigonometric function (e.g. $tan(sin(t))$) so as far as I know, it has no closed form of antiderivative. Therefore, I am only able to use numerical integration.
– lyhuehue01
Nov 25 at 3:13
Then, in such a situation, the inescapable approach is the numerical one.
– Jon
Nov 25 at 13:27