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From: Ivan Werning on 28 Feb 2010 16:28
Often the solution to an ODE with x(0) given requires solving
x'(t)=f(x(t),u(t),t)
where we compute u(t) from some function
u(t)=g(x(t),t).

An example is the maximum principle in optimal control: here x(t) includes the states and co-states of the system and u(t) is the control, and g= argmin of the Hamiltonian function.

When I use ode45 to solve such a system I am able to compute the solution x(t) at given points or save it as a structure. This is very convenient.

But often the object of interest is u(t) itself. So, is there a way to save u(t) similarly, during the ODE solution? Or must I re-evaluate g(x(t),t) myself, once the solution x(t) is computed? The latter seems wasteful; especially in the optimal control context when g is computed by minimizing the Hamiltonian using fmincon or similar. Moreover, it doesn't give me u(t) in the convenient (and fast to evaluate) structure output.

-Ivan
From: Roy on 19 Mar 2010 13:37
On Feb 28, 5:28 pm, "Ivan Werning" <iwerni...(a)yahoo.com> wrote:
> Often the solution to an ODE with x(0) given requires solving
> x'(t)=f(x(t),u(t),t)
> where we compute u(t) from some function
> u(t)=g(x(t),t).
>
> An example is the maximum principle in optimal control: here x(t) includes the states and co-states of the system and u(t) is the control, and g= argmin of the Hamiltonian function.
>
> When I use ode45 to solve such a system I am able to compute the solution x(t) at given points or save it as a structure. This is very convenient.
>
> But often the object of interest is u(t) itself. So, is there a way to save u(t) similarly, during the ODE solution? Or must I re-evaluate g(x(t),t) myself, once the solution x(t) is computed? The latter seems wasteful; especially in the optimal control context when g is computed by minimizing the Hamiltonian using fmincon or similar. Moreover, it doesn't give me u(t) in the convenient (and fast to evaluate) structure output.
>
> -Ivan

The problem is that ode45 is going to call f.m many more times than
are going to be used in the final answer, for reasons two reasons:

- ode45 is using runge kutta and therefore evaluating f.m several
times per accepted time step
- for purposes of adaptive stepsize control, ode45 often computes
values that are discarded as the step size is adjusted.



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