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I'm from econ. That is I'm not conceptually familiar with the underlying math of dynamical systems. When I usually deal with dynamic systems it's of the form $$ \max_u\int{F(t,x(t),u(t))}\\ \text{s.t.}~\dot{x}(t)=f(t,x(t),u(t))\\ x(0)=x_0 $$

where $x(t)\in\mathbb{R}^n$ is the state and $u(t)\in\mathbb{R}^m$ the control vector respectively.

However, I try to model a system where the state actually depends on the control, i.e. $x(t,u(t))$. Does it change the analysis substantially? Or does it work the usual way $$ H=F(t,x(t,u(t)),u(t))+\lambda(t)f(t,x(t,u(t)),u(t)) $$ with FOCs $$ H_u=0\quad\text{and}\quad H_x=-\dot{\lambda} $$ Appreciate help or a good read advice.

Simon
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  • Some clarification is warranted, because it's not clear here how exactly your $x$ depends on the control $u$. Does it depend on it only through the time evolution function $f$, or can it be affected directly by the current value of $u$ somehow? (If it can, then you can use the Dirac delta to reduce it back to the previous case, but this gets rather technical/subtle.) – Ian Dec 16 '14 at 22:32
  • So how are you supposed to optimize the cost function without being able to influence the state through the control? – Rubi Shnol Sep 09 '15 at 09:28

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Usually the state always depends on the input. Otherwise, changing the input would have no effect on the state! We control the states of the systems with the control inputs! This is the case because $x(t) = x(t_0) + \int_{t_0}^t f(t,u(t),x(t) dt$ and the RHS of this equation is dependend on u(t)!

You have to be aware that $H_u$ are the paritial and not the total derivatives, that means you consider x and t constant when calculating $\frac{\partial H}{\partial u}$

You also have to be aware that the solution might not be the optimal one. For some dynamic systems it however be optimal! See therfore: http://en.wikipedia.org/wiki/Pontryagin%27s_minimum_principle

What do you mean by "FOCs"?

Rufio
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  • FOC is short for first order (optimality) condition. That is, it is the condition that the gradient must be zero at a max or min of a smooth function. – Ian Dec 16 '14 at 22:29