I was reading a document with the following example

Let us consider the minimization of


Subject to the constraint

$\dot{x}(t)=x(t-1)+u(t),\quad x(0)=1$.

And then it just shows a table with the different values of $t,x(t),u(t), \phi(t)$

Results for Example 1 with fixed penalty constant parameter $\mu=1.5$

  1. Is the construction of the Hamiltonian the same as when working without delays?.
  2. How to state the Pontryagin Maximum Principle for this case?

Any book or document on this subject that you can suggest to me?

The Hamiltonian, according to what he read, would be like that


And the attached equations would be

$\dfrac{d\lambda}{dt}(t)=-\dfrac{\partial H}{\partial x(t)}(t)-\chi_{[0,t-1]}(t)\dfrac{\partial H}{\partial x(t-1)}(t+1)$

$\dfrac{d\lambda}{dt}(t)=-x(t)-\chi_{[0,t-1]}(t)\lambda(t+1)$, where $\chi_{[0,t-1]}$ is the characteristic function.

Finally, What I just indicated, is it correct?

  • $\begingroup$ How is $x(t)$ defined on the time interval $-1<t<0$, also $1$? $\endgroup$ – Kwin van der Veen May 15 '18 at 16:54
  • $\begingroup$ @KwinvanderVeen I don't understand your question. $\endgroup$ – VarúAnselmo Sui May 15 '18 at 21:57
  • $\begingroup$ If $x(t-1)$ is not defined for $0<t<1$ (or equivalently $x(t)$ on the interval $-1<t<0$) then $\dot{x}(t) is also not defined on the interval $0<t<1$. $\endgroup$ – Kwin van der Veen May 16 '18 at 4:34
  • $\begingroup$ Also, what is $\mu$? $\endgroup$ – Kwin van der Veen May 16 '18 at 10:28
  • $\begingroup$ $\mu$ is a fixed penalty constant parameter $\endgroup$ – VarúAnselmo Sui May 17 '18 at 1:26

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.