I was reading a document with the following example
Let us consider the minimization of
$\phi(u)=\frac{1}{2}\int_0^2[x^2(t)+u^2(t)]dt$.
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 $\mu=1.5$
- Is the construction of the Hamiltonian the same as when working without delays?.
- 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
$H(t,x,y,u)=\frac{1}{2}[x^2(t)+u^2(t)]+\lambda[x(t-1)+u(t)]$.
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?