# Optimal Control of Delay Differential Equations

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 $\mu=1.5$

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

$H(t,x,y,u)=\frac{1}{2}[x^2(t)+u^2(t)]+\lambda[x(t-1)+u(t)]$.

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?

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