i've never worked on optimal control problem before and have an issue with this problem :
Let $x_0\in \mathbb{R}^2_+$ and a constant $\lambda \neq0$ be fixed.
I want to minimize,
$\qquad$ over final time $T>0$,
$\qquad$ over absolutely continuous trajectories $x(\cdot)$ taking values in $\mathbb{R}^2_+$ with $x(0)=x_0$, $x(T)=z$
the cost function
$\qquad \qquad J_T(x;z) = \displaystyle \sup_{\theta = (\theta_1,\theta_2) \in \mathcal{C}^\infty\left([0,T];\mathbb{R^2}\right) } \int_0^T \left[ \langle x'(t)\:,\: \theta(t)\rangle\: - \: \lambda(e^{\theta_1(t)}-1)\\ \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad - x_1(t)x_2(t)(e^{\theta_2(t)-\theta_1(t)}-1) - x_2(t)(e^{-\theta_2(t)}-1)\right] dt$
where $z$ is left free in some convex but unbounded domain of $\mathbb{R}^2_+$.
My questions are :
1) can we have the sufficient conditions for optimality ?
i mean could we prove that there exist $\qquad x^*$, $T^*$ and $z^*$ so that
$\qquad \qquad \qquad J_{T^*}(x^*;z^*) \leq J_T(x;z)$ $\qquad \forall \: x,z,T$
2) Assuming optimality, that is $\displaystyle \inf_{x,z,T} J_T(x;z) <+\infty$,
do we have
$\displaystyle \qquad \qquad \lim_{||z||\to +\infty} \inf_{x,T} J_T(x;z) = +\infty$ ?
I know that this could sound very abstract or perhaps a little ambiguous.
in my context, the initial value
$\qquad x_0\quad $ is a stable attractor ( of dynamical system)
$\qquad \qquad \qquad \qquad \left\{
\begin{array}{lcl}
y_1'(t) &=& \lambda - y(t)y_2(t)\\
y_1'(t) &=& y_1(t)y_2(t) - y_2(t)
\end{array}
\right.$
$\qquad z\quad $ are target points on the boundary of the attraction domain of $x_0$
$\qquad J_T(x;z)$ is the energy needed to go from $x_0$ (the attractor) to $z$ (at the boundary of attraction domainn) by following the trajectory $x$.
Noting that (integration by part : x(T) = z and x(0)=x_0 )
$\qquad \qquad J_T(x;z) =
\displaystyle \sup_{\theta = (\theta_1,\theta_2) \in \mathcal{C}^\infty\left([0,T];\mathbb{R^2}\right) }\Bigg\{ \langle z\:,\:\theta(T)\rangle - \langle x_0\:,\:\theta(0)\rangle - \int_0^T \langle x(t)\:,\: \theta'(t)\rangle\:dt\\
\qquad \qquad \qquad \qquad \qquad - \int_0^T \lambda(e^{\theta_1(t)}-1) - x_1(t)x_2(t)(e^{\theta_2(t)-\theta_1(t)}-1) - x_2(t)(e^{-\theta_2(t)}-1) \: dt\Bigg\}$
I really want to know if the more $z$ is far away from $x_0$ the more the energy to reach it from $x_0$ would be important? that's the aim of my question 2). Or perhaps i can have a condition that implies it?
For me it seems reasonable to imagine that if the energy to go from $x_0$
to the boundary is finished, the optimal trajectory could not be too far from $x_0$ ??
The problem can be rewrite as classical Bolza optimal control problem. i could give the representation if it could help.
Any book which can help?? Thanks in advance !!!