State Pontryagin's maximum principle (PMP) for the problem of minimizing

$$\int_{0}^{T} c(x(t), u(t)) d t+K(x(T))$$

where $x(t) \in \mathbb{R}^{n}, u(t) \in \mathbb{R}^{m}, d x / d t=a(x(t), u(t))$; here, $x(0)$ and $T$ are given, and $x(T)$ is unconstrained.

Consider the two-dimensional problem in which $d x_{1} / d t=x_{2}, d x_{2} / d t=u$, $c(x, u)=\frac{1}{2} u^{2}$ and $K(x(T))=\frac{1}{2} q x_{1}(T)^{2}, q>0$. Show that, by use of a variable $z(t)=x_{1}(t)+x_{2}(t)(T-t)$, one can rewrite this problem as an equivalent one-dimensional problem.

Use PMP to solve this one-dimensional problem, showing that the optimal control can be expressed as $u(t)=-q z(T)(T-t)$, where $z(T)=z(0) /\left(1+\frac{1}{3} q T^{3}\right)$.

Express $u(t)$ in a feedback form of $u(t)=k(t) z(t)$ for some $k(t)$.

Suppose that the initial state $x(0)$ is perturbed by a small amount to $x(0)+\left(\epsilon_{1}, \epsilon_{2}\right)$. Give an expression (in terms of $\epsilon_{1}, \epsilon_{2}, x(0), q$ and $T$ ) for the increase in minimal cost.