Explain how transversality conditions can be helpful when employing Pontryagin's Maximum Principle to solve an optimal control problem.

A particle in $\mathbb{R}^{2}$ starts at $(0,0.5)$ and follows the dynamics

$$\dot{x}=u \sqrt{|y|}, \quad \dot{y}=v \sqrt{|y|}, \quad t \in[0, T],$$

where controls $u(t)$ and $v(t)$ are to be chosen subject to $u^{2}(t)+v^{2}(t)=1$.

Using Pontryagin's maximum principle do the following:

(a) Find controls that minimize $-y(1)$;

(b) Suppose we wish to choose $T$ and the controls $u, v$ to minimize $-y(T)+T$ under a constraint $(x(T), y(T))=(1,1)$. By expressing both $d y / d x$ and $d^{2} y / d x^{2}$ in terms of the adjoint variables, show that on an optimal trajectory,

$$1+\left(\frac{d y}{d x}\right)^{2}+2 y \frac{d^{2} y}{d x^{2}}=0$$