Area-constrained quadratic control cost

For a closed planar path with $\dot{x}=u$, $\dot{y}=v$ and $\dot{z}=uy-vx$, the signed area satisfies $z(1)=-2$ times the usual oriented area. Periodic Wirtinger and Cauchy-Schwarz inequalities give $|z(1)|\le \frac{1}{2\pi }{\int }_0^1(u^2+v^2)dt$. Thus terminal $z(1)=1$ requires cost at least $\pi$. A clockwise circular path traversed once at constant speed attains equality, proving the global minimum rather than merely a stationary Pontryagin extremal.