= 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 $|z(1)|\le\frac1{2\pi}\int_0^1(u^2+v^2)dt$. 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.
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