A strong-constraint 4DVAR objective can have two local minima

Since $u_1=\mathrm{\Psi }(u_0)=u_0$, the one-variable objective from [F1] is $J(u)=\frac{1}{2}{u}^2+\frac{1}{2}(1-u^2{)}^2=\frac{1}{2}-\frac{1}{2}{u}^2+\frac{1}{2}{u}^4\mathrm{.}$

Differentiating gives $J^{\mathrm{\prime }}(u)=u(2u^2-1)$. Its only stationary points are $0$ and $\pm 1\mathrm{/}\sqrt{2}$. The derivative changes from negative to positive at each of $\pm 1\mathrm{/}\sqrt{2}$, so both are strict local minima. The objective is even, $J(\pm 1\mathrm{/}\sqrt{2})=3\mathrm{/}8<J(0)=1\mathrm{/}2$, and $J(u)\to \mathrm{\infty }$ as $\mathrm{\mid }u\mathrm{\mid }\to \mathrm{\infty }$. Every global minimum is therefore attained at a stationary point, and these two distinct points are both global minimizers. Thus the minimizer is not unique, so an iterative routine's finite definition gives no guarantee that it returns a designated unique solution.