Consider the following performance index: $$J=\cfrac{1}{2}u_1^2+\cfrac{1}{2}u_2^2+\cfrac{1}{2}u_3^2+p_1\cdot u_1+p_2\cdot u_2+p_3\cdot u_3$$ Suppose $u_1$, $u_2$ and $u_3$ are the design variables and $p_1$, $p_2$ and $p_3$ are the parameters of the problem. Additionaly, $u_1^2+u_2^2+u_3^2\leq\bar{u}$ and $p_1^2+p_2^2+p_3^2\geq\bar{u}$ where $\bar{u}$ is a constant parameter.
How to find an analytical solution for $\min_{u_1,u_2,u_3}J$ by considering the above-mentioned constraints and definitions? Does it have any unique non-trivial solution?
This kind of problem may arise in constrained optimal control problems.
Example: For the following bounded input optimal control problem $$\dot{z}=Az+u; \|u\|\leq\bar{u}$$ with the performance index $V=\cfrac{1}{2}\int{(u^Tu)}dt$ the optimal control command $u^*$ can be found by considering Hamiltonian $\mathscr{H}(u)=\cfrac{1}{2}u^Tu+p^T(Az+u)$. Since the control variable is bounded then it should be found so that $\mathscr{H}(u^*)<\mathscr{H}(u)$ for $z^*$, $p^*$ and all $u$. This leads to: $$\cfrac{1}{2}{u^*}^T{u^*}+{p^*}^T{u^*}<\cfrac{1}{2}u^Tu+{p^*}^Tu$$
If the control was unbounded the solution would be $u^*=-p^*$. For the problem with bounded control variable, $u^*=-p^*$ is also the solution when $\|p\|<\bar{u}$. What is the solution when $\|p\|\geq\bar{u}$?
Note that $J(u) = {1 \over 2} u^T u + p^T u = {1 \over 2} (\|u+p\|^2 - \|p\|^2)$.
We look for solutions of $\min \{ {1 \over 2 } \|u+p\|^2 | \|u\|^2 \le \bar{u} \}$, or letting $x=u+p$, the equivalent $\min \{ \|x\|^2 | \|x-p\| \le \bar{u} \} = \min_{x \in \bar{B}(p,\bar{u})} \|x\|^2$.
It is straightforward to see that the unique solution of the latter problem is given by $x^* = 0$, if $\|p\| \le \bar{u}$ and $x^*= (1-{\bar{u} \over \|p\|}) p$, if $\|p\| > \bar{u}$.
Hence the solution is $u^*= x^*-p$. Explicitly, the solution is $u^* = -p$, if $\|p\| \le \bar{u}$ and $u^*=-{\bar{u} \over \|p\|} p$, if $\|p\| > \bar{u}$.
Nearest point computations appear in many guises in optimization.