Suppose I have the following system relating force to flux in a magnetic actuator \begin{align} \vec{F} &= \mathbb{C}\vec{\Phi^2}\\ \vec{F} &= \begin{bmatrix} F_x\\ F_y \end{bmatrix}\\ \mathbb{C} &= c \begin{bmatrix} -1 & 0 & 1 & 0\\ 0 & 1 & 0 & -1 \end{bmatrix}\\ \vec{\Phi^2}&= \begin{bmatrix} \Phi_w^2\\ \Phi_n^2\\ \Phi_e^2\\ \Phi_s^2\\ \end{bmatrix} \end{align} The forces are known as is the constant $c$ in the $\mathbb{C}$ matrix. If I use the Moore-Penrose inverse to solve for $\vec{\Phi^2}$, I will obtain a solution that minimizes it's L2-norm (least-squares fit solution). But suppose each flux term is a summation of a bias flux term and a coil flux term (i.e. $\Phi_w = \Phi_{wb}+\Phi_{wc}$) and I want to minimize the norm of the vector of coil fluxes: \begin{align} \vec{\Phi_c} = \begin{bmatrix} \Phi_{wc}\\ \Phi_{nc}\\ \Phi_{ec}\\ \Phi_{sc}\\ \end{bmatrix} \end{align} Each term in $\vec{\Phi^2}$ has the structure $\Phi^2 = (\Phi_c^2+2\Phi_c\Phi_b+\Phi_b^2)$. How do I go about finding the solution that minimizes the L2-norm of $\vec{\Phi_c}$ (i.e. minimize $\Sigma\Phi_{ic}^2$)? Is there a name for this kind of optimization so I can research it further?
Clarification of different vector definitions: \begin{align} \vec{\Phi} &= \begin{bmatrix} \Phi_{w}\\ \Phi_{n}\\ \Phi_{e}\\ \Phi_{s}\\ \end{bmatrix} = \vec{\Phi_c}+\vec{\Phi_b} = \begin{bmatrix} \Phi_{wc}+\Phi_{wb}\\ \Phi_{nc}+\Phi_{nb}\\ \Phi_{ec}+\Phi_{eb}\\ \Phi_{sc}+\Phi_{sb}\\ \end{bmatrix}\\ \vec{\Phi^2} &= \begin{bmatrix} \Phi_w^2\\ \Phi_n^2\\ \Phi_e^2\\ \Phi_s^2\\ \end{bmatrix} = \begin{bmatrix} \Phi_{wc}^2+2\Phi_{wc}\Phi_{wb}+\Phi_{wb}^2\\ \Phi_{nc}^2+2\Phi_{nc}\Phi_{nb}+\Phi_{nb}^2\\ \Phi_{ec}^2+2\Phi_{ec}\Phi_{eb}+\Phi_{eb}^2\\ \Phi_{sc}^2+2\Phi_{sc}\Phi_{sb}+\Phi_{sb}^2\\ \end{bmatrix}\\ \end{align}