This article is dedicated to the optimization for N×N parabolic system with mixed boundary conditions beneath conjugation conditions and including Schrödinger operator according to the maximum principle. Through Green’s formula, the properties of the first eigenvalues and the Lax-Milgram lemma, we provide evidence for the solutions to these systems’ existence and uniqueness. The maximum principle and the existence of a solution for a given linear parabolic system with variable coefficients are also topics of study.