This work concerns with the study of the continuous classical quaternary boundary optimal control problem or, for brief quaternary boundary optimal control problem (QBOCP) controlling by quaternary nonlinear hyperbolic system (QNLHS).  The theorem for existence a unique quaternary state vector solution (QSVS) for the weak formulation (WFO) of the QNLHS is proved in an infinite dimensional space via the method of Galerkin, and the help of the Aubin’s Theorem with given a continuous boundary control quaternary vector (CBCQV). The continuity of the operator of Lipschitz between the quaternary state vector and the conforming quaternary boundary control vector is demonstrated. The existence theorem of a continuous boundary optimal control quaternary vector (CBOCQV) which is minimized the objective function (OF), and is controlled by the QNLHS is demonstrated in an infinite dimensional space.