In order to develop a new analysis technique for distributed parameter systems (DPS), the array that is a generalization of the vector-matrix is introduced, and the array algebra is constructed, which generalizes the various concepts in the conventional vector-matrix theory. A wide class of DPS is approximately transformed into array dynamical systems by use of the finite element method or the finite difference method. In the array system formulation, the spatial structure of the DPS is preserved, and the formulation is accordingly suitable for the utilization of the spatial-state-distribution pattern (SSDP), which is in a sense the essence of the DPS, to the control. An array optimization problem for an array dynamical system is formulated, and the necessary optimality condiiion is derived by use of the array algebra. The linear quadratic (LQ) problem for a linear array dynamical system is investigated, and the resultant optimal control is the array version of the well-known LQ optimal control and realizes a sort of feedback of the SSDP of the DPS. The array theory developed generalizes the vector-matrix theory and prepares a technique for the system analysis for a wide class of DPS in a unified method and for developing the SSDP feedback control for DPS.