Array Optimization and its Application to Control Problems

In order to develope a new analysis technique for distributed parameter systems(DPS), the array that is a generalization of vector-matrix is introduced, and the array algebra is constructed, which generalizes the concepts in the conventional vector systems into those in the array systems. An array optimization problem for an array dynamical system is formulated, and the necessary optimality condition is derived by use of the array algebra. Array dynamical systems are obtained when a wide class of DPS are approximated using FEM or difference method, and express the state distribution of the DPS. An LQ problem for a linear array dynamical system is investigated, and the optimal control is obtained, which is the array version of the well-known LQ optimal control and realizes a sort of feedback of the state distribution pattern of the DPS The array introduced in this paper generalizes the vector-matrix theory and provides a technique for developing more advanced pattern feedback control for DPS.