Abstract
Existence theorems are proved for multidimensional Lagrange control systems, in which the state is a function of several independent variables in a fixed domain, while in contrast, the control is a function of only some of the independent variables (that is, u is a lower-dimensional control). As usual in Lagrange problems, the cost functional is a multiple integral, the state equation is a system of partial differential equations, with assigned boundary conditions, and constraints may be imposed on the values of the state and control variables.
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