The paper presents an extension of discrete $2\mathrm{D}$ systems of Roesser and Fornasini–Marchesini types for the continuous case. Formulas for solution representation of linear continuous $2\mathrm{D}$ systems are obtained, including the possibility of expanding the methods of operational calculus by constructing corresponding integral transformations with respect to the spatial variable. Some linear-quadratic optimization problems are considered and the corresponding optimality conditions based on the operator setting are stated. Also, some form of the optimal control representation is presented.

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