We introduce in this paper a decomposition method for optimal control of distributed systems of evolution. It is based on decomposition of domains, which can be overlapping or non overlapping. It is the overlapping case which is presented here. The method is completely general. Penalty arguments are also used. They can be "replaced" by Lagrange multipliers techniques, as shown in papers by O. PIRONNEAU and the Author. Similar decomposition techniques have been introduced hy the Author in [2]... [5] for problems of stabilization. Numerical results will be presented in subsequent papers written in collaboration with R. GLOWINSKI, J. PERIAUX and O. PIRONNEAU.
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