Abstract
This paper discusses the optimal control issue for elliptic kxk cooperative fractional systems. The fractional operators are proposed in the Laplace sense. Because of the nonlocality of the Laplace fractional operators, we reformulate the issue as an extended issue on a semi-infinite cylinder in Rk+1. The weak solution for these fractional systems is then proven to exist and be unique. Moreover, the existence and optimality conditions can be inferred as a consequence.