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This paper develops a new set of necessary and sufficient conditions for the stability of differential linear repetitive processes, based on application of the Kalman-Yakubovich-Popov lemma. These new conditions reduce the problem of determining the stability of an example to checking for the existence of a solution of a set of linear matrix inequalities. A relatively easy extension to enable stabilizing control law design, with additional perfromance specifications if required, is established. The inclusion of extra design specifications is developed for the case of regional constraints on the eigenvalues of state matrix and a finite frequency range design. Finally, a possible application in iterative learning control is briefly discussed.