In many technological processes a fundamental stage involves the mixing of two or more fluids. As a result, the design of optimal mixing protocols is a problem of both fundamental and practical importance. In this paper, the authors formulate a prototypical mixing problem in a control framework, where the objective is to determine the sequence of fluid flows that will maximize entropy. By developing the appropriate ergodic-theoretic tools for the determination of entropy of periodic sequences, they derive the form of the protocol which maximizes entropy among all of the possible periodic sequences composed of two shear flows orthogonal to each other. The authors discuss the relevance of their results in the interpretation of previous studies of mixing protocols.