In this work, the optimal synthesis and control structure design (CSD) problems for flexible heat exchanger networks (HENs) are integrated into a new sequential methodology. The proposed approach relies, on one hand, on a convexification and outer-approximation strategy to solve the synthesis stage and, on the other hand, on the sum of squared deviations (SSD) method for the optimal CSD. These methods guarantee the optimality of the synthesis process, as well as the proper operation of the HEN in several operating points. The first stage of the proposed approach, which focuses on the flexible HEN synthesis problem, considers both temperature and flow rate modifications in the inlet streams. A multiperiod synthesis formulation is proposed where critical points are iteratively incorporated to fulfill the flexibility requirements. Because the problem size and the nonconvexities increase when additional critical points are considered, both the convexification of nonlinear terms and an outer approximation strategy are used to guarantee the optimality of the solutions at this stage. The second stage handles the decisions associated with the design of the control structure. This stage is critical because the network is required to work in a wide range of operating points. If the classical CSD method based on the wellknown relative gain array (RGA) is applied, and only the nominal operating point is considered, such requirements are not fullfiled. In fact, this work demonstrates that such classical CSD approaches are not sufficient to operate the HEN in the range of variation considered by the multiperiod synthesis phase. As an alternative method, the application of the SSD approach to multiple operating points is proposed. Thus, several optimal control structures are developed to ensure the operability of the HEN. Three academic case studies are presented to illustrate the application of the proposed methodology.