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We extend the framework of sequential action control to systems of partial differential equations which can be posed as abstract linear control problems in a Hilbert space. We follow a late-lumping approach and show that the control action can be explicitly obtained from variational principles using adjoint information. Moreover, we analyze the closed-loop system obtained from the SAC feedback for quadratic stage costs. We apply this theory prototypically to an unstable heat equation and verify the results numerically.
