New sequences of monotonically increasing sets are introduced, for linear discrete-time systems subject to input and state constraints. The elements of the set sequences are controlled invariant and admissible regions of stabilizability. They are generated from the iterative application of the inverse reachability mapping, its geometric generalization, called the inverse directional reachability mapping, and mappings constructed by parts of the one-step inverse reachability and the one-step inverse directional reachability set. The four proposed set sequences converge to the maximal region of stabilizability.
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