CAD.3:4.4 - Construct a continuation, not only one permitted command

A permissible pair (z,v) must also lead somewhere from which work can continue. For the nominal update z^+=A_pz+B_pv, seek a set Z such that every admitted z has at least one permitted v with z^+\in Z.

One direct construction repeatedly removes states without such a successor:

Z_{n+1}= \{z\in Z_n:\exists v,\ (z,v)\in S,\ A_pz+B_pv\in Z_n\}.

An exact fixed point is controlled invariant under the stated model. A finite iteration gives an outer candidate, not automatically an invariant set; test the successor condition before relying on it. Gridding a continuous set needs an approximation-error argument.

An alternative is to exhibit a smaller terminal set T and an explicit admissible terminal law. A finite plan ending in T then has a feasible continuation by appending that law. For receding-horizon use, the standard shift argument also needs the actual next nominal state, retained feasible tail, modeled uncertainty and implementation to match the argument. A measurement-based reset must preserve the initialization and feasibility conditions or use an independently qualified recovery.

If a new optimization fails or misses its deadline, continuing a previously feasible tail can be legitimate only while its assumptions still hold. “Use the previous plan” is not a universal fallback.