arXiv:2610.02151v2 Announce Type: replace Abstract: This paper addresses simultaneous input and output constraint satisfaction for a class of multi-input linear time-invariant systems with state feedback and integral action. A control barrier function (CBF)-based governor modifies the reference command while preserving the nominal feedback controller. Necessary and sufficient conditions are derived under which the governor is feasible at every state of a prescribed operating set and ensures simultaneous input and output constraint satisfaction, forward invariance, and bounded closed-loop solutions. The paper also shows that these conditions need not be met for certain choices of the CBF governor parameters due to conflicts between input and output constraints. A systematic design procedure is proposed that first searches for feasible free parameters and, if necessary, proposes a relaxation in the input constraint that restores feasibility. A companion paper provides numerical examples and counterexamples illustrating these results.