Critical-region boundaries are hyperplanes iff the extremum occurs at a fixed endpoint, because the costate remains linear in the initial state at any fixed time.
Multi-parametric model-based control: theory and applications
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On Linear Critical-Region Boundaries in Continuous-Time Multiparametric Optimal Control
Critical-region boundaries are hyperplanes iff the extremum occurs at a fixed endpoint, because the costate remains linear in the initial state at any fixed time.