On Exact Critical-Region Boundaries in Continuous-Time Multiparametric Optimal Control Problems
Abstract
Continuous-time multiparametric model predictive control solves the optimal control problem once, offline, for every initial state, and divides the initial-state space into critical regions, each with its own control law and sequence of active constraints. A boundary between two regions is either a hyperplane, given by one linear equation, or curved. The curved boundaries have not been computed to date; a classifier has been trained instead to distinguish the regions, so the partition has been known only approximately. In this paper we show that a curved boundary can be computed exactly, without solving any optimal control problem. The method rests on one property. Once the sequence of active constraints is fixed, together with the instants inside it at which a constraint becomes active or inactive (the switching times) and the instant at which the change of the neighboring region first appears (the event time), every condition on a boundary is a linear equation in the initial state, so the computation of a boundary, curved or not, is exact. We fix one of these instants, a switching time or the event time, and solve the linear equations that correspond to it for the corresponding initial state on the boundary; stepping the fixed instant along the horizon yields the whole boundary. A hyperplane is the case in which no instant is left to fix. We illustrate the method on the Newell-Lee evaporator, which has seven critical regions and eight boundaries, four hyperplanes and four curved, computing one boundary of each type in full with the adjacent regions at its ends.
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BibTeX
@misc{lamakani2026exact,
title = {{On Exact Critical-Region Boundaries in Continuous-Time Multiparametric Optimal Control Problems}},
author = {Lida Lamakani and Efstratios N. Pistikopoulos},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.32043},
eprint = {2610.32043},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.32043},
}
RIS
TY - UNPB
TI - On Exact Critical-Region Boundaries in Continuous-Time Multiparametric Optimal Control Problems
AU - Lida Lamakani
AU - Efstratios N. Pistikopoulos
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2610.32043
UR - https://arxiv.org/abs/2610.32043
AB - Continuous-time multiparametric model predictive control solves the optimal control problem once, offline, for every initial state, and divides the initial-state space into critical regions, each with its own control law and sequence of active constraints. A boundary between two regions is either a hyperplane, given by one linear equation, or curved. The curved boundaries have not been computed to date; a classifier has been trained instead to distinguish the regions, so the partition has been known only approximately. In this paper we show that a curved boundary can be computed exactly, without solving any optimal control problem. The method rests on one property. Once the sequence of active constraints is fixed, together with the instants inside it at which a constraint becomes active or inactive (the switching times) and the instant at which the change of the neighboring region first appears (the event time), every condition on a boundary is a linear equation in the initial state, so the computation of a boundary, curved or not, is exact. We fix one of these instants, a switching time or the event time, and solve the linear equations that correspond to it for the corresponding initial state on the boundary; stepping the fixed instant along the horizon yields the whole boundary. A hyperplane is the case in which no instant is left to fix. We illustrate the method on the Newell-Lee evaporator, which has seven critical regions and eight boundaries, four hyperplanes and four curved, computing one boundary of each type in full with the adjacent regions at its ends.
ER -