On Linear Critical-Region Boundaries in Continuous-Time Multiparametric Optimal Control

TL;DR

This paper characterizes when boundaries between critical regions in continuous-time multiparametric optimal control are hyperplanes, showing they occur only when extremal conditions are at fixed endpoints, and provides an analytical partition for a third-order system.

Contribution

It proves a necessary and sufficient condition for hyperplane boundaries in continuous-time control, and analytically derives the complete critical-region partition for a specific third-order system.

Findings

Boundaries are hyperplanes if extremum is at initial or terminal time.

When extremum shifts with initial condition, boundaries are generally curved.

Continuous-time partition remains unchanged under discretization, unlike the discrete case.

Abstract

When an optimal control problem is solved for all possible initial conditions at once, the initial-state space splits into critical regions, each carrying a closed-form control law that can be evaluated online without solving any optimization. This is the multiparametric approach to explicit control. In the continuous-time setting, the boundaries between these regions are determined by extrema of Lagrange multipliers and constraint functions along the optimal trajectory. Whether a boundary is a hyperplane, computable analytically, or a curved manifold that requires numerical methods has a direct effect on how the partition is built. We show that a boundary is a hyperplane if and only if the relevant extremum is attained at either the initial time or the terminal time, regardless of the initial condition. The reason is that the costate is a linear function of the initial state at any fixed time, so when the extremum is tied to a fixed endpoint, the boundary condition is linear and the boundary normal follows directly from two matrix exponentials and a linear solve. When the extremum occurs at a time that shifts with the initial condition, such as a switching time or an interior stationary point, the boundary is generally curved. We demonstrate the result on a third-order system, obtaining the complete three-dimensional critical-region partition analytically for the first time in this problem class. A comparison with a discrete-time formulation shows how sharply the region count grows under discretization, while the continuous-time partition remains unchanged.