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Multiparametric continuous-time optimal control via Pontryagin's Maximum Principle: explicit solutions and comparisons with discrete-time formulations

Lida Lamakani, Efstratios N. Pistikopoulos

Abstract

Model predictive control offers a powerful framework for managing constrained systems, but its repeated online optimization can become computationally prohibitive. Multiparametric programming addresses this challenge by precomputing optimal solutions offline, enabling real-time control through simple function evaluation. While extensively developed for discrete-time systems, this approach suffers from combinatorial growth in solution complexity as discretization is refined. This paper presents a systematic continuous-time multiparametric framework for linear-quadratic optimal control that directly solves Pontryagin's optimality conditions without discretization artifacts. Through two illustrative examples, we demonstrate that continuous-time formulations yield solutions with substantially fewer critical regions than their discrete-time counterparts. Beyond this reduction in partition complexity, the continuous-time approach provides deeper insight into system dynamics by explicitly identifying switching times and eliminating discretization artifacts that obscure the true structure of optimal control policies. Knowledge of the switching structure also accelerates online optimization methods by providing analytical information about the solution topology. Clear step-by-step algorithms are provided for identifying switching structures, computing parametric switching times, and constructing critical regions, making the continuous-time framework accessible for practical implementation.

Multiparametric continuous-time optimal control via Pontryagin's Maximum Principle: explicit solutions and comparisons with discrete-time formulations

Abstract

Model predictive control offers a powerful framework for managing constrained systems, but its repeated online optimization can become computationally prohibitive. Multiparametric programming addresses this challenge by precomputing optimal solutions offline, enabling real-time control through simple function evaluation. While extensively developed for discrete-time systems, this approach suffers from combinatorial growth in solution complexity as discretization is refined. This paper presents a systematic continuous-time multiparametric framework for linear-quadratic optimal control that directly solves Pontryagin's optimality conditions without discretization artifacts. Through two illustrative examples, we demonstrate that continuous-time formulations yield solutions with substantially fewer critical regions than their discrete-time counterparts. Beyond this reduction in partition complexity, the continuous-time approach provides deeper insight into system dynamics by explicitly identifying switching times and eliminating discretization artifacts that obscure the true structure of optimal control policies. Knowledge of the switching structure also accelerates online optimization methods by providing analytical information about the solution topology. Clear step-by-step algorithms are provided for identifying switching structures, computing parametric switching times, and constructing critical regions, making the continuous-time framework accessible for practical implementation.
Paper Structure (22 sections, 2 theorems, 79 equations, 6 figures, 11 tables, 2 algorithms)

This paper contains 22 sections, 2 theorems, 79 equations, 6 figures, 11 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Continuous-Time Optimal Control Structure
  3. Continuous-Time Optimal Control Problem
  4. Assumptions
  5. Pontryagin's Minimum Principle Conditions
  6. Switching Points and Jump Conditions
  7. Multiparametric Continuous-Time Optimal Control
  8. Parametric Formulation
  9. Parametric Switching Structures
  10. Critical Regions and Explicit Solutions
  11. Illustrative Examples
  12. Example 1: One-State Integrating System with Mixed Output and Input Constraints
  13. Problem Statement
  14. Continuous-Time Multiparametric Solution
  15. Discrete-Time Counterpart
  16. ...and 7 more sections

Key Result

Proposition 2.1

Consider a continuous-time optimal control problem with path constraints of order zero (depending only on $x$, $u$, and $t$). Assume continuous dynamics without impulsive terms and no switching costs. Then, for a prescribed arc sequence, the switching times can be determined by enforcing the jump co together with the constraint activity conditions. Under these assumptions, Hamiltonian continuity f

Figures (6)

  • Figure 1: Schematic illustration of switching points and constrained arcs. The times $t_{st}$ and $t_{sx}$ denote the entry and exit points of a constrained arc, respectively. Changes in the active constraint set are accompanied by corresponding changes in the associated Lagrange multipliers.
  • Figure 2: Interaction between Alg. \ref{['alg:mpdo']} (global exploration) and Alg. \ref{['alg:param_switch']} (local switching-time fitting).
  • Figure 3: Continuous-time critical-region map for Example 2.
  • Figure 4: Discrete-time critical-region maps for Example 2: (a) $N=5$ (11 critical regions); (b) $N=10$ (21 critical regions).
  • Figure 5: Optimal control trajectories for Example 2 at $x_0 = (-0.95, -1.65)$: continuous-time (blue) versus discrete-time with $N=5$ (orange) and $N=20$ (green).
  • ...and 1 more figures

Theorems & Definitions (6)

  • Proposition 2.1: State-matching characterization of switching points
  • proof
  • Remark 2.1
  • Lemma 3.1: Existence, uniqueness, and continuity of switching time
  • proof
  • Remark 3.1