Conditions for Normality of Pontryagin's Maximum Principle in the Absence of State Constraints
Abstract
Pontryagin's Maximum Principle is said to be "normal" for the problem at hand when it has a solution (maximizer u*) that explicitly involves the problem's cost function, making the maximum principle helpful (while in the opposite situation it is of no help). The aim of the present paper is, for Lagrange optimal control problems without state constraints, to identify classes of problems for which the Principle is normal. General conditions are introduced that guarantee normality, and classes of problems that satisfy these conditions are identified. Such classes include problems for which the constrained (by bounds on the control values) "augmented" reachable set, with cost along an upward vertical axis, has a convex lower-boundary function and the optimal terminal state lies in the relative interior of the reachable state set, with the terminal constraint within the affine hull of that set. Such classes include problems for which the constrained (by bounds on the values of the control signal) "augmented" (with the cost value along an upward "vertical axis") reachable set has a "convex" lower boundary for which the optimal terminal state lies in the relative interior of the set of reachable states. Special attention is given to Lagrange steering problems: sufficient conditions are obtained on a nested sequence of sub-classes of such problems, culminating with the class of steering problems with scalar control signal and linear time-invariant (LTI) state dynamics and objective integrand, for which normality is guaranteed whenever all eigenvalues of the state dynamics matrix A are real (e.g., if A is symmetric or triangular or the system is overdamped) or have small enough imaginary parts; simple 2 x 2 examples show that, in contrast with the situation in the finite-dimensional case, this property does not extend to arbitrary A matrices.
Related
- On the Pontryagin Maximum Principle for Rough Stochastic Differential Equations
- Pontryagin maximum principle for non-smooth state-constrained control problems over Wasserstein spaces
- Sub-Gaussian Concentration and Entropic Normality of the Maximum Likelihood Estimator
- Strong Consistency and Asymptotic Normality of the Maximum Likelihood Estimator in r-Largest Generalized Extreme-Value Models
- Maximum principle for quasilinear SPDE’s on a bounded domain without regularity assumptions
BibTeX
@misc{tits2026conditions,
title = {{Conditions for Normality of Pontryagin's Maximum Principle in the Absence of State Constraints}},
author = {André Tits and Pierre-Antoine Absil and Noé Tits},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.05743},
eprint = {2611.05743},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.05743},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Conditions for Normality of Pontryagin's Maximum Principle in the Absence of State Constraints
AU - André Tits
AU - Pierre-Antoine Absil
AU - Noé Tits
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.05743
UR - https://arxiv.org/abs/2611.05743
AB - Pontryagin's Maximum Principle is said to be "normal" for the problem at hand when it has a solution (maximizer u*) that explicitly involves the problem's cost function, making the maximum principle helpful (while in the opposite situation it is of no help). The aim of the present paper is, for Lagrange optimal control problems without state constraints, to identify classes of problems for which the Principle is normal. General conditions are introduced that guarantee normality, and classes of problems that satisfy these conditions are identified. Such classes include problems for which the constrained (by bounds on the control values) "augmented" reachable set, with cost along an upward vertical axis, has a convex lower-boundary function and the optimal terminal state lies in the relative interior of the reachable state set, with the terminal constraint within the affine hull of that set. Such classes include problems for which the constrained (by bounds on the values of the control signal) "augmented" (with the cost value along an upward "vertical axis") reachable set has a "convex" lower boundary for which the optimal terminal state lies in the relative interior of the set of reachable states. Special attention is given to Lagrange steering problems: sufficient conditions are obtained on a nested sequence of sub-classes of such problems, culminating with the class of steering problems with scalar control signal and linear time-invariant (LTI) state dynamics and objective integrand, for which normality is guaranteed whenever all eigenvalues of the state dynamics matrix A are real (e.g., if A is symmetric or triangular or the system is overdamped) or have small enough imaginary parts; simple 2 x 2 examples show that, in contrast with the situation in the finite-dimensional case, this property does not extend to arbitrary A matrices.
ER -