# Lipschitzian Regularity of Minimizers for Optimal Control Problems with Control-Affine Dynamics

Lipschitzian Regularity of Minimizers for Optimal Control Problems with Control-Affine Dynamics We study the Lagrange Problem of Optimal Control with a functional $\int_{a}^{b}L\left(t,\,x\left(t\right),\,u\left(t\right)\right)\,dt$ and control-affine dynamics $\dot{x}$ = f(t,x) + g(t,x)u and (a priori) unconstrained control u∈ \bf R m . We obtain conditions under which the minimizing controls of the problem are bounded—a fact which is crucial for the applicability of many necessary optimality conditions, like, for example, the Pontryagin Maximum Principle. As a corollary we obtain conditions for the Lipschitzian regularity of minimizers of the Basic Problem of the Calculus of Variations and of the Problem of the Calculus of Variations with higher-order derivatives. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics and Optimization Springer Journals

# Lipschitzian Regularity of Minimizers for Optimal Control Problems with Control-Affine Dynamics

, Volume 41 (2) – Apr 1, 2025
18 pages

/lp/springer_journal/lipschitzian-regularity-of-minimizers-for-optimal-control-problems-ImRJ5t4b3Y
Publisher
Springer-Verlag
Copyright © Inc. by 2000 Springer-Verlag New York
Subject
Mathematics; Calculus of Variations and Optimal Control; Optimization; Systems Theory, Control; Theoretical, Mathematical and Computational Physics; Mathematical Methods in Physics; Numerical and Computational Physics, Simulation
ISSN
0095-4616
eISSN
1432-0606
D.O.I.
10.1007/s002459911013
Publisher site
See Article on Publisher Site

### Abstract

We study the Lagrange Problem of Optimal Control with a functional $\int_{a}^{b}L\left(t,\,x\left(t\right),\,u\left(t\right)\right)\,dt$ and control-affine dynamics $\dot{x}$ = f(t,x) + g(t,x)u and (a priori) unconstrained control u∈ \bf R m . We obtain conditions under which the minimizing controls of the problem are bounded—a fact which is crucial for the applicability of many necessary optimality conditions, like, for example, the Pontryagin Maximum Principle. As a corollary we obtain conditions for the Lipschitzian regularity of minimizers of the Basic Problem of the Calculus of Variations and of the Problem of the Calculus of Variations with higher-order derivatives.

### Journal

Applied Mathematics and OptimizationSpringer Journals

Published: Apr 1, 2025

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