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On the Strong Subregularity of the Optimality Mapping in an Optimal Control Problem with Pointwise Inequality Control

N P Osmolovskii1, V M Veliov2

  • 1Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland.

Applied Mathematics and Optimization
|March 20, 2023
PubMed
Summary

This study establishes sufficient conditions for strong metric subregularity (SMsR) in optimal control problems. A second-order sufficient optimality condition ensures this SMsR property, addressing the two-norm-discrepancy challenge.

Keywords:
Control constraintMayer’s problemOptimal controlOptimizationmetric subregularity

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Area of Science:

  • Optimal Control Theory
  • Mathematical Optimization
  • Nonsmooth Analysis

Background:

  • The Pontryagin maximum principle is crucial for solving optimal control problems.
  • Strong metric subregularity (SMsR) is essential for analyzing the stability and sensitivity of solutions.
  • Pointwise control constraints, defined by inequalities, are common in practical optimal control applications.

Purpose of the Study:

  • To derive sufficient conditions for the strong metric subregularity (SMsR) of the optimality mapping.
  • To analyze Mayer-type optimal control problems with inequality constraints on controls.
  • To investigate the relationship between second-order sufficient optimality conditions and SMsR.

Main Methods:

  • Application of the local Pontryagin maximum principle.
  • Analysis of optimality mappings under pointwise control constraints.
  • Utilizing second-order sufficient optimality conditions.
  • Addressing the two-norm-discrepancy using two norms in the control space.

Main Results:

  • Sufficient conditions for SMsR of the optimality mapping are presented.
  • The second-order sufficient optimality condition for a weak local minimum is shown to be sufficient for a version of SMsR.
  • The analysis successfully handles the two-norm-discrepancy inherent in the problem formulation.

Conclusions:

  • The findings provide valuable theoretical insights into the regularity properties of optimal control solutions.
  • The established conditions enhance the understanding of solution stability for problems with control constraints.
  • This work contributes to the advancement of optimization theory and its applications in control systems.