Maximum Principle and Second-Order Optimality Conditions in Control Problems with Mixed Constraints

Aram Arutyunov,Fernando Pereira,Dmitry Karamzin

doi:10.3390/axioms11020040

Aram Arutyunov, Fernando Pereira + Show 1 more

Open Access

https://doi.org/10.3390/axioms11020040

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Journal: Axioms	Publication Date: Jan 20, 2022
Citations: 1	License type: CC BY 4.0

Affiliation: University of Porto, Russian Academy of Sciences

Abstract

This article concerns the optimality conditions for a smooth optimal control problem with an endpoint and mixed constraints. Under the normality assumption, which corresponds to the full-rank condition of the associated controllability matrix, a simple proof of the second-order necessary optimality conditions based on the Robinson stability theorem is derived. The main novelty of this approach compared to the known results in this area is that only a local regularity with respect to the mixed constraints, that is, a regularity in an ε-tube about the minimizer, is required instead of the conventional stronger global regularity hypothesis. This affects the maximum condition. Therefore, the normal set of Lagrange multipliers in question satisfies the maximum principle, albeit along with the modified maximum condition, in which the maximum is taken over a reduced feasible set. In the second part of this work, we address the case of abnormal minimizers, that is, when the full rank of controllability matrix condition is not valid. The same type of reduced maximum condition is obtained.

Highlights

In this article, second-order necessary optimality conditions for an optimal control problem with mixed equality and inequality constraints are investigated
Under the normality condition, which is ensured by the full rank of the controllability matrix, a rather simple proof of the optimality conditions is proposed based on Robinson’s theorem on the metric regularity for set-valued mappings
This means that some reduced cone of Lagrange multipliers is invoked, which is defined by using the index of the quadratic form of the Lagrange function; see, e.g., [1,2,3]

Summary

Introduction

Second-order necessary optimality conditions for an optimal control problem with mixed equality and inequality constraints are investigated. The control process (x, u) is said to satisfy the maximum principle provided that there exists a vector λ = (λ0, λ1, λ2) ∈ (R1+k1+k2 )∗, where λ0 ≥ 0 and λ1 ≥ 0, an absolutely continuous vector-valued function ψ ∈ W1,∞([0, 1]; (Rn)∗), and a measurable, essentially bounded, vector-valued function ν ∈ L∞([0, 1]; (Rq)∗) of which the j-th component is nonnegative for j = 1, . Under the regularity condition given in Definition 3, the multipliers ψ, and ν are uniquely defined by the vector λ, where (λ, ψ, ν) is the set of Lagrange multipliers corresponding to (x, u) in view of the maximum principle. This assertion follows from the Euler–Lagrange equation.

Normality Condition

Abnormal Case

Conclusions