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23 pages, 359 KB

Open AccessArticle

Pontryagin’s Maximum Principle for Optimal Control Problems Governed by Integral Equations with State and Control Constraints

by Hugo Leiva and Marcial Valero

Symmetry 2025, 17(12), 2088; https://doi.org/10.3390/sym17122088 - 5 Dec 2025

Cited by 1 | Viewed by 1343

Abstract

This paper proves a new lemma that characterizes controllability for linear Volterra control systems and shows that the usual controllability assumption for the variational linearized system near an optimal pair is superfluous. Building on this, it establishes a Pontryagin-type maximum principle for Volterra optimal control with general control and state constraints (fixed terminal constraints and time-dependent state bounds), where the cost combines a terminal term with a state-dependent and integral term. Using the Dubovitskii–Milyutin framework, we construct conic approximations for the cost, dynamics, and constraints and derive necessary optimality conditions under mild regularity: (i) a classical adjoint system when only terminal constraints are present and (ii) a Stieltjes-type adjoint with a non-negative Borel measure when pathwise state constraints are active. Furthermore, under convexity of the cost functional and linear Volterra dynamics, the maximum principle becomes a sufficient criterion for global optimality (recovering the classical sufficiency in the differential case). The differential case recovers the classical PMP, and an SIR example illustrates the results. A key theme is symmetry/duality: the adjoint differentiates in the state while the maximum condition differentiates in the control, reflecting operator transposition and the primal–dual geometry of Dubovitskii–Milyutin cones. Full article

(This article belongs to the Special Issue Nonlinear Differential and Integral Equations and Their Infinite Systems)

11 pages, 271 KB

Open AccessArticle

Legendre–Clebsch Condition for Functional Involving Fractional Derivatives with a General Analytic Kernel

by Faïçal Ndaïrou

Fractal Fract. 2025, 9(9), 588; https://doi.org/10.3390/fractalfract9090588 - 8 Sep 2025

Viewed by 974

Abstract

Fractional calculus of variations for a broad class of fractional operators with a general analytic kernel function is considered. Using techniques from variational analysis, we derive first- and second-order necessary optimality conditions, namely the Euler–Lagrange equation, the Weierstrass necessary condition, the Legendre condition, and finally the Legendre–Clebsch condition. Our results are new in the sense that the Euler–Lagrange equation is based on duality theory, and thus build up only with left fractional operators. The Weierstrass necessary condition is a variant of strong necessary optimality condition, and it is derived from maximum condition of Pontryagin for this general analytic kernels. The Legendre–Clebsch condition is obtained under normality assumptions on data because of equality constraints. Full article

(This article belongs to the Special Issue Fractional Differential Operators with Classical and New Memory Kernels, Second Edition)

31 pages, 506 KB

Open AccessArticle

A Distinguished Subgroup of Compact Abelian Groups

by Dikran Dikranjan, Wayne Lewis, Peter Loth and Adolf Mader

Axioms 2022, 11(5), 200; https://doi.org/10.3390/axioms11050200 - 24 Apr 2022

Cited by 5 | Viewed by 3300

Abstract

Here “group” means additive abelian group. A compact group G contains

δ

–subgroups, that is, compact totally disconnected subgroups

Δ

such that

G / Δ

is a torus. The canonical subgroup

Δ (G)

of G that is the sum of all [...] Read more.

Here “group” means additive abelian group. A compact group G contains

δ

–subgroups, that is, compact totally disconnected subgroups

Δ

such that

G / Δ

is a torus. The canonical subgroup

Δ (G)

of G that is the sum of all

δ

–subgroups of G turns out to have striking properties. Lewis, Loth and Mader obtained a comprehensive description of

Δ (G)

when considering only finite dimensional connected groups, but even for these, new and improved results are obtained here. For a compact group G, we prove the following:

Δ (G)

contains

tor (G)

, is a dense, zero-dimensional subgroup of G containing every closed totally disconnected subgroup of G, and

G / Δ (G)

is torsion-free and divisible;

Δ (G)

is a functorial subgroup of G, it determines G up to topological isomorphism, and it leads to a “canonical” resolution theorem for G. The subgroup

Δ (G)

appeared before in the literature as

td (G)

motivated by completely different considerations. We survey and extend earlier results. It is shown that td, as a functor, preserves proper exactness of short sequences of compact groups. Full article

(This article belongs to the Special Issue Advance in Topology and Functional Analysis——In Honour of María Jesús Chasco's 65th Birthday)

13 pages, 286 KB

Open AccessReview

Advances in the Theory of Compact Groups and Pro-Lie Groups in the Last Quarter Century

by Karl H. Hofmann and Sidney A. Morris

Axioms 2021, 10(3), 190; https://doi.org/10.3390/axioms10030190 - 17 Aug 2021

Cited by 4 | Viewed by 3803

Abstract

This article surveys the development of the theory of compact groups and pro-Lie groups, contextualizing the major achievements over 125 years and focusing on some progress in the last quarter century. It begins with developments in the 18th and 19th centuries. Next is from Hilbert’s Fifth Problem in 1900 to its solution in 1952 by Montgomery, Zippin, and Gleason and Yamabe’s important structure theorem on almost connected locally compact groups. This half century included profound contributions by Weyl and Peter, Haar, Pontryagin, van Kampen, Weil, and Iwasawa. The focus in the last quarter century has been structure theory, largely resulting from extending Lie Theory to compact groups and then to pro-Lie groups, which are projective limits of finite-dimensional Lie groups. The category of pro-Lie groups is the smallest complete category containing Lie groups and includes all compact groups, locally compact abelian groups, and connected locally compact groups. Amongst the structure theorems is that each almost connected pro-Lie group G is homeomorphic to

R^{I} \times C

for a suitable set I and some compact subgroup C. Finally, there is a perfect generalization to compact groups G of the age-old natural duality of the group algebra

R [G]

of a finite group G to its representation algebra

R (G, R)

, via the natural duality of the topological vector space

R^{I}

to the vector space

R^{(I)}

, for any set I, thus opening a new approach to the Hochschild-Tannaka duality of compact groups. Full article

(This article belongs to the Special Issue Advance in Topology and Functional Analysis——In Honour of María Jesús Chasco's 65th Birthday)

4 pages, 208 KB

Open AccessArticle

The Tubby Torus as a Quotient Group

by Sidney A. Morris

Axioms 2020, 9(1), 11; https://doi.org/10.3390/axioms9010011 - 20 Jan 2020

Cited by 1 | Viewed by 3201

Abstract

Let E be any metrizable nuclear locally convex space and

\hat{E}

the Pontryagin dual group of E. Then the topological group

\hat{E}

has the tubby torus (that is, the countably infinite product of copies of the circle group) as a [...] Read more.

Let E be any metrizable nuclear locally convex space and

\hat{E}

the Pontryagin dual group of E. Then the topological group

\hat{E}

has the tubby torus (that is, the countably infinite product of copies of the circle group) as a quotient group if and only if E does not have the weak topology. This extends results in the literature related to the Banach–Mazur separable quotient problem. Full article

(This article belongs to the Collection Topological Groups)

7 pages, 220 KB

Open AccessEditorial

An Overview of Topological Groups: Yesterday, Today, Tomorrow

by Sidney A. Morris

Axioms 2016, 5(2), 11; https://doi.org/10.3390/axioms5020011 - 5 May 2016

Viewed by 6197

Abstract

It was in 1969 that I began my graduate studies on topological group theory and I often dived into one of the following five books. My favourite book “Abstract Harmonic Analysis” [1] by Ed Hewitt and Ken Ross contains both a proof of the Pontryagin-van Kampen Duality Theorem for locally compact abelian groups and the structure theory of locally compact abelian groups.[...] Full article

(This article belongs to the Special Issue Topological Groups: Yesterday, Today, Tomorrow)

13 pages, 159 KB

Open AccessArticle

The Duality between Corings and Ring Extensions

by Florin F. Nichita and Bartosz Zielinski

Axioms 2012, 1(2), 173-185; https://doi.org/10.3390/axioms1020173 - 10 Aug 2012

Cited by 3 | Viewed by 6052

Abstract

We study the duality between corings and ring extensions. We construct a new category with a self-dual functor acting on it, which extends that duality. This construction can be seen as the non-commutative case of another duality extension: the duality between finite dimensional [...] Read more.

(This article belongs to the Special Issue Hopf Algebras, Quantum Groups and Yang-Baxter Equations)

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