Differential Inclusions Research Articles

Approximate controllability results for second-order non-local neutral impulsive differential inclusions with damping

This paper is mainly concerned with the approximate controllability of damped second-order neutral impulsive differential inclusions with non-local conditions in Banach spaces. We study our primary outcomes by using the theoretical concepts about cosine and sine functions of operators, Bohnenblust and Karlin's fixed point theorem. Using principles and ideas from the theory of the cosine family of operators and the fixed-point approach, we verify the existence of mild solutions for the given system. A new set of sufficient conditions is formulated and proved for the approximate controllability of second-order differential inclusions under the assumption that the associated linear part of the system is approximately controllable. Finally, an application is presented to illustrate our theoretical results.

A Recurrent Neural Network Approach for Constrained Distributed Fuzzy Convex Optimization.

This article investigates a class of constrained distributed fuzzy convex optimization problems, where the objective function is the sum of a set of local fuzzy convex objective functions, and the constraints include partial order relation and closed convex set constraints. In undirected connected node communication network, each node only knows its own objective function and constraints, and the local objective function and partial order relation functions may be nonsmooth. To solve this problem, a recurrent neural network approach based on differential inclusion framework is proposed. The network model is constructed with the help of the idea of penalty function, and the estimation of penalty parameters in advance is eliminated. Through theoretical analysis, it is proven that the state solution of the network enters the feasible region in finite time and does not escape again, and finally reaches consensus at an optimal solution of the distributed fuzzy optimization problem. Furthermore, the stability and global convergence of the network do not depend on the selection of the initial state. A numerical example and an intelligent ship output power optimization problem are given to illustrate the feasibility and effectiveness of the proposed approach.

Existence of solutions for second-order differential inclusions associated with highly non-semicontinuous multifunctions

Existence of solutions for second-order differential inclusions associated with highly non-semicontinuous multifunctions

Antiperiodic Solutions for Impulsive ω-Weighted ϱ–Hilfer Fractional Differential Inclusions in Banach Spaces

In this article, we construct sufficient conditions that secure the non-emptiness and compactness of the set of antiperiodic solutions of an impulsive fractional differential inclusion involving an ω-weighted ϱ–Hilfer fractional derivative, D0,tσ,v,ϱ,ω, of order σ∈(1,2), in infinite-dimensional Banach spaces. First, we deduce the formula of antiperiodic solutions for the observed problem. Then, we give two theorems regarding the existence of these solutions. In the first, by using a fixed-point theorem for condensing multivalued functions, we show the non-emptiness and compactness of the set of antiperiodic solutions; and in the second, by applying a fixed-point theorem for contraction multivalued functions, we prove the non-emptiness of this set. Because many types of famous fractional differential operators are particular cases from the operator D0,tσ,v,ϱ,ω, our results generalize several recent results. Moreover, there are no previous studies on antiperiodic solutions for this type of fractional differential inclusion, so this work is novel and interesting. We provide two examples to illustrate and support our conclusions.

Access to Labour and “Differential Inclusion” for Young Venezuelan Migrants in Ecuador

Young migrants and refugees provide important inputs concerning production and social reproduction mechanisms (care work) that reproduce unequal but highly profitable patterns of accumulation around the world. The expansion of globalisation and neoliberalism has deepened such social dynamics, leading to a “multiplication of labour.” The diversification and heterogenization of migrant labour within neoliberal frameworks raise ethical and human rights concerns, including issues related to fair wages, working conditions, and access to social protections. After a neoliberal era in Latin America, the emergence of post‐development politics in the region led to increased efforts to address the needs of these populations. This article seeks to contribute to debates about “differential inclusion” in South–South migration and access to labour and social protection by analysing a specific case study of young Venezuelans, a recently growing phenomenon that has a great impact on the region. Ecuador is noteworthy because it hosts one of the largest populations of Venezuelan migrants and refugees and has adopted a human rights perspective in conjunction with public investments in social policy, health, and education. Despite efforts to legalize the work status of Venezuelan migrants, a more restrictive policy began to be implemented in 2019 that limited their access to formal labour and social protection. Within these complex dynamics of differential inclusion, instead of seeing these young migrants and refugees as victims, we analyse their resilience strategies of accessing social provisions while coping with informality and irregular status, as well as conditions of multiplication of labour. Using four real‐life stories as examples taken from a larger ethnographic study, we illustrate how dynamics of differential inclusion intersect with the gender, age, and legal status of young Venezuelans in Ecuador. The case studies are complemented with structural explanations from 6,000 household surveys collected between 2017 and 2020.

Evolution inclusions with non-fixed times of impulses

In this paper, we study evolution inclusions on Gelfand triple with not fixed times of impulses. We assume that the right-hand side satisfies some mild conditions to prove existence of solutions. Then we prove a continuous dependence of the solution set on the initial conditions, impulsive surfaces and the right-hand side under one-sided Lipschitz condition. Illustrative examples of partial differential inclusions are provided.

Novel Caputo fractional differential approach and hybrid control scheme for finite‐time synchronization of nonidentical delayed reaction–diffusion neural networks

AbstractThis work focuses on the finite‐time synchronization (F‐TS) for nonidentical delayed neural networks (DNNs) including Caputo fractional partial differential operator and reaction–diffusion terms. A novel F‐TS lemma is proposed by constructing a new Caputo differential inequality. Applying the new lemma and designing two hybrid controllers with time delay and the sign function, the F‐TS criteria on the introduced Caputo reaction–diffusion nonidentical DNNs under the Neumann boundary condition are derived by making use of the Filippov differential inclusion, Green's theorem, and fractional Razumikhin theorem. The conditions are expressed through the algebraic inequality which can greatly decrease the computation in checking the F‐TS performance. Moreover, the validity and correctness of the F‐TS results are illustrated by selecting various orders, spatial positions, and diffusion parameters.

On first and second-order perturbed differential inclusions governed by maximal monotone operators

In this paper we establish, in a separable Hilbert space, a result asserting the existence of absolutely continuous solutions for a system made up of a first-order differential inclusion governed by time and state-dependent maximal monotone operators; and an ordinary differential equation. From this result, we derive existence of absolutely continuous solutions to a second-order differential inclusion governed by time and state-dependent maximal monotone operators.

Non-degenerate necessary conditions for non-convex differential inclusions

This paper concerns dynamic optimization problems with pathwise state constraints, in which the dynamics are formulated as a differential inclusion. First order necessary conditions of optimality are available for these problems. Foremost among these conditions is the Euler Lagrange inclusion (combined with a transversality and Weierstrass conditions), which improves on earlier conditions such as Clarke's Hamiltonian inclusion. The most recent version of the Euler Lagrange inclusion (which we call the standard Euler Lagrange condition) has been validated under unrestrictive conditions, which allow the velocity sets of the differential inclusion to be non-convex. If the initial state is fixed and located in the boundary of the state constraint set then the Euler Lagrange inclusion conveys no information about minimizers. This is the degeneracy phenomenon for state constrained dynamic optimization. The non-degenerate necessary conditions literature provides additional conditions, to supplement the standard necessary conditions. These typically identify some properties of the boundary values of the maximized Hamiltonian which ensure non-triviality of the necessary conditions, under a simple controllability hypothesis. Non-degenerate conditions, incorporating the Euler Lagrange inclusion are currently available only under the hypothesis that the velocity sets are convex. Using new perturbation techniques, we derive a non-degenerate Euler Lagrange condition that is valid for non-convex velocity sets. These conditions bring into line non-degenerate optimality conditions for differential inclusion problems with those for problems involving controlled differential equations which have previously been validated for non-convex velocity sets.

Second order hemivariational inequality driven by evolution differential inclusion to a dynamic thermoviscoelastic contact problem

In this paper we study an abstract system which consists of a hyperbolic hemivariational inequality coupled with a differential evolution inclusion involving a history-dependent operator in Banach spaces. A hybrid iterative system corresponding to the hemivariational inequality is introduced. Combining the Rothe method, a feedback iterative technique, and a surjectivity result for pseudomonotone operators, we establish existence and a priori estimate for solutions to an approximate problem. Next, through a limiting procedure for solutions of the hybrid iterative system, we show the existence of a mild solution to the original problem. Finally, we apply the main results to a dynamic contact problem in thermoviscoelasticity.

Properties of set-valued Young integrals and Young differential inclusions generated by sets of Hölder functions

The present studies concern properties of set-valued Young integrals generated by families of β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta $$\\end{document}-Hölder functions and differential inclusions governed by such a type of integrals. These integrals differ from classical set-valued integrals of set-valued functions constructed in an Aumann’s sense. Integrals and inclusions considered in the manuscript contain as a particular case set-valued integrals and inclusions driven by a fractional Brownian motion. Our study is focused on topological properties of solutions to Young differential inclusions. In particular, we show that the set of all solutions is compact in the space of continuous functions. We also study its dependence on initial conditions as well as properties of reachable sets of solutions. The results obtained in the paper are finally applied to some optimality problems driven by Young differential inclusions. The properties of optimal solutions and their reachable sets are discussed.

Existence of solution for a second-order differential inclusion with a deviated nonlocal condition

In this paper, we study the existence of solution for the second-order differential inclusion With the local deviated nonlocal condition. Where and φ is deviated given function. As an application the problem with local and integral condition Will be considered.

Viability for impulsive stochastic differential inclusions driven by fractional Brownian motion

In this paper, we prove necessary and sufficient conditions for the viability for an impulsive stochastic functional differential inclusion driven by a fractional Brownian motion. The viable property is of interest since it reflects the stability of the model under consideration. The fractional Brownian motion provides a memory effect to the model. Whereas, the appearance of the impulsive factor introduces jumps to the solutions and is new to the analysis of this type. Hence our results are new even in the special case of stochastic differential equation setting.

A closed-measure approach to stochastic approximation

This paper introduces a new method to tackle the issue of the almost sure convergence of stochastic approximation algorithms defined from a differential inclusion. Under the assumption of slowly decaying step-sizes, we establish that the set of essential accumulation points of the iterates belongs to the Birkhoff centre associated with the differential inclusion. Unlike previous works, our results do not rely on the notion of asymptotic pseudotrajectories, predominant technique to address the convergence problem. They follow as a consequence of Young's superposition principle for closed measures. This perspective bridges the gap between Young's principle and the notion of invariant measure of set-valued dynamical systems introduced by Faure and Roth. Also, the proposed method allows for obtaining sufficient conditions under which the velocities locally compensate around any essential accumulation point.

On a Non-Convex Lagrange Optimal Control Problem

Abstract In this paper, we are concerned with an iterative differential inclusion governed by the time-dependent maximal monotone operator with perturbation. The approach to solve our problem is based on the Yosida approximation technique. The theoretical result is applied to prove an existence result for a Lagrange optimal control problem without assumptions concerning convexity.

Set-valued rigid-body dynamics for simultaneous, inelastic, frictional impacts

Robotic manipulation and locomotion often entail nearly-simultaneous collisions—such as heel and toe strikes during a foot step—with outcomes that are extremely sensitive to the order in which impacts occur. Robotic simulators and state estimation commonly lack the fidelity and accuracy to predict this ordering, and instead pick one with a heuristic. This discrepancy degrades performance when model-based controllers and policies learned in simulation are placed on a real robot. We reconcile this issue with a set-valued rigid-body model which generates a broad set of outcomes to simultaneous frictional impacts with any impact ordering. We first extend Routh’s impact model to multiple impacts by reformulating it as a differential inclusion (DI), and show that any solution will resolve all impacts in finite time. By considering time as a state, we embed this model into another DI which captures the continuous-time evolution of rigid-body dynamics, and guarantee existence of solutions. We finally cast simulation of simultaneous impacts as a linear complementarity problem (LCP), and develop an algorithm for tight approximation of the post-impact velocity set with probabilistic guarantees. We demonstrate our approach on several examples drawn from manipulation and legged locomotion, and compare the predictions to other models of rigid and compliant collisions.

Fixed-Point Approaches for Multi-Valued Contraction Mappings in a Novel Space With an Application

The vital goals of this manuscript are to combine metric-like spaces with S-metric spaces under a control function to obtain a new space called the controlled S-metric-like spaces (CSMLSs, for short). Under this name, many fixed-point (FP) results have been obtained for multi-valued mappings (MVMs). In addition, we present several non-trivial examples to back up our statements. The results obtained generalize and unify many results in the same direction. Finally, to support and test the adequacy of the theoretical results, the existence of the solution to the differential inclusion problem (DIP) was studied as a type of application.

On the solutions of a Sturm-Liouville type system of differential inclusions with nonlocal integral boundary conditions

We consider a Sturm-Liouville type system of differential inclusions with nonlocal integral boundary conditions and we obtain an existence result for this problem in the case when the set-valued maps have nonconvex values.

Mild Solutions for w-Weighted, Φ-Hilfer, Non-Instantaneous, Impulsive, w-Weighted, Fractional, Semilinear Differential Inclusions of Order μ ∈ (1, 2) in Banach Spaces

The aim of this work is to obtain novel and interesting results for mild solutions to a semilinear differential inclusion involving a w-weighted, Φ-Hilfer, fractional derivative of order μ∈(1,2) with non-instantaneous impulses in Banach spaces with infinite dimensions when the linear term is the infinitesimal generator of a strongly continuous cosine family and the nonlinear term is a multi-valued function. First, we determine the formula of the mild solution function for the considered semilinear differential inclusion. Then, we give sufficient conditions to ensure that the mild solution set is not empty or compact. The desired results are achieved by using the properties of both the w-weighted Φ-Laplace transform, w-weighted ψ-convolution and the measure of non-compactness. Since the operator, the w-weighted Φ-Hilfer, includes well-known types of fractional differential operators, our results generalize several recent results in the literature. Moreover, our results are novel because no one has previously studied these types of semilinear differential inclusions. Finally, we give an illustrative example that supports our theoretical results.

Errata of solution of differential inclusion problem in controlled S-metric spaces via new multivalued fixed point theorem

Errata of solution of differential inclusion problem in controlled S-metric spaces via new multivalued fixed point theorem