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In this paper, we propose a generalized framework for developing learning-rate-free momentum stochastic gradient descent (SGD) methods in the minimization of nonsmooth nonconvex functions, especially in training nonsmooth neural networks. Our framework adaptively generates learning rates based on the historical data of stochastic subgradients and iterates. Under mild conditions, we prove that our proposed framework enjoys global convergence to the stationary points of the objective function in the sense of the conservative field, hence providing convergence guarantees for training nonsmooth neural networks. Based on our proposed framework, we propose a novel learning-rate-free momentum SGD method (LFM). Preliminary numerical experiments reveal that LFM performs comparably to the state-of-the-art learning-rate-free methods (which have not been shown theoretically to be convergent) across well-known neural network training benchmarks. [ABSTRACT FROM AUTHOR]

In this paper, the solutions of a class of switch dynamical systems are investigated. The right-hand side of the underlying equations is discontinuous with respect to the state variable. The discontinuity is represented by jump discontinuous functions such as signum or Heaviside functions. In this paper, a novel approach of the solutions of this class of discontinuous equations is presented. The initial value problem is restated as a differential inclusion via Filippov's regularization, after which, via the approximate selection results, the differential inclusion is transformed into a continuous, single-valued differential equation. Besides its existence, a sufficient uniqueness condition, the strengthened one-sided Lipschitz Condition, is also introduced. The important issue of the numerical integration of this class of equations is addressed, emphasizing by examples the errors that could appear if the discontinuity problem is neglected. The example of a mechanical system, a preloaded compliance system, is considered along with other examples. [ABSTRACT FROM AUTHOR]

In this paper, we consider the approximate controllability of semilinear differential inclusion featuring multivalued nonlinearities with nonconvex values. Initially, we explore the approximate controllability result by assuming the underlying state space to be a super-reflexive Banach space. Then, we relax the super-reflexivity condition and prove the approximate controllability in reflexive Banach spaces by employing the relaxation technique. Incorporating multivalued maps with nonconvex values adds complexity to the studied differential inclusions yet enhances their practical applicability, serving as the core motivation for this research endeavor. We conclude by presenting an illustrative example that satisfies the criteria outlined in this paper. [ABSTRACT FROM AUTHOR]

This corrigendum is concerned with the technical preliminary [1, Lemma 1]. Unfortunately, its proof contains a mistake which ultimately renders its conclusion erroneous. In this note, we provide a corrected version of the latter, and show that this modification has no impact on the other results of [1] while incurring very benign changes in none but two series of computations throughout the manuscript. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new coupled system of differential inclusions involving with Hadamard fractional orders. By applying a fixed point theorem for three operators containing $ x\in{AxBx+Cx} $ in Banach algebras, we get an existence result for the discussed system via multi-valued maps in a Banach space. An example is provided to support the validation of the theoretical result achieved.

The existence of two positive solutions for an elliptic differential inclusion is established, assuming that the nonlinear term is an upper semicontinuous set-valued mapping with compact convex values having subcritical growth. Our approach is based on variational methods for locally Lipschitz functionals. As a consequence, a multiplicity result for elliptic Dirichlet problems having discontinuous nonlinearities is pointed out.

In this study, we use the dynamic programming method introduced by Mirică (2004) to solve the well-known war game of attrition and attack as formulated by Isaacs (1965). By using this modern approach, we extend the classical framework to explore optimal strategies within the differential game setting, offering a complete, comprehensive and theoretically robust solution. Additionally, the study identifies and analyzes feedback strategies, which represent a significant advancement over other strategy types in game theory. These strategies dynamically adapt to the evolving state of the system, providing more robust solutions for real-time decision-making in conflict scenarios. This novel contribution enhances the application of game theory, particularly in the context of warfare models, and illustrates the practical advantages of incorporating feedback mechanisms into strategic decision-making. The admissible feedback strategies and the corresponding value function are constructed through a refined application of Cauchy's Method of characteristics for stratified Hamilton–Jacobi equations. Their optimality is proved using a suitable Elementary Verification Theorem for the associated value function as an argument for sufficient optimality conditions. [ABSTRACT FROM AUTHOR]

For differential inclusions and hemivariational inequalities driven by anisotropic differential operators, we establish the existence of generalized variational solutions and weak solutions. The main novelty consists in allowing that the driving operators might not satisfy any ellipticity condition, which is achieved for the first time in the anisotropic and nonsmooth context. The approach is based on a finite dimensional approximation process. [ABSTRACT FROM AUTHOR]

The existence of two positive solutions for an elliptic differential inclusion is established, assuming that the nonlinear term is an upper semicontinuous set-valued mapping with compact convex values having subcritical growth. Our approach is based on variational methods for locally Lipschitz functionals. As a consequence, a multiplicity result for elliptic Dirichlet problems having discontinuous nonlinearities is pointed out. [ABSTRACT FROM AUTHOR]

In this paper, we focus on investigating the existence of mild solutions and asymptotically almost periodic mild solutions for a class of partial differential inclusions. These inclusions involve a forcing multivalued function that relies on implicit spatial derivatives of the state variable. We introduce a novel approach to simplify the complexities associated with singularities when taking the α -norm. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the approximate controllability of control problems governed by evolution hemivariational inequalities in super-reflexive Banach spaces by preassuming the approximate controllability of the associated linear system. We first show that the original problem is connected with a differential inclusion problem involving the Clarke subdifferential operator, and then we prove the approximate controllability of the original problem via the approximate controllability of the differential inclusion problem. The paper offers a unique solution to a challenge introduced by assuming the underlying space X as a super-reflexive Banach space, which presents issues of convexity due to the nonlinear nature of the duality map involved in the expression of the control. Such issues are not present when X is a separable Hilbert space. Therefore, the paper's novelty is its successful resolution of the convexity problem, paving the way for approximate controllability of the evolution hemivariational inequality in which X is a super-reflexive Banach space. [ABSTRACT FROM AUTHOR]

Using Italy as a case study, this paper extends the debate on refugees’ differential inclusion in the European labour market by considering their spaces for living and social reproduction. Departing from the notion of ‘labour regime’ and the definition of a Dormitory Labour Regime, the paper argues that we are assisting in the emergence of a Camp Labour Regime – a situation in which the form of accommodation (i.e. refugee reception facilities) plays a pivotal role in the organisation of production, representing a nodal point to foster labour concentration, wage reduction and labour intensification, thereby ‘fashioning’ refugee labour as both cheap and flexible, although at the same time constituting spaces of opportunity for refugee workers. Therefore, the paper illustrates the role played by the three main actors – the state, the companies and the refugees themselves – in coproducing and partaking in this emerging Camp Labour Regime. [ABSTRACT FROM AUTHOR]

For feedback systems governed by fractional semilinear differential inclusions and a sweeping process in a Hilbert space, controllability conditions are found. For the proof, topological methods of nonlinear analysis for multivalued condensing maps are used. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new coupled system of differential inclusions involving with Hadamard fractional orders. By applying a fixed point theorem for three operators containing x ∈ A x B x + C x in Banach algebras, we get an existence result for the discussed system via multi-valued maps in a Banach space. An example is provided to support the validation of the theoretical result achieved. [ABSTRACT FROM AUTHOR]

The paper deals with the game problem of approach for a conflict-controlled system in a finite-dimensional Euclidean space at a fixed moment of time. The approximate calculation of the solvability sets for the considered approach game is studied. A method is proposed for approximate calculation of solvability sets on the basis of a unification model, which supplements Krasovskii's unification method in the theory of differential games. [ABSTRACT FROM AUTHOR]

Turkey, as the largest refugee-hosting country, provides insights into the externalization of European borders and the internalization of its own borders. Oscillating between open-door and strict-border policies, alongside the political inactiveness of public authorities and CSOs governing Syrian mobility, Turkey reveals crucial dynamics. Focusing on Syrian refugees' housing in a refugee hub of Izmir, this study unravels internal bordering and explores its rationales and the involvement of non-state actors. Syrian refugees encounter three internal bordering mechanisms affecting their housing inclusion: selective overpricing, ethnic filtering and arbitrary interrogations. These differential inclusion practices, led by market actors (landlords and realtors), ethnic networks, and street-level bureaucrats (mukhtars and municipal officers), are driven by the prevalence of anti-refugee sentiment in Turkey, where Syrians are perceived as unwanted or undeserving. The internal bordering mechanisms are rooted in ethnic, economic, cultural, and security-based rationales, influenced by the personal interests and socio-economic backgrounds of the involved actors. [ABSTRACT FROM AUTHOR]

We use some recent developments in dynamics programming method to obtain the complete and theoretically justified solution of a differential game formulated and studied in Chen and Qiu (Abstr Appl Anal 2014:1–8, 2014. https://doi.org/10.1155/2014/837431) using the Lanchester equation. We use a certain refinement of Cauchy's method of characteristics for stratified Hamilton–Jacobi equations to describe a large set of admissible trajectories and identify a domain on which the value function exists and is generated by a pair of admissible feedback strategies. Its optimality is justified by using one of the well-known verification theorems as an argument for sufficient optimality conditions. [ABSTRACT FROM AUTHOR]

We are concerned in this paper with a new class of coupled problems by differential inclusions involving time and state-dependent maximal monotone operators with perturbations. The operators vary (in the sense of the pseudo-distance) in an absolutely continuous way in time, and in a Lipschitz-continuous way in state. By a discretization technique, we state the well-posedness (existence and uniqueness of the solution) in a real separable Hilbert space. Applications to optimal control and to fractional theory are provided. [ABSTRACT FROM AUTHOR]

In this paper we consider a mathematical model of control system in the form differential inclusion. The problem of controllability of this system under the condition of mobility of the terminal set is researched. For this model of a dynamic system we define a notion of the M-controllability set. Using the methods of the theory of differential inclusions and multivalued maps, the structural and topological properties of the M-controllability set are studied. [ABSTRACT FROM AUTHOR]

By adding a scalar parameter to the classical Lyapunov matrix inequality, the underlying structure turns to be richer: Partial order of stability sets is introduced. This is then used to improve estimates of trajectories associated with differential inclusions. Technically, the spectrum of all matrices, satisfying a given Lyapunov inequality, lies within a special disk in the right-half plane: Under inversion such a disk is mapped onto itself. As a by-product, it is shown that these disks are a natural tool to understanding the Matrix-Sign-Function iteration scheme, used in matrix computations. Hyper-Lyapunov inclusions are formulated through Matrix-Quadratic-Form Inequalities and so are the analogous Hyper-Stein sets of matrices whose spectrum lies within sub -unit disks. [ABSTRACT FROM AUTHOR]

Concerns with 'vulnerability' increasingly proliferate in global and regional pacts, international and domestic legislations, and policy discourses and practices regarding migration and international protection. Also in France, vulnerability governance has made its inroads, and policy documents hail vulnerability considerations as a strengthening of the politics of reception and integration of asylum seekers and a means to improve accommodation and care. Drawing on ethnographic fieldwork in Marseille, I argue that vulnerability governance, situated within the context of securitization of migration and budget constraints in the reception system, leads to a 'differential inclusion', which is partial, conditional and precarious. By examining the understanding and operationalization of vulnerability within French migration legislation, policies, and governance practices, the study exposes how normative constructs of gender and sexuality inform the identification and hierarchization of vulnerable persons. Ethnographic evidence illustrates how these norms are perpetuated by governance actors, including civil society, and sometimes strategically mobilized or internalized by asylum seekers in their quest for recognition and assistance. In conclusion, the article highlights how protection-seeking migrants also contest the authorities' understanding and operationalization of vulnerability. Through protests and legal actions, they expose the state's role in producing and differentially distributing vulnerability through abandonment and destitution. [ABSTRACT FROM AUTHOR]

This paper concerns the existence of absolutely continuous solutions for a class of perturbed second-order differential inclusions governed by time and state dependent maximal monotone operators. The perturbation depends on all the variables and their values are nonconvex. [ABSTRACT FROM AUTHOR]

The current work focuses on the existence of absolutely continuous solutions to an evolution problem involving time and state-dependent maximal monotone operators. The wellposedness result is proved under an anti-monotone condition on the domain of the operators. Then, a minimization problem subject to this evolution problem is studied. [ABSTRACT FROM AUTHOR]

Signs of ageing become apparent only late in life, after organismal development is finalized. Ageing, most notably, decreases an individual’s fitness. As such, it is most commonly perceived as a non-adaptive force of evolution and considered a by-product of natural selection. Building upon the evolutionarily conserved age-related Smurf phenotype, we propose a simple mathematical life-history trait model in which an organism is characterized by two core abilities: reproduction and homeostasis. Through the simulation of this model, we observe (1) the convergence of fertility’s end with the onset of senescence, (2) the relative success of ageing populations, as compared to non-ageing populations, and (3) the enhanced evolvability (i.e. the generation of genetic variability) of ageing populations. In addition, we formally demonstrate the mathematical convergence observed in (1). We thus theorize that mechanisms that link the timing of fertility and ageing have been selected and fixed over evolutionary history, which, in turn, explains why ageing populations are more evolvable and therefore more successful. Broadly speaking, our work suggests that ageing is an adaptive force of evolution.

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. [ABSTRACT FROM AUTHOR]

Canada's national migration regimes, which are witnessing an increase in temporary foreign workers, underscore the contentious nature of inclusion for migrant farmworkers in smaller communities. While migrant farmworkers are entitled to benefits, their access to services and social support is limited. Even though settlement initiatives that welcome and support immigrant newcomers have shifted and intensified toward smaller towns and communities across Canada, support initiatives and the benefits for which temporary migrant farmworkers qualify have received far less attention and resources. Moreover, the COVID-19 pandemic not only highlighted the importance of migrant farmworkers to agricultural industries in Canada, given their designation as "essential" workers, but also catalyzed a significant shift in how communities can support workers. This paper focuses on a migrant-intensive region in southwestern Ontario. It utilizes a qualitative constructivist approach to interviews with 30 migrant farmworkers and 32 service providers to examine the support and services available to workers. Employing the conceptual framework of differential inclusion, this study analyzes the nature of support and highlights how temporary migrant farmworkers are embedded in relations of inclusion and exclusion. This study also explores the potential and capacity of community-based support initiatives to foster inclusion for workers in migrant-intensive communities and the emergence of a support framework during COVID-19. Given the rise in the migrant population in smaller cities and rural communities across Canada, this study aims to address the limited scholarship on the support landscape for migrant farmworkers in these regions. [ABSTRACT FROM AUTHOR]

We show the existence of solutions satisfying two point, three point or four point boundary conditions, for the differential inclusion (φ(x′(t)))′ ∈ F(t,x(t)), where F(·, ·) is a compact lower semi-continuous multi-valued map and φ is a homeomorphism function. [ABSTRACT FROM AUTHOR]

This article discusses differential inclusion as it relates to mobility in Europe through migrants' experiences of the closure of the European Union (EU) Schengen borders during the COVID-19 pandemic. Based on 36 comparative online interviews with three groups of migrants – Erasmus students, asylum seekers and seasonal workers – the article empirically investigates how differential inclusion is reflected in migrants' perceptions of border closures and the impact of border closures on international mobility. Drawing on the concept of differential inclusion, I examine the divergent border mobilities in a moment of crisis. In the interviews, migrants' reflections on borders are informed either by their own perception of borders, their surprise at the lack of awareness of borders for other migrants, or the realisation that closed borders are crossed for capitalist economic demands under high health risks. Taking this as its basis, the article makes two arguments. First, that preexisting differential inclusion exacerbated during border closures in a global health emergency. Second, that borders are not concrete but flexible in (im)mobilising people according to capitalist economic demands. In this way, the article contributes to an understanding of the process of rebordering that took place during COVID-19 and in which borders remained spaces of differentiation. [ABSTRACT FROM AUTHOR]

This paper proposes an admittance control scheme for robots equipped with joint-level position controllers involving deadtime. Its main feature is an elaborate discrete-time jerk limiter, which limits the third derivative of the position command sent to the controller. The jerk limiter is designed to suppress undesirable oscillation especially when the robot is in contact with a stiff environment. The controller is designed as a differential inclusion involving normal cones in the continuous-time domain, and its discrete-time algorithm is derived by the implicit Euler discretization. The presented controller was validated with experiments using a collaborative robot UR3e of Universal Robots, which has a deadtime of 6 ms in the velocity-command mode. [ABSTRACT FROM AUTHOR]

Based on fixed point theory for condensing operators, an initial value problem for semilinear differential inclusions of fractional order in Banach spaces is studied. It is assumed that the linear part of the inclusion generates a family of cosine operator functions and the nonlinear part is a multivalued map with nonconvex values. Local and global existence theorems for mild solutions of the initial value problem are proved. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In depictions of the Swedish welfare regime, there is a belief concerning the existence of a strong link between the universalism of rights and residency. But with tougher migration laws, temporary and time-limited residence permits, the settlement of different groups and access to housing and other social rights have been made more difficult. Public housing is often used as a means of providing housing for refugees, where time-limited subletting contracts are frequently used, writing off tenants’ rights to possession. While there is a strong tradition of universal public rental housing without means testing in the Swedish housing supply, the institutional practices of housing provision for refugees deviate from universal features. In this paper, we examine the outcome of housing provision for refugees in one municipal setting. Our study is based on qualitative interviews with 19 officials and politicians about arguments and justifications concerning the housing practices in use. The analysis also includes interviews with 11 refugees with varying immigration status, focusing on their experiences of housing solutions and tenant rights. Our results show that the municipal responsibility to provide housing for refugees reciprocates immigration control, which produces not only variation in fulfilment of decent housing, but also differential inclusion based on a locally practised categorization of housing rights.

The article investigates the phenomenon of precarious labour within the Finnish wild berry industry, focusing on the socio‐legal dimensions that enable short‐term “just‐in‐time” migration, primarily from Thailand, for the berry season. Since the initial 2005 recruitment of Thai citizens to engage in forest berry picking for the Finnish berry industry, the industry has become heavily reliant on migrant labour. At the same time, the pickers’ situation exemplifies a case of unregulated labour, as pickers are categorised as a group outside of labour laws in Finland. By asking how this “non‐work”—berry picking without labour rights—has repeatedly been justified on a policy level, the article provides a case study that unpacks the creation of a racialised migrant labour force through a statecraft of differential inclusion, in an arrangement regarded to advance rural economies. Empirically, the article draws on an analysis of policy documents through which a particular kind of temporary migration corridor is administered.

This paper deals with the research of solutions of bounded variation (BV) to evolution inclusion coupled with a time and state dependent maximal monotone operator. Different problems are studied: existence of solutions, unicity of the solution, existence of periodic and bounded variation right continuous (BVRC) solutions. Second-order evolution inclusions and fractional (Caputo and Riemann–Liouville) differential inclusions are also considered. A result of the Skorohod problem driven by a time- and space-dependent operator under rough signal and a Volterra integral perturbation in the BRC setting is given. The paper finishes with some results for fractional differential inclusions under rough signals and Young integrals. Many of the given results are novel. [ABSTRACT FROM AUTHOR]

The paper deals with systems of ordinary differential equations containing in the right-hand side controls which are discontinuous in phase variables. These controls cause the occurrence of sliding modes. If one uses one of the well-known definitions of the solution of discontinuous systems, then the motion of an object while being on some surface can be described in terms of differential inclusions. With the help of the previously developed apparatus for solving differential inclusions, a method is constructed for finding the trajectories of a system moving in a such a mode. Since some of frequently used discontinuous controls contain nonsmooth functions of phase variables, the paper pays special attention to study the differential properties of such systems. At the end of the paper controls of a slightly different, in contrast to the classical, type are considered which have useful differential properties, and a method is constructed for solving systems with such controls considered both before hitting the required surface and moving in its vicinity. [ABSTRACT FROM AUTHOR]

This paper studies the interplay between the concepts of error bounds and the Kurdyka–Łojasiewicz (KL) inequality on Hadamard manifolds. To this end, we extend some properties and existence results of a solution for differential inclusions on Hadamard manifolds. As a second contribution, we show how the KL inequality can be used to obtain the convergence of the gradient method for solving convex feasibility problems on Hadamard manifolds. The convergence results of the alternating projection method are also established for cyclic and random projections on Hadamard manifolds and, more generally, CAT(0) spaces. [ABSTRACT FROM AUTHOR]

The aim of the present paper is to state and discuss the well-posedness of a new evolution inclusion governed by the subdifferential of a function φ perturbed both by a Carathéodory mapping and by an integral forcing term. The integrand of the forcing term depends on two time-variables. We prove the existence and uniqueness of local and global solution, assuming that φ is primal lower regular and the perturbation terms satisfy some standard conditions. The results are applied to the study of non-regular electrical circuits with tranmission line, Zener diode and inductor. A second application is concerned with semilinear heat equation with memory. These applications illustrate an additional novelty of our paper. [ABSTRACT FROM AUTHOR]

The purpose of this text is to propose an attempt of an extension of the Filippov-Ważewski Relaxation Theorem for a certain class of fractional differential inclusions. The classical result devoted to ordinary differential inclusions is a part of the qualitative theory: a description of the relationship between the solutions to the differential inclusion and the convexified differential inclusion was given. There exist several generalizations of that result and in this note a method is proposed so that the range of some parameters is extended. An example of a possible application may arise in control theory and the question is whether it is possible to have the same reachable set economizing the set of controls. [ABSTRACT FROM AUTHOR]

The paper explores the differential inclusion of a special form. It is supposed that the support function of the set in the right-hand side of an inclusion may contain the maximum of the finite number of continuously differentiable (in phase coordinates) functions. It is required to find a trajectory that would satisfy differential inclusion with the boundary conditions prescribed and simultaneously lie on the surface given. Such problems arise in practical modeling discontinuous systems and in other applied ones. The initial problem is reduced to a variational one. It is proved that the resulting functional to be minimized is superdifferentiable. The necessary minimum conditions in terms of superdifferential are formulated. The superdifferential (or the steepest) descent method in a classical form is then applied to find stationary points of this functional. Herewith, the functional is constructed in such a way that one can verify whether the stationary point constructed is indeed a global minimum point of the problem. The convergence of the method proposed is proved. The method constructed is illustrated by examples. [ABSTRACT FROM AUTHOR]

We define a new class of -condensing multivalued maps, where is a vector measure of noncompactness and is a bounded positive linear operator whose spectral radius Some properties of such multivalued maps are studied. In particular, it is shown that the topological degree theory and classical fixed point theorems can be extended to this class of multimaps. An analog of the theorem on a diagonal representation of a condensing multimap is proved. It is demonstrated that the translation multioperator along trajectories of a system of semilinear differential inclusions in Banach spaces is -condensing, where is the vector Hausdorff measure of noncompactness. [ABSTRACT FROM AUTHOR]

In this paper, we study a Cauchy type problem in Banach spaces for various classes of functional inclusions with causal multioperators. Based on the topological degree theory for condensing multimaps, we prove a global theorem on the existence of trajectories for systems governed by functional inclusions. As an application, we obtain generalizations of existence theorems for a Cauchy type problem for semilinear second order differential inclusions of this type and semilinear differential inclusions of the fractional order 1 < q < 2. [ABSTRACT FROM AUTHOR]