A reinterpretation of set differential equations as differential equations in a Banach space

Set differential equations are usually formulated in terms of the Hukuhara differential, which implies heavy restrictions for the nature of a solution. We propose to reformulate set differential equations as ordinary differential equations in a Banach space by identifying the convex and compact subsets of $\R^d$ with their support functions. Using this representation, we demonstrate how existence and uniqueness results can be applied to set differential equations. We provide a simple example, which can be treated in support function representation, but not in the Hukuhara setting

Fuzzification of Fractal Calculus

In this manuscript, fractal and fuzzy calculus are summarized. Fuzzy calculus in terms of fractal limit, continuity, its derivative, and integral are formulated. The fractal fuzzy calculus is a new framework that includes fractal fuzzy derivatives and fractal fuzzy integral. In this framework, fuzzy number-valued functions with fractal support are the solutions of fractal fuzzy differential equations. Different kinds of fractal fuzzy differential equations are given and solved

A decomposition theorem for fuzzy set–valued random variables

In this paper, a decomposition theorem for a (square integrable) fuzzy random variable FRV is proposed. The paper is mainly divided in two part. In the first part, for any FRV X, we define the Hukuhara set as the family of (deterministic) fuzzy sets C for which the Hukuhara difference X 96HC exists almost surely; in particular, we prove that such a family is a closed (with respect to different well known metrics) convex subset of the family of all fuzzy sets. In the second part, we prove that any square integrable FRV can be decomposed, up to a random translation, as the sum of a FRV Y and an element C\u2032 chosen uniquely (thanks to a minimization argument) in the Hukuhara set. This decomposition allows us to characterize all fuzzy random translation; in particular, a FRV is a fuzzy random translation if and only if its Aumann expectation equals C\u2032 (given by the above decomposition) up to a deterministic translation. Examples and open problems are also presented

On the Mitigation of Late Stage Redesign in Mechatronics Using Integrated Approaches

RÉSUMÉ Les systèmes mécatronique combinent des éléments issus du génie mécanique, électrique, contrôle et logiciel. Due à la nature multi-domaine de ces systèmes, il est nécessaire de s’assurer d’un processus de conception optimal afin de réduire le temps et le cout de développement. De ce fait, cette thèse s’intéresse aux boucles de re-conception tard durant le processus de développement. Ces boucles peuvent être causé entre autres par des interactions négatives qui affectent la performance et l’intégration des composantes et sous-systèmes et l’incertitude dans les paramètres du système. Premièrement, cette thèse propose une nouvelle méthode de modélisation qui permet d’identifier et d’évaluer les dépendances durant les phases initiales de conception. Cette méthode est ensuite utilisée dans la création d’un index qui permet de représenter le niveau total de dépendances négative du système. L’index est ensuite utilisé dans l’évaluation multicritère, ce qui permet de choisir des systèmes étant plus faciles à concevoir. Finalement, une méthode de modélisation qui permet de considérer de façon concurrente les dépendance positive et négative est présenté. Par la suite, cette thèse propose d’utiliser les nombres flous afin de traiter l’incertitude des paramètres. En premier lieu, la thèse montre que les nombres flous peuvent être utilisé afin de simuler le comportement d’un système mécatronique sujet à de l’incertitude. De plus, une méthode de conception utilisant la simulation floue est proposée afin de concevoir les systèmes mécatronique de façon robuste. De plus, les nombres floues permettent de déterminer la stabilité du système, ce qui permet le développement d’une méthodologie de conception robuste totalement intégré, qui considère à la fois l’aspect physique et contrôle du système.----------ABSTRACT Mechatronic systems are highly integrated devices, with elements from mechanical, electrical, software and control engineering. It is thus necessary to ensure a streamlined design process to reduce development time and cost. Consequently, this thesis researches on the issue of late stages redesigns in mechatronics. The late stages redesigns may occur due to problems while integrating the different components and subsystems. Two causes of these redesigns are unpredicted negative interactions between the elements of the system, and inadequate performance due to uncertainties. To deal with the issue of negative interactions, this thesis first suggests a modeling method that enables to identify and assess negative dependencies early during the design process. It is shown that the modeling method can be efficiently used to detect dependencies that would be detrimental to the system’s performance and which may require more design effort. Then, based on this modeling method, an index representing the total level of negative dependencies present within the system is proposed. The index is shown to be able to predict decrease of performance due to the negative dependencies and can thus be used as a valuable criterion during decision making. Finally, a modeling method to handle concurrently positive and negative dependencies is suggested. This modeling method is shown to have an impact on the currently existing complexity metrics and should thus allow to better represent the reality of the design. Furthermore, to deal with the issue of uncertainties affecting the performance of the system, this thesis proposes a design methodology using fuzzy numbers. First, it is shown that fuzzy numbers can be used to model and simulate the uncertain behavior of mechatronic systems while being computationally efficient. Then a robust design methodology is presented and shown to be effective in optimizing a mechatronic system while reducing the uncertainties in the performance. Furthermore, based on the use of fuzzy numbers in the modeling of the mechatronic system, it is shown that it is possible to determine the stability of the device under uncertainties. Finally, a fully integrated robust design methodology is presented, which consider both control and design parameters selection, and which can be used to mitigate late stages redesigns due to improper performance. In sum, this thesis investigates and suggests multiple integrated design solutions to mitigate late stages redesigns in the mechatronic design process

Interval-valued upside potential and downside risk portfolio optimisation

A novel interval optimisation approach is developed to include imprecise forecasts into the portfolio selection process for investors measuring upside potential and downside risk as deviations from a target return. Crisp scenarios are substituted by interval scenarios and the resulting interval optimisation problem is solved in a tractable manner by means of a bi-objective formulation exploiting a partial order relation between intervals. Four utility case studies involving assets from the F.T.S.E. M.I.B. Index are considered to illustrate how impreciseness can be efficiently handled in portfolio management

Modified Artificial Neural Networks For Solving Fuzzy Differential Equations

In this paper, we introduce a novel approach based on modified  neural networks  to solve fuzzy differential equations. Using modified  neural network makes that training points should be selected over an open interval  without training the network in the range of first and end points. Therefore, the calculating volume involving computational error is reduced. In fact, the training points depending on the distance selected for training neural network are converted to similar points in the open interval  by using a new approach, then the network is trained in these similar areas. In comparison with existing similar neural networks proposed model provides solutions with high accuracy. The proposed method is illustrated by three numerical examples. Keywords: Fuzzy  differential  equation, Modified  neural  network, Feed-forward  neural  network, BFGS Teqnique, Hyperbolic tangent   function

Mixed Variational Inequality Interval-valued Problem: Theorems of Existence of Solutions

In this article, our efforts focus on finding the conditions for the existence of solutions of Mixed Stampacchia Variational Inequality Interval-valued Problem on Hadamard manifolds with monotonicity assumption by using KKM mappings. Conditions that allow us to prove the existence of equilibrium points in a market of perfect competition. We will identify solutions of Stampacchia variational problem and optimization problem with the interval-valued convex objective function, improving on previous results in the literature. We will illustrate the main results obtained with some examples and numerical results