Interval Cauchy problem with a second type Hukuhara derivative

Abstract

In this paper we consider interval differential equations. Such the equations can be appropriate in modeling of dynamical systems under presence of uncertainty of parameters. We study an interval initial value problem with a second type Hukuhara derivative. By an example of real-world application we indicate the advantages of the usage of such a kind of interval-valued derivative. A continuous dependence of the solution on initial value and right-hand side of the equation is shown. The existence of approximate local solutions is proven, and then it is used in the derivation of existence of at least one local solution to interval Cauchy problem with second type Hukuhara derivative. The compactness of solutions set is also stated. Finally, the explicit formulae for local solutions to linear interval differential equations are provided.

Introduction

The Interval Differential Equations (IDEs), understood as the uncertain ordinary differential equations, can be typical for many applications in physics or engineering as soon as uncertainties and tolerances of parameters are taken into account to reflect realistic information available on dynamical systems. Thinking about a physical problem which is transformed into a deterministic problem of ordinary differential equations we cannot usually be sure that this modeling is perfect. Especially, if the data (e.g. initial value) are not known precisely but only through some measurements the intervals which cover the data are determined. Therefore there appear problems of differential equations with uncertainty. Such uncertainty can be expressed in terms of intervals and ordinary differential equations are replaced with IDEs which are more appropriate tool for modeling with uncertainties. It is also clear that uncertainty expressed by an interval is smaller as soon as the diameter of this interval (i.e. difference between the greatest and the smallest element of the interval) is smaller.

The notion of Hukuhara derivative of set-valued mapping, introduced by Hukuhara [19], is strictly connected with the theory of IDEs and set differential equations (SDEs) and it possesses some advantages. For example, if the set-valued mapping is single-valued then the Hukuhara derivative reduces to the ordinary vector derivative. Therefore, in the case of ${\mathbb{R}}^d$-valued mappings, the results obtained in the SDEs setting become the corresponding results of ordinary differential equations. However in the IDEs and SDEs set-up we have only semilinear metric space to work with instead of normed linear space employed in the usual study of ordinary differential equations. The investigations of SDEs have been started by de Blasi and Iervolino [6] and form currently a wide branch of generalization of ordinary differential equations (see e.g. research articles [1], [6], [7], [12], [13], [14], [15], [16], [22], [23], [24], [26], [27], [28], [31], [32], [33], [34], [35], [38], [39] and monograph [25] and references therein).

A solution to IDEs or SDEs (i.e. an interval-valued or set-valued mapping), with Hukuhara derivative defined as in [19], is the Hukuhara differentiable mapping. This implies that the diameter of the solution values is a nondecreasing function of time (see e.g. [25]). From the applications point of view this property can be sometimes inconvenient because it means practically that uncertainty, contained in a model of physical system which is described by IDE or SDE, can only grow as times goes by. Hence the successive values of modeled phenomenon are covered by nondecreasing (in the sense of the diameter) intervals or sets of tolerance.

To overcome this situation Stefanini and Bede [37] introduced a concept of generalized Hukuhara differentiability of interval-valued mapping, which allows them to obtain the solutions of IDEs with decreasing diameter of solutions values. In [37], under assumption that the right-hand side of IDE satisfies the Lipschitz condition, the existence and uniqueness of two solutions has been proved: the one for IDE with classical Hukuhara derivative and the second one for IDE with (ii)-gH type of Hukuhara derivative (which is called, in this paper, the second type Hukuhara derivative and described in Definition 2.1 later on). The work [37] starts the investigations of IDEs with second type Hukuhara derivative and [9], [29] contain some further studies in this subject.

In this paper we focus our research on the interval Cauchy problem with second type Hukuhara derivative as the results concerning that with classical Hukuhara derivative are known. In the new setting of IDEs we utilize some interval integral equations which cause that many times throughout the paper we have to guarantee existence of appropriate Hukuhara differences. Such problems do not appear in the context of IDEs with classical Hukuhara derivative. We present the theoretical studies on existence of at least one local solution to IDEs with second type Hukuhara derivative and prove the stabilities of solutions under change of initial value and right-hand side of the equation.

The paper is organized as follows. In Section 2 we collect some necessary facts about Hukuhara difference, about the notions of differentiation and integration of interval-valued functions which will be used in the rest of the article. In Section 3 we present a real-world application of IDE with second type of Hukuhara derivative. Then continuous dependence of the solution on initial value and right-hand side of equation is shown. Further the existence of approximate local solutions is stated. This lead us to the existence of at least one local solution for IDEs with second type of Hukuhara derivative. For the set of solutions we obtain a topological property of compactness. Finally, we provide the explicit formulae of unique second type Hukuhara differentiable local solutions to linear IDEs.