The article considers the Cauchy problem for a linear set-valued differential equation with the Hukuhara derivative and derives an analytical formula for its solution.

The article presents a conformal fractional-fractal derivative for set-valued mappings, unifying conformal fractional and fractal derivatives. Key properties are derived. Analytical solutions are obtained for linear set-valued Cauchy problems, both without impulses and with impulsive effects, involving this derivative.

The paper presents various derivatives of set-valued mappings,their main properties and how they are related to each other.Next, we consider Cauchy problems with linear homogeneousset-valued differential equations with different types ofderivatives (Hukuhara derivative, PS-derivative andBG-derivative). It is known that such initial value problems withPS-derivative and BG-derivative have infinitely many solutions.Two of these solutions are called basic. These are solutions suchthat the diameter function of the solution section is amonotonically increasing (the first basic solution) or monotonicallydecreasing (the second basic solution) function. However, the secondbasic solution does not always exist. We provideconditions for the existence of basic solutions of such initialvalue problems. It is shown that their existence depends on thetype of derivative, the matrix of coefficients on the right-handand the type of the initial set. Model examples are considered.

We consider a linear set-valued differential equation with generalized derivative and variable matrix. The conditions for the existence of solutions are presented. We determine the shapes of their cross sections at each time in the analytic form. The results are illustrated by model examples.

The article explores a linear set-valued differential equation featuring both conformable fractional and generalized conformable fractional derivatives. It presents conditions for the existence of solutions and provides analytical expressions for the shape of solution sections at different time points. Model examples are employed to illustrate the results.

The article considers the control linear differential equation with Hukuhara derivative and the problem of moving a set-valued object to a target set, that is, when at some point in time the cross section of a set-valued solution of the system is contained in the target set. The solvability conditions for this problem are obtained, as well as the time and controls that guarantee the fulfillment of the termination process condition. It is shown that in some cases the given time and controls will be optimal. The results of the article are illustrated by model examples.

We consider two linear set-valued integral equations, establish the conditions for the existence of their solutions, and determine, in the analytic form, the shape of their sections at any time. The results are illustrated by model examples.

Discusses the synthesis of a distributed oil field development control system. The article indicates the significance of the modeling object at the moment, highlights the main parameters of the simulated object, then compiles a mathematical model of the system, which is then implemented programmatically, as well as analyzes the control object and synthesizes the control system with a distributed high-precision regulator. To solve the problems set in the work, methods of mathematical and computer modeling, methods of analyzing and synthesizing systems with distributed parameters, the foundations of the theory of building algorithms and programs were used. This work based on the report work on the graduation paper [1].

We consider a time-optimal problem for a set-valued linear control system in the case where a section of the solution of the system coincides with a target set. For this problem, we establish both the solvability conditions and the optimal time and optimal controls. The results are illustrated by model examples.

We discuss various definitions and properties of the derivatives of set-valued mappings. We also consider a linear set-valued differential equation and investigate the problem of existence of solutions of this equation with Hukuhara derivative, PS-derivative, and BG-derivative. The obtained results are illustrated by model examples.

УДК 517.9 Розглядається задача оптимальної швидкодії для лінійної керованої багатозначної системи у випадку, коли переріз розв'язку цієї системи збігається з цільовою множиною. Отримано умови розв'язності даної задачі, а також оптимальний час та оптимальні керування. Результати проілюстровано на модельних прикладах.

In this article one optimal control problem when the system behavior is described by linear fuzzy differential equations is considered. The conditions of its solvability are formulated, and the optimal time and optimal controls are obtained.

The article discusses various definitions of the derivative of a set-valued mapping and their properties. Also, a linear set-valued differential equation is considered and the existence of solutions for this equation with Hukuhara derivative, Plotnikov-Skripnik derivative and Bede-Gal derivative is investigated.

The article presents some definitions of derivatives for set-valued mappings and their properties. A linear set-valued differential equation is considered and conditions for the existence of basic solutions are given. Subsequently, one optimal control problem is considered, when the system behavior is described by linear set-valued differential equations.

We consider a multivalued discrete system and study its properties and the existence of its solution.

Recently, many authors have considered questions of the existence, uniqueness, and properties of solutions of set-valued differential and integro-differential equations, higher order equations, and have investigated impulse and control systems in the framework of the theory of set-valued equations. Obviously, obtaining all these results would be impossible without the development of the theory of set-valued analysis. In the latter, new definitions of the derivative have appeared for set-valued mappings, which, unlike the previously used Hukuhara derivative, made it possible to differentiate set-valued mappings whose diameter is not only a non decreasing function. As a result, set-valued differential equations were considered whose solutions are set-valued mappings whose diameter is not a monotonic function. This article discusses the new formulation of the optimal control problem (the time-optimality problem) that became possible due to these new derivatives and differential equations, as well as a method for solving this problem.

We substantiate an averaging scheme for integrodifferential inclusions on a bounded interval.

In the introduction of this article, we gave a brief overview of publications on the theory of set-valued equations. Next, we considered slow-fast systems of set-valued differential equations and substantiated the possibility of applying the averaging method for approximate resolution or investigation of the properties of solutions of such systems. c ©2019 World Academic Press, UK. All rights reserved.

In this paper, we consider two types of set-valued Volterra–Hammerstein integral equations and prove the existence and uniqueness theorem.