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.

Розглянуто лінійне множиннозначне диференціальне рівняння із узагальненою похідною та змінною матрицею. Наведено умови існування розв’язків і отримано в аналітичному вигляді форму їхніх перерізів у кожний момент часу. Результати проілюстровано модельними прикладами.

УДК 517.9 Розглянуто різні означення похідної множиннозначного відображеннята їхні властивості. Вивчається лінійне множиннозначне диференціальне рівняння та досліджується існування розв'язків цього рівняння з похідною Хукухари, PS-похідноюта BG-похідною. Отримані результати проілюстровано на модельних прикладах.

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.

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.

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 applicability of a three-step averaging scheme to set-valued differential equations with generalized derivative.

We develop the ideas of the method of averaging for some classes of fuzzy systems (fuzzy differential equations with delay, fuzzy differerntial equations with pulsed action, fuzzy integral equations, fuzzy differential inclusions, and differential inclusions with fuzzy right-hand sides without and with pulsed action).

The integral equations are encountered in various fields of science and in numerous applications, including elasticity, plasticity, heat and mass transfer, oscillation theory, fluid dynamics, filtration theory, electrostatics, electrodynamics, biomechanics, game theory, control, queuing theory, electrical engineering, economics, and medicine. In this paper the fuzzy integral equation is considered and the existence and uniqueness theorem, the theorem of continuous dependence on the right-hand side and initial fuzzy set are proved. Also the possibility of using the scheme of full averaging for fuzzy integral equation with a small parameter is considered.

We substantiate the possibility of application of the method of averaging on a finite interval to impulsive differential inclusions with fuzzy right-hand sides containing a small parameter. In the case of periodic right-hand sides, it is shown that the estimate can be improved.

We consider a generalized set-valued differential equation with generalized derivative and prove the theorems on existence and uniqueness of its solution for the cases of interval-valued and set-valued mappings.