# Plotnikov A. V.

### One multivalued discrete system and its properties

Komleva T. A., Plotnikov A. V., Plotnikova L. I.

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 11. - pp. 1519-1524

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

### A Procedure of Complete Averaging for Fuzzy Differential Inclusions on a Finite Segment

Ukr. Mat. Zh. - 2015. - 67, № 3. - pp. 366-374

We justify the applicability of the method of complete averaging on a finite segment for differential inclusions with fuzzy right-hand sides containing a small parameter.

### Conditions for the Existence of Local Solutions of Set-Valued Differential Equations with Generalized Derivative

Plotnikov A. V., Skripnik N. V.

Ukr. Mat. Zh. - 2013. - 65, № 10. - pp. 1350–1362

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.

### Systems of control over set-valued trajectories with terminal quality criterion

Arsirii A. V., Plotnikov A. V.

Ukr. Mat. Zh. - 2009. - 61, № 8. - pp. 1142-1147

We consider the optimal control problem with terminal quality criterion in which the state of a system is described by a set-valued mapping, and an admissible control is a summable function. We describe an algorithm that approximates the admissible control function by a piecewise-constant function and prove theorems on the closeness of the corresponding trajectories and the values of quality criteria.

### Necessary and sufficient conditions of optimality in the problems of control with fuzzy parameters

Molchanyuk I. V., Plotnikov A. V.

Ukr. Mat. Zh. - 2009. - 61, № 3. - pp. 384-390

We study the problem of high-speed operation for linear control systems with fuzzy right-hand sides. For this problem, we introduce the notion of optimal solution and establish necessary and sufficient conditions of optimality in the form of the maximum principle.

### Differential equations with set-valued solutions

Komleva T. A., Plotnikov A. V., Skripnik N. V.

Ukr. Mat. Zh. - 2008. - 60, № 10. - pp. 1326–1337

Some special space of convex compact sets is considered and notions of a derivative and an integral for multivalued mapping different from already known ones are introduced. The differential equation with multivalued right-hand side satisfying the Caratheodory conditions is also considered and the theorems on the existence and uniqueness of its solutions are proved. In contrast to O. Kaleva's approach, the given approach enables one to consider fuzzy differential equations as usual differential equations with multivalued solutions.

### Integro-differential systems with fuzzy noise

Plotnikov A. V., Vasilkovskaya V. S.

Ukr. Mat. Zh. - 2007. - 59, № 10. - pp. 1322–1330

For a controlled integro-differential equation with fuzzy noise, we introduce the notions of a fuzzy bundle of trajectories and a fuzzy reachability set and prove some properties of fuzzy bundles.

### On some properties of bundles of trajectories of a controlled bilinear inclusion

Komleva T. A., Plotnikov A. V.

Ukr. Mat. Zh. - 2004. - 56, № 4. - pp. 484–494

We consider a differential bilinear inclusion with control and present conditions under which the reachability set for this inclusion is compact.

### Differentiation of Multivalued Mappings. $T$-Derivative

Ukr. Mat. Zh. - 2000. - 52, № 8. - pp. 1119-1126

We consider several approaches to the differentiation of multivalued mappings and introduce a new definition of derivative (*T*-derivative), which generalizes the Hukuhara derivative.

### Integro-differential equations with multivalued solutions

Plotnikov A. V., Tumbrukaki A. V.

Ukr. Mat. Zh. - 2000. - 52, № 3. - pp. 359-367

We prove a theorem on the unique existence of a classical solution of an integro-differential equation with Hukuhara derivative. We also justify an averaging scheme for equations of this type in the standard form.

### Asymptotic investigation of equations of controlled motion with multivalued trajectories

Ukr. Mat. Zh. - 1990. - 42, № 10. - pp. 1409–1412

### Averaging of equations of controlled motion with multiple-valued trajectories

Ukr. Mat. Zh. - 1987. - 39, № 5. - pp. 657–659