Research output: Contribution to journal › Article › peer-review
Градиентный метод решения некоторых типов дифференциальных включений. / Fominyh, Alexander Vladimirovich; Karelin, Vladimir Vital Evich; Polyakova, Lyudmila Nickolaevna.
In: ТРУДЫ ИНСТИТУТА МАТЕМАТИКИ И МЕХАНИКИ УРО РАН, Vol. 26, No. 1, 2020, p. 256-273.Research output: Contribution to journal › Article › peer-review
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TY - JOUR
T1 - Градиентный метод решения некоторых типов дифференциальных включений
AU - Fominyh, Alexander Vladimirovich
AU - Karelin, Vladimir Vital Evich
AU - Polyakova, Lyudmila Nickolaevna
PY - 2020
Y1 - 2020
N2 - We discuss some classes of problems with differential inclusions, for which an efficient algorithm based on the gradient method is developed. The first part of the paper describes an algorithm for solving differential inclusions with a free or a fixed right end and a convex continuous multivalued mapping that admits a support function with a continuous derivative with respect to the phase coordinates. This algorithm reduces the problem under consideration to the problem of minimizing a certain functional in a function space. For this functional, the Gâteaux gradient is obtained and necessary and, in some cases, sufficient minimum conditions are found. Further, the gradient descent method is applied to the functional. In the second part of the paper, the developed approach is illustrated by solving three main classes of differential inclusions: (1) a differential inclusion obtained from a control system with a variable control domain depending on the phase coordinates, (2) a differential inclusion containing the direct sum, union, or intersection of convex sets in the right-hand side, (3) a linear interval system of ODEs considered as a differential inclusion.
AB - We discuss some classes of problems with differential inclusions, for which an efficient algorithm based on the gradient method is developed. The first part of the paper describes an algorithm for solving differential inclusions with a free or a fixed right end and a convex continuous multivalued mapping that admits a support function with a continuous derivative with respect to the phase coordinates. This algorithm reduces the problem under consideration to the problem of minimizing a certain functional in a function space. For this functional, the Gâteaux gradient is obtained and necessary and, in some cases, sufficient minimum conditions are found. Further, the gradient descent method is applied to the functional. In the second part of the paper, the developed approach is illustrated by solving three main classes of differential inclusions: (1) a differential inclusion obtained from a control system with a variable control domain depending on the phase coordinates, (2) a differential inclusion containing the direct sum, union, or intersection of convex sets in the right-hand side, (3) a linear interval system of ODEs considered as a differential inclusion.
KW - Differential inclusion
KW - Gradient descent method
KW - Gâteaux gradient
KW - Linear interval system
KW - Support function
KW - Variable control domain
UR - http://www.scopus.com/inward/record.url?scp=85090516247&partnerID=8YFLogxK
UR - https://www.mendeley.com/catalogue/58934f5f-7c42-3589-b74c-abcc976c20c2/
U2 - 10.21538/0134-4889-2020-26-1-256-273
DO - 10.21538/0134-4889-2020-26-1-256-273
M3 - статья
VL - 26
SP - 256
EP - 273
JO - ТРУДЫ ИНСТИТУТА МАТЕМАТИКИ И МЕХАНИКИ УРО РАН
JF - ТРУДЫ ИНСТИТУТА МАТЕМАТИКИ И МЕХАНИКИ УРО РАН
SN - 0134-4889
IS - 1
ER -
ID: 60424664