33369

Автор(ы): 

Автор(ов): 

1

Параметры публикации

Тип публикации: 

Книга (брошюра, монография, стандарт)

Название: 

VISCOSITY SOLUTION FOR DESIGN OF CONTROL NONLINEAR UNCERTAIN SYSTEMS

Сведения об издании: 

1-ое издание

Сведения об издании: 

1-ое издание

ISBN/ISSN: 

ISBN 978-5-98862-232-1

Город: 

  • Москва

Издательство: 

  • Грифон

Год издания: 

2015

Объём, стр.: 

48
Аннотация
Successful implementation of obtained theoretical results in a number of problems is connected with solving of partial first-order differential equations. Such partial derivative equations appear under solving of a great number of theoretical and applied problems of mathematics, mechanics, physics, biology, chemistry, engineering, control, etc. Such equations are Hamilton-Jacobi equation in theoretical mechanics [1], Bellman equation in theory of optimal control [2], Isaacs equation [3], eikonal equation in geometrical optics [4], Burgers and Hopf limit equations in gas dynamics and hydrodynamics [5], etc. The method of characteristics proposed in the first half of the 19th century by O. Cauchy for solving boundary problems for such equations reduces integrating of partial first-order derivative equations to integrating of a system of ordinary differential equations. This method is based on the fact that invariance of graph of the classical solution for a boundary problem is relative to the characteristics. However, in case of partial derivative nonlinear equation, smooth solution exists only locally [6]. In 1950-1970s a lot of mathematicians paid much attention to generalized solutions of Hamilton-Jacobi and other types of equations [7, 8]. Developed methods mainly based on integral methods and integral properties of generalized solutions. In early 1980s a concept of viscosity solution was introduced the existence of which was proved by method of disappearing viscosity [9]. The method is also being developed at present time. The researches pay attention to analytical, constructive and numerical methods of construction of viscosity solutions [10] and application of theoretical results to solving of various applied problems. Another well-known concept of the generalized solution based on idempotent analysis was proposed in works by V.P. Maslov and his disciples. By means of this approach linearizing convex problems, Hamilton-Jacobi equations with a convex Hamiltonian and their applications to problems of mathematical physics are studied. Optimal control problems and differential games are connected one way or another with a search for solutions of Hamilton-Jacobi-Bellman, Isaaks equations. To solve such equations, constructive and numerical methods (including grid ones) were developed [13, 14, 15]. An important result of the theory of minimax solutions of partial first-order derivative equations being a base for differential game theory is proving the equivalence of concepts of minimax and viscosity solutions [16]. Within the frameworks of minimax solution concept originating from the theory of position differential games [17], [18] developed by school of N. N. Krasovsky on the base of minimax evaluations and operations, theorems of existence and uniqueness, correctness and content-richness of minimax solution concept for various types of boundary problems of partial first-order derivative equations were proved. Despite available theoretical results in this area, the issue of Hamilton-Jacobi-Isaacs equation solution in the problems of differential games with non-linear indefinite dynamic objects in the rate of their functioning persists and is important today.

Библиографическая ссылка: 

Афанасьев В.Н. VISCOSITY SOLUTION FOR DESIGN OF CONTROL NONLINEAR UNCERTAIN SYSTEMS. 1-ое издание. М.: Грифон, 2015. – 48 с.