Автор(ы): Афанасьев В. Н. (ИПУ РАН, Лаборатория 45)Автор(ов): 1 Параметры публикацииТип публикации: Книга (брошюра, монография, стандарт)Название: VISCOSITY SOLUTION FOR DESIGN OF CONTROL NONLINEAR UNCERTAIN SYSTEMSСведения об издании: 1-ое изданиеСведения об издании: 1-ое изданиеISBN/ISSN: ISBN 978-5-98862-232-1Город: МоскваИздательство: ГрифонГод издания: 2015Объём, стр.: 48 АннотацияSuccessful 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 , Bellman equation in theory of optimal control , Isaacs equation , eikonal equation in geometrical optics , Burgers and Hopf limit equations in gas dynamics and hydrodynamics , 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 . 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 . The method is also being developed at present time. The researches pay attention to analytical, constructive and numerical methods of construction of viscosity solutions  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 . Within the frameworks of minimax solution concept originating from the theory of position differential games ,  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.