发帖
雷达卡
At present, in order to resolve problems of ecology and to save mineralresources for future population generations, it is quite necessary to knowhow to maintain nature arrangement in an efficient way.It is possible to achieve a rational nature arrangement when analyzingsolutions to problems concerned with optimal control of distributedsystems and with optimization of modes in which main ground mediumprocesses are functioning (motion of liquids, generation of temperaturefields, mechanical deformation of multicomponent media). Such analysisbecomes even more difficult because of heterogeneity of the region that isclosest to the Earth surface, and thin inclusions/cracks in it exert theiressential influence onto a state and development of the mentionedprocesses, especially in the cases of mining.Many researchers, for instance, A.N. Tikhonov - A.A. Samarsky [121],L. Luckner - W.M. Shestakow [65], Tien-Mo Shih, K.L. Johnson [47],E. Sanchez-Palencia [94] and others stress that it is necessary to considerhow thin inclusions/cracks exert their influences onto development of theseprocesses, while such inclusions differ in characteristics from main mediato a considerable extent (moisture permeability, permeability to heat, bulkdensity or shear strength may be mentioned).XllAn influence exerted from thin interlayers onto examined processes istaken into account sufficiently adequately by means of various constraints,namely, by the conjugation conditions [4, 8, 10, 15, 17-20, 22-26, 38, 44,47, 52, 53, 68, 76, 77, 81, 83, 84, 90, 95, 96-100, 112-114, 117, 123].The mathematical models include the (partial differential) equations thatdescribe states of components in multicomponent media and haveboundary (object-medium interaction) and initial conditions. And theconjugation conditions, specified on median surfaces of thin inclusions andbased on the main laws of conservation, are added to them. Such anapproach generates the new mathematical problem classes, and a problemsolution makes it possible for first-type discontinuities to be present onconjugation condition specification surfaces.It should be noted that, in 1980s, the problem of construction ofcomputation algorithm with a higher-order accuracy was resolved ingeneral for elliptic, parabolic and hyperbolic equations and for elasticitytheory equation systems with boundary and initial conditions [see 16, 43,54, 55, 71, 78, 79, 91, 92, 119, 124 and other ones]. However, thecorrectness of these problem classes with conjugation conditions was notinvestigated and the efficient algorithms, used to solve them numerically,were also absent.Some simple problems from the above-mentioned families were solvedanalytically. From the mechanical point of view, the energy functionalswere obtained for deformed solids with inclusions of a low rigidity. Whenthe conjugation conditions were considered, the penalty method was usedby some authors. There are also the works, where an equation of a state isextended to a solution discontinuity surface by means of the Diracfunction.Unlike these works, the authors of the present monograph propose to usethe respective classes of the discontinuous functions in order to investigateboundary-value and initial boundary-value problems with partialXlllderivatives and conjugation conditions [18, 19, 21, 96-100, 112]. Thiscircumstance allows to create the classical energy functionals and weaklystated problems specified on such function classes. The computationalgorithms with an enhanced problem discretization accuracy order aredeveloped for the mentioned group of problem classes with conjugationconditions. This is done when proceeding from the application of thefinite-element method functions that allow discontinuity. As for adiscretization step order, the accuracy of such algorithms is not worse thanthe accuracy of the similar ones and known for the respective problemclasses with smooth solutions.The authors of the present monograph show the existence of a uniquegeneralized solutions for such problem classes, and a unique solution on asubspace is demonstrated for the Neumann problem. Such unique solutionscontinuously depend on the disturbances including the right-hand sides ofequations, conjugation conditions, boundary conditions. Therefore, it ispossible to prove the existence of the unique optimal controls as for theJ.L. Lions' quadratic cost functionals.The contents of the proposed authors' monograph is given mainly in theirworks [101-111].It should also be noted that the basic fundamental results were obtained inthe theory of optimal control in the works by L.S. Pontryagin,V.P. Boltyansky, R.V. Gamkrelidze, E.F. Mishchenko [85, 42], J. Warga[126], A.A. Feldbaum [40], R. Bellman [5], N.N. Krasovski [51],B.N. Pshenichnyi [87, 88], V.M. Tikhomirov [120] and by other authors.
提升卡
置顶卡
沉默卡
变色卡
抢沙发
千斤顶
显身卡
京公网安备 11010802022788号
