Items related to Optimal Control of Distributed Systems with Conjugation...
Synopsis
At present, in order to resolve problems of ecology and to save mineral resources for future population generations, it is quite necessary to know how to maintain nature arrangement in an efficient way. It is possible to achieve a rational nature arrangement when analyzing solutions to problems concerned with optimal control of distributed systems and with optimization of modes in which main ground medium processes are functioning (motion of liquids, generation of temperature fields, mechanical deformation of multicomponent media). Such analysis becomes even more difficult because of heterogeneity of the region that is closest to the Earth surface, and thin inclusions/cracks in it exert their essential influence onto a state and development of the mentioned processes, 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 consider how thin inclusions/cracks exert their influences onto development of these processes, while such inclusions differ in characteristics from main media to a considerable extent (moisture permeability, permeability to heat, bulk density or shear strength may be mentioned). Xll An influence exerted from thin interlayers onto examined processes is taken 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].
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Product Description
This work develops the methodology according to which classes of discontinuous functions are used in order to investigate a correctness of boundary-value and initial boundary-value problems for the cases with elliptic, parabolic, pseudoparabolic, hyperbolic, and pseudohyperbolic equations and with elasticity theory equation systems that have nonsmooth solutions, including discontinuous solutions. With the basis of this methodology, the monograph shows a continuous dependence of states, namely, of solutions to the enumerated boundary-value and initial boundary-value problems (including discontinuous states) and a dependence of solution traces on distributed controls and controls at sectors of n-dimensional domain boundaries and at n-1-dimensional function-state discontinuity surfaces (i.e., at mean surfaces of thin inclusions in heterogeneous media). Such an aspect provides the existence of optimal controls for the mentioned systems with J.L. Lions' quadratic cost functionals.
Review
From the reviews of the first edition:
"The book is devoted to a special class of linear-quadratic optimal control problems governed by partial differential equations … . the book gives a clear enough understanding of properties of optimal control problems for the basic cases of nonstandard conjugation conditions (including controls) on the interface between the two domains." (Uldis Raitums, Zentralblatt MATH, Vol. 1080, 2006)
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