By K.A. Lurie

This publication represents a longer and considerably revised model of my earlierbook, optimum regulate in difficulties ofMathematical Physics,originally released in Russian in 1975. approximately 60% of the textual content has been thoroughly revised and significant additions were incorporated that have produced a essentially new textual content. My target was once to modernize the presentation but in addition to maintain the unique effects, a few of that are little identified to a Western reader. the assumption of composites, that is the center of the trendy conception of optimization, was once initiated within the early seventies. The reader will locate right here its implementation within the challenge of optimum conductivity distribution in an MHD-generatorchannel flow.Sincethen it has emergedinto an intensive thought that's present process a continual improvement. The booklet doesn't fake to be a textbook, neither does it supply a scientific presentation of the speculation. really, it displays an idea which I ponder as basic within the smooth method of optimization of dis tributed platforms. Bibliographical notes,though huge, don't fake to be exhaustive to boot. My thank you are because of ProfessorJean-Louis Armand and ProfessorWolf Stadler whose pleasant tips in translating and sharpening the textual content was once so beneficial. i'm indebted to Mrs. Kathleen Durand and Mrs. Colleen Lewis for the challenging activity of typing huge parts of the manuscript.

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**Sample text**

We take an ellipse D with center at the point P of the region G and semiaxes a and b oriented along the unit vectors a and p, respectively; the ellipse is made of a homogeneous conductor of resistivity U. We assume that the ellipse D is placed in a homogeneous field of currents j that form an angle cp with the axis p (Fig. 2) and flow in an infinite conducting medium with constant resistivity u. The components of the current density J within the ellipse are determined by the expressions'i" 10< = • a +b .

23) f1(Z, K , v, q)J dq r To prove thi s, it is sufficient to introduce additional dependent variables-the vector Zo = (zn +\ .. , zn+m) and the vector Zl = ( zn+m+\ ... , z n+2m- I)_ by means of the expressions* v · Zo = [« , V· 21 = L, n+ i _ Z Xk n +m + i Z Xk - zn+k = 0 (i, k = 1, ... , m ) 0 (i, k = 1, .. , er » : (k = m (m 2 = 1, ... " 1 Cartesian coordinate s and = r Zo = (zn +\ zn+2), 0 are the plane s Xm = con st. > JO, n+2 _ zx. 2). 22). 25) may be similarly transformed. 25) (in the latter case, the missing relations are obtained from the necessary conditions for a minimum).

Finally, in a number of cases the region G is not specified in advance; to be specific, we assume that an (open) part er of its boundary r is not known a priori but must be determined together with the control functions . 17) are not specified. 32 Chapter 2 The set a may have a non empty intersection with the sets 'Yj(lij) on which the conditions of the above types are specified; the boundary au of the set a is assumed known. 19) obtained in this manner the concept of a generalized solution. (x); obviously, our previously introduced set D(G) c D(R m ) .