A double ellipsoid heat source model was proposed by Dr, I. Goldak in this investigation. A mathematical model used by Goldak for weld heat sources was based on a Gaussian distribution of power destiny. It is necessary to note, the function of Gaussian or normal distribution is the continuous smooth function.
In 2011-th professor Tzarkov A.V. (MSTU named N. Bauman) used the double ellipsoid weld pool model in order to analyze the inﬂuence of weld pool geometry on solidiﬁcation crack formation. In a result of the investigation he found that the curvature of rear part weld pool has great inﬂuence on high temperature strain distribution and consequently on solidiﬁcation process with hot cracking formation.
In this report the problem of control of solidiﬁcation crack formation in welded plates is considered. In this problem the welding source is determined (in) dependence on the preset conﬁguration and curvature of the rear part weld pool. A double ellipsoid model of weld pool with preset semi-axes a, c1, c2 may be used for the ﬁrst approximation of preset weld pool conﬁguration. It is an inverse problem which may be more efﬁciently solved as optimal control problem. The Function of welding source as a controlling function obtained in the result of solution is determined in a class of piecewise continuous functions which is more common class and the continuous-smooth functions are special partial case of common class. Recent methods of optimal control which use for solution of optimal control problems are require to present the controlling functions in class of piecewise constant functions. Laser, inﬂuence, electron beam, plasma, arc and submerged arc are arc are welding sources with high concentrated energy. A mathematical models of these of these welding sources may be introduced in class of piecewise in class of piecewise continuous with an efﬁcient accuracy.