Zaremba [1] and Marchaud [2] carried out the first investigations of differential equations with a multivalued right-hand side. The authors of these works generalized the results of the theory of differential equations to a more general case, namely, to differential equations with a multi-valued right-hand side. Marchaud and Zaremba extended the class of admissible solutions to continuous functions and, instead of the ordinary derivative, used generalized concepts of derivative, i.e., contingencies (paratingencies), which are multi-valued mappings.For the next 25 years there were no work~ devoted to this problem because of the absence of applications. At the beginning of the 1960s, Wazewski [3,4] and Filippov [5, 6] published a series of works, in which they obtained fundamentally new results concerning the existence and properties of solutions of differential equations with a multi-valued right-hand side (differential inclusions). One of the most important results was the establishment of a relationship between differential inclusions and optimal control problems, and this led to rapid development of the theory of differential inclusions.Indeed, it is customary to write the equation describing the motion of a controlled object as the differential equationwhere u E U E comp (R '~) is a control vector. Under quite general assumptions, the controlled system (1) is equivalent to the differential inclusion e F(t, x(t)),where F(t, x) = f(t, x, U).h solution of the differential inclusion (2) means an absolutely continuous function x(t) that satisfies (2) for almost all t. The investigation of differential inclusions requires the study of the properties of multi-valued functions, i.e., the development of the whole mathematical apparatus for multi-valued functions.The reader can find an extensive bibliography concerning these investigations, e.g., in [7][8][9]. Suppose that comp (R '~) is the collection of all nonempty compact subsets of the space R '~ and cony (R '~) is the collection of all nonempty convex compact subsets of R '~. We denote by S~(a) and St(A) the closed r-neighborhood of the point a and the set A, respectively, by p(A,B) = min lira-bll aEA,bEB the Euclidean distance between the sets A, B E comp (1~'~), by ,8(A, B) = max p(a, B) aEA the semi-distance of the set A from the set B, and by h(A, B) = max{~(A, B),~(B, A)} the Hausdorff distance between the sets A and B.

High reactivity of powdered aluminium together with its relatively low price are the main reasons for the application of this metal as a reducing agent for the reduction of oxides and fluorides of rare and rare earth metals by the aluminothermal method. This technology is widely used in the production of master alloys for ferrous metallurgy. The "out-of-furnace" process (i.e. without external heat access) is the prevailing technology for metallic niobium production worldwide. However, aluminothermal reduction for industrial production of metallic tantalum has been limited because of the high required temperature, over 2800 °C.In this paper the results of simultaneous reduction of tantalum(V) and iron(III) oxides by calcium-aluminothermal "out-offurnace" process are presented. Powdered aluminium and metallic calcium were used as reductants. The pickup of tantalum at smeltings with the mass of raw materials over 1 kg reached 94%. The metal was obtained in the form of compact ingot; separation of metallic and slag phases was excellent. Crushed melt underwent refining remelting in an electron beam furnace.For thermodynamic studies and optimization of the aluminothermal reduction processes of Ta 2 O 5 and Fe 2 O 3 in the temperature range of 1000-3000 °C a licensed computer simulation program "HSC Chemistry 6.1, 2007", elaborated by the company Outotec Oy, was applied.

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