For families of continuous controlled dynamic input-output processes, we obtain and discuss necessary and sufficient conditions for their differential realization in the class of quasilinear nonstationary differential equations of state in a Banach space.

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A universal regression-tensor approach is developed in the mathematical modeling of optimal parameters of chemical-technological process of complex mechanical products. The testing of developed algorithms was performed on the example of multi-factorial process of low-temperature sulfur-chromium plating of precision mechanical parts. V. A. Rusanov et al. 1208 nonlinear regression modeling [6] (with simultaneous reduction of dimension of used pseudo-matrix). According to this applied interpretation of theoretical results of the article [6] is studied in details below; it is interesting in connection with other applications of method of least squares-interpolation where generalizations of algorithmic solutions proposed in [6] can manifest. Keywords Covariant Tensors of Final Valence, Identification of Nonlinear Vector Regression, Optimization of Metal CoatingsThe approbation of theoretical apparatus of nonlinear vector regression in the article stands the problem of optimization (as a way of technological calculation) of characteristics of multi-factorial chemical-technological process. Payment of optimization of process of low-temperature sulfur-chromium plating of precised mechanical products is taken as example. The Statement of the Problem of Modeling of Optimal Chemical-Technological ProcessLet R-field of real numbers, n R -n-dimensional vector space over R with Euclidian norm . n R ,