Optimization of Cauchy problem for partial differential inclusions of parabolic type

Optimization of Cauchy problem for partial differential inclusions of parabolic type is considered, and sufficient condition for optimality is derived. For derivation of sufficient conditions both for convex and nonconvex partial differential inclusions, the apparatus of locally conjugate mappings is used. Besides, some special limiting conditions such as the equality of the coordinate-wise limits of the conjugate variables and their derivatives are formulated. The obtained results are generalized to the multidimensional cases. The linear problem illustrates the fact that some of the conjugate variables in concrete problems can be eliminated.