This work develops the methodology according to which classes of discontinuous functions are used in order to investigate a correctness of boundary-value and initial boundary-value problems for the cases with elliptic, parabolic, pseudoparabolic, hyperbolic, and pseudohyperbolic equations and with elasticity theory equation systems that have nonsmooth solutions, including discontinuous solutions.With the basis of this methodology, the monograph shows a continuous dependence of states, namely, of solutions to the enumerated boundary-value and initial boundary-value problems (including discontinuous states) and a dependence of solution traces on distributed controls and controls at sectors of n-dimensional domain boundaries and at n-1-dimensional function-state discontinuity surfaces (i.e., at mean surfaces of thin inclusions in heterogeneous media). Such an aspect provides the existence of optimal controls for the mentioned systems with J.L. Lions' quadratic cost functionals.