Statement of nonlinear thermal problem for the system of conjugate elements

This article is devoted to the studying of strongly and weakly invariant sets with respect to dynamical control systems with unbounded velocity sets. The concepts of weakly invariant and weakly preinvariant sets in relation to impulsive control system are introduced, together with infinitesimal conditions in a form of systems of proximal Hamilton-Jacobi inequalities. Applications of some family of weakly preinvariance, which sets to inner estimations of the integral funnel and necessary global optimality conditions for impulsive processes, are given.