Abstract. The concept of physical solution (on a finite time interval) is introduced. This concept is based on the natural energy equalities and continuous dependence of the state functions on the time variable in the phase space for classes of autonomous evolutionary variational inequalities on convex cones with non-linear non-monotonic (in the general case) maps. We use the classical penalty method for the approximate search of such solutions. For this solutions we justify the possibility of a global description of the behavior of such systems in the natural phase space in the topology of strong convergence, by using finite algorithms (up to a small parameter).

Keywords: evolutionary variational inequality, physical solution, energy equality, strong continuity.