Finite-approximate controllability of second-order semilinear evolution equations with Clarke subdifferential

In this paper, we investigate finite-approximate controllability for a class of second-order semilinear evolution inclusions involving Clarke subdifferentials in Hilbert spaces. Under the stated compactness, growth and continuity assumptions, we prove that approximate controllability of the corresponding linear second-order system implies finite-approximate controllability of the nonlinear inclusion. Our approach is based on cosine operator family theory, Clarke subdifferential analysis, the Kakutani–Fan–Glicksberg set-valued fixed-point theorem, and a controllability functional combined with convex subdifferential calculus.