The authors consider an optimal control problem for a class of nonlinear second order infinite-dimensional systems in the setting of the Gelfand triple \(X\hookrightarrow H\hookrightarrow X^*\) with injection being continuous and compact. The problem they consider can be formulated as follows: \[ \begin{cases} J(u, x)= \int_ I L(t, x, \dot x, u)dt\to\inf\\ \text{subject to}:\\ \ddot x+ A(t, \dot x)+ Bx= f(t, x)u,\;x(0)= x_ 0\in X,\;\dot x(0)= x_ 1\in H,\;u(t)\in U(t)\text{ a.e.},\end{cases} \] where \(U\) is a measurable set-valued map with values in \(2^ Y\), \(Y\) being a separable reflexive Banach space, and \(A\) is a nonlinear operator from \(I\times X\to X^*\), \(B\in {\mathcal L}(X, X^*)\), \(f: I\times H\to {\mathcal L}(Y,H)\).
Under some suitable assumptions the authors prove the existence of optimal controls for the problem as stated. This result is an extension of an earlier work of N. S. Papageorgiou [Glas. Mat., III. Ser. 27, No. 2, 297-311 (1992; Zbl 0806.49009)].