On the nonuniqueness and instability of solutions of tracking-type optimal control problems

We study tracking-type optimal control problems that involve a non-affine, weak-to-weak continuous control-to-state mapping, a desired state \begin{document}$ y_d $\end{document} , and a desired control \begin{document}$ u_d $\end{document} . It is proved that such problems are always nonuniquely solvable for certain choices of the tuple \begin{document}$ (y_d, u_d) $\end{document} and instable in the sense that the set of solutions (interpreted as a multivalued function of \begin{document}$ (y_d, u_d) $\end{document} ) does not admit a continuous selection.