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 $y_d$, and a desired control $u_d$. It is proved that such problems are always nonuniquely solvable for certain choices of the tuple $(y_d,u_d)$ and instable in the sense that the set of solutions (interpreted as a multivalued function of $(y_d,u_d)$) does not admit a continuous selection.