  A counterexample to uniqueness of global minimizers of semilinear optimal
control problems is given. The lack of uniqueness occurs for a special choice
of the state-target in the cost functional. Our arguments show also that, for
some state-targets, there exist local minimizers, which are not global. When
this occurs, gradient-type algorithms may be trapped by the local minimizers,
thus missing the global ones. Furthermore, the issue of convexity of quadratic
functional in optimal control is analyzed in an abstract setting.
  As a Corollary of the nonuniqueness of the minimizers, a nonuniqueness result
for a coupled elliptic system is deduced.
  Numerical simulations have been performed illustrating the theoretical
results.
  We also discuss the possible impact of the multiplicity of minimizers on the
turnpike property in long time horizons.
