  In this paper, we study second-order necessary and sufficient optimality
conditions of Karush--Kuhn--Tucker-type for locally optimal solutions in the
sense of Pareto to a class of multi-objective optimal control problems with
mixed pointwise constraint. To deal with the problems, we first derive
second-order optimality conditions for abstract multi-objective optimal control
problems which satisfy the Robinson constraint qualification. We then apply the
obtained results to our concrete problems. The proofs of obtained results are
direct, self-contained without using scalarization techniques.
