This note discusses the concepts of convex control systems and convex optimal control problems. We study control systems governed by ordinary differential equations in the presence of state and target constraints. Our note is devoted to the following main question: under which additional assumptions is a "sophisticated" constrained optimal control problem equivalent to a "simple" convex minimization problem in a related Hilbert space. We determine some classes of convex control systems and show that, for suitable cost functionals and constraints, optimal control problems for these classes of systems correspond to convex optimization problems. The latter can be reliably solved using standard numerical algorithms and effective regularization schemes. In particular, we propose a conceptual computational approach based on gradient-type methods and proximal point techniques.