In this paper we study multi-objective control problems that give rise to equivalent convex optimization problems. me develop a uniform treatment of such problems by showing their equivalence to linear programming problems with equality constraints and an appropriate positive cone, We present some specialized results on duality theory, and we apply them to the study of three multi-objective control problems : the optimal l(1) control with time-domain constraints on the response to some fixed input, the mixed H-2/l(1)-control problem, and the l(1) control with magnitude constraint on the frequency response, What makes these problems complicated is that they are often equivalent to infinite-dimensional optimization problems, The characterization of the duality relationship between the primal and dual problem allows us to derive several results, These results establish connections with special convex problems (linear programming or linear matrix inequality problems), uncover finite-dimensional structures in tile optimal solution, when possible, and provide finite-dimensional approximations to any degree of accuracy when the problem does not appear to have a finite-dimensional structure, To illustrate the theory and highlight its potential, several numerical examples are presented.