This paper presents an approximation scheme for optimal control problems using finite-dimensional linear programs and interval analysis. This is done in two parts. Following Vinter approach (SIAM J Control Optim 31(2):518–538, 1993) and using occupation measures, the optimal control problem is written into a linear programming problem of infinite-dimension (weak formulation). Thanks to Interval arithmetic, we provide a relaxation of this infinite-dimensional linear programming problem by a finite dimensional linear programming problem. A proof that the optimal value of the finite dimensional linear programming problem is a lower bound to the optimal value of the control problem is given. Moreover, according to the fineness of the discretization and the size of the chosen test function family, obtained optimal values of each finite dimensional linear programming problem form a sequence of lower bounds which converges to the optimal value of the initial optimal control problem. Examples will illustrate the principle of the methodology.