A dynamic programming approach is considered for a class of minimum problems with impulses. The minimization domain consists of trajectories satisfying an ordinary differential equation whose right-hand side depends not only on a measurable control v but also on a second control u and on its time derivative d. For this reason, the control u, and the differential equation are called impulsive. The value function of the considered minimum problem turns out to depend on the time, the state, the u variable, and the variation allowed to the impulsive control. It is shown that the value function satisfies, in a generalized sense, a dynamic programming equation (DPE), which is obtained from a dynamic programming principle involving space-time trajectories. Moreover the value function is the unique map-solving equation (DPE) satisfying either an inequality condition or a supersolution condition at each point of the boundary. Incidentally this extends a result by Barren, Jensen, and Menaldi [Nonlinear Anal., 21 (1993), pp. 241-268], where the impulsive control is scalar monotone and the corresponding vector field is independent of the state variable. Next, a maximum principle is proved, and the well-known relationship between adjoint variables and value function is suitably extended to impulsive control systems. A fully elaborated example concludes the paper.