The number and type of boundary conditions to be used in the numerical modeling of fluid mechanics problems is normally chosen according to a simplified analysis of the characteristics, and also from the experience of the modeler. The problem is harder at inflow/ outflow boundaries which are, in most cases, artificial boundaries, so that a bad decision about the boundary conditions to be imposed may affect the precision and stability of the whole computation. For inviscid flows, the analysis of the sense of propagation in the normal direction to the boundaries gives the number of conditions to be imposed and, in addition, the conditions that are ''absorbing" for the waves impinging normally to the boundary. In practice, it amounts to counting the number of positive and negative eigenvalues of the advective flux Jacobian projected onto the normal. The problem is even harder when the number of incoming characteristics varies during the computation, and the correct treatment of these cases poses both mathematical and practical problems. One example considered here is a compressible flow where the flow regime at a certain part of an inlet/outlet boundary can change from subsonic to supersonic and the flow can revert. In this work the technique for dynamically imposing the correct number of boundary conditions along the computation, using Lagrange multipliers and penalization, is discussed and several numerical examples are presented.