Fuel injection system for an internal combustion engine

A fuel injection system having one high-pressure fuel pump and one fuel injection valve communicating therewith for each engine cylinder. A piston of the fuel pump defines a work chamber which communicates with a pressure chamber of the fuel injection valve, which valve has an injection valve member by which injection openings are controlled, and which valve is movable in an opening direction counter to a closing force by the pressure prevailing in the pressure chamber. A first control valve controls, a connection of the pump work chamber to a relief chamber and a second control valve controls, a connection of a control pressure chamber, communicating with the pump work chamber and defined by a control piston, to a relief chamber. The control piston, controls a flow cross section of the connection of the control pressure chamber to the relief chamber as a function of the stroke of the control piston in such a way that when the injection valve member is in a closing direction, a larger flow cross section is uncovered than when the injection valve member is disposed in an open position.