Consideration was given to the systems of n quasilinear equations with first-order partial derivatives, two independent variables, and an arbitrary number of numerical control parameters. From the application standpoint, the independent variables play the part of time and space, and the parameters are replaced either by the control functions or the solutions of the inverse problem. The geometrical formalization ascending to Georg Friedrich Bernhard Riemann and enabling one to assign to a quasilinear system a field of linear operators on the corresponding vector bundle is applicable to the systems under consideration. The criterion for diagonalizbility, that is, reduction to the Riemann invariants of quasilinear systems with control parameters by state transformations, was established in terms of this field.