Abstract <p> In this note, we single out some promising classes of differential-algebraic equations(DAEs) with nonlinearity of hysteresis type modeled by a sweeping process. DAEs is a wellrecognized and extensively studied area of the modern applied mathematics, arisen as a naturalgeneralization of the concept of ordinary differential equations (ODEs). The unsolvability measurewith respect to the derivatives for some DAE is an integer that is called the index of the DAE.The analysis is carried out under the assumption of the existence of a structural form withseparated “differential” and “algebraic” subsystems. This structural form is equivalent to the initialsystem in the sense of solution, and the operator that transformes the DAE into the structuralform possesses the left inverse operator. Finding the structural form is constructive and does notuse a change of variables. In addition, the problem of consistency of initial data is solvedautomatically. Systems of DAEs are attracting more and more attention due to mathematicalmodeling problems in many applied domains: automated control theory, optimal control withmixed constraints, mechanics, chemical kinetics, hydrodynamics, thermal engineering, etc. Thesystems under investigation arise in modeling various physical processes, in particular, in electricalcircuits with hysteresis phenomena. For such a DAE, we design an equivalent structural form (inthe sense of solutions). Necessary and sufficient conditions for the existence and uniqueness of asolution to an initial value problem and controllability are proved. Illustrative examples are givenin the conclusions.</p>