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 well recognized and extensively studied area of the modern applied mathematics, arisen as a natural generalization of the concept of ordinary differential equations (ODEs). The unsolvability measure with 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 with separated “differential” and “algebraic” subsystems. This structural form is equivalent to the initial system in the sense of solution, and the operator that transformes the DAE into the structural form possesses the left inverse operator. Finding the structural form is constructive and does not use a change of variables. In addition, the problem of consistency of initial data is solved automatically. Systems of DAEs are attracting more and more attention due to mathematical modeling problems in many applied domains: automated control theory, optimal control with mixed constraints, mechanics, chemical kinetics, hydrodynamics, thermal engineering, etc. The systems under investigation arise in modeling various physical processes, in particular, in electrical circuits with hysteresis phenomena. For such a DAE, we design an equivalent structural form (in the sense of solutions). Necessary and sufficient conditions for the existence and uniqueness of a solution to an initial value problem and controllability are proved. Illustrative examples are given in the conclusions.
P. S. Petrenko (Sun,) studied this question.