Abstract:
In this note, we single out some promising classes of differential-algebraic equations (DAEs) with non-linearity 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 which transformes the DAE into the structural form possesses the left inverse operator. The finding of the structural form is constructive and do not use a change of variables. In addition the problem of consistency of the initial data is solved automatically. 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 (with a 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 conclusion.
Fulltext:
PDF file (1732 kB)
First page:
PDF file