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A note on differential-algebraic systems with impulsive and hysteresis phenomena

Pavel Petrenko, Olga Samsonyuk, Maxim Staritsyn
In this note, we single out some promising classes of differential-algebraic equations (DAEs) with hysteresis phenomena, and propose their meaningful generalizations. We consider D Es of index 2 having two features: i) non-linearity of hysteresis type modeled by a sweeping process, and ii) impulsive control represented by a bounded signed Borel measure. For such a DAE we design an equivalent structural form, based on the Kronecker-Weierstrass transformation, and prove a necessary and sufficient condition for the existence and uniqueness of a solution to an initial value problem. We propose a notion of generalized solution to a DAE as a realization of impulsive trajectory relaxation. This relaxation is described by a dynamical system with states of bounded variation and can be equivalently represented as a system of “ordinary” DAEs.
CYBERNETICS AND PHYSICS, Vol.9, No.1, 2020, 51-56, https://doi.org/10.35470/2226-4116-2020-9-1-51-56
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