Open AccessA note on differential-algebraic systems with impulsive and hysteresis phenomena
Author(s) -
P. S. Petrenko,
Olga N. Samsonyuk,
Maxim Staritsyn
Publication year - 2020
Publication title -
cybernetics and physics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.267
H-Index - 8
eISSN - 2226-4116
pISSN - 2223-7038
DOI - 10.35470/2226-4116-2020-9-1-51-56
Subject(s) - mathematics , uniqueness , bounded function , hysteresis , ordinary differential equation , algebraic number , realization (probability) , relaxation (psychology) , pure mathematics , type (biology) , differential (mechanical device) , initial value problem , mathematical analysis , differential equation , physics , biology , thermodynamics , psychology , social psychology , ecology , statistics , quantum mechanics
n 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.