Abstract
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.
Highlights
Differential-algebraic equations (DAEs) is a wellrecognized and extensively studied area of the modern applied mathematics, arisen as a natural generalization of the concept of ODE
We investigate and promote a very specific and poorly-studied class of DAEs, where hysteresis phenomena, represented through sweeping processes [Kunze and Marques, 2000, Moreau, 1977], are combined with impulsive behavior of the modeled dynamic process
The models, overviewed in the previous sections, can be reformulated under the framework of so-called differential variational inequalities (DVIs) [Pang and Stewart, 2008]. Such a fact is not surprising in the nonimpulsive setup, where DVIs are known to cover a plenty of dynamic complementarity problems, sweeping processes, and differential algebraic equations [Adly, Haddad, and Thibault, 2014, Brokate, Krejcı, and Schnabel, 2004, Brokate and Sprekels, 1996, Facchinei and Pang, 2003, Krejcı, 1991, Moreau, 1977]
Summary
Differential-algebraic equations (DAEs) is a wellrecognized and extensively studied area of the modern applied mathematics, arisen as a natural generalization of the concept of ODE. [Brogliato, 1999, Song, Krauss, Kumar and Dupont, Stewart, 2000, Heemels, Schumacher, and Weiland, 2000]), as well as in modeling of electric circuits with hysteresis phenomena [Adly, Haddad, and Thibault, 2014, Acary, Bonnefon, and Brogliato, 2011]. We investigate and promote a very specific and poorly-studied class of DAEs, where hysteresis phenomena, represented through sweeping processes [Kunze and Marques, 2000, Moreau, 1977], are combined with impulsive behavior of the modeled dynamic process. The main motivation for this study is due to the above mentioned models of electric circuits, where impulses do naturally appear, see, e.g., [Acary, Bonnefon, and Brogliato, 2011]
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