A NOTE ON DIFFERENTIAL-ALGEBRAIC SYSTEMS WITH IMPULSIVE AND HYSTERESIS PHENOMENA

In this note, we single out some promising classes of differential-algebraic equations (DAEs) with hysteresis phenomena, and propose their meaningful generalizations. We consider DAEs of index 2 having two fea-tures: 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.


Introduction
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. An interest to the framework of DAEs was attracted by multiple applications in rigidbody dynamics with friction and hybrid control engineering (see, e.g. [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, andThibault, 2014, Acary, Bonnefon, andBrogliato, 2011].
In this paper, we investigate and promote a very specific and poorly-studied class of DAEs, where hysteresis phenomena, represented through sweeping processes [Kunze andMarques, 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].

DAEs with hysteresis
Consider the following DAE of index 2 paired with a sweeping process of a rate independent hysteresis type (modeled by the play operator Sprekels, 1996, Krejcí, 1991]): Here, T = [t 0 , t 1 ] is a given time interval, x(t), y(t) ∈ R n ; Q(t) = x(t) − Z is a "moving set", where Z is a given closed convex subset of R n , and N Q u denotes the normal cone to a closed convex set Q at a point u; A, B, C ∈ R (n×n) are given constant matrices, wherein A is singular, while the pair (A, B) is assumed to be regular. Recall that a pair (A, B) of matrices is called regular, if there exists λ ∈ R n such that det(λA − B) = 0. By a solution of (1), (2) we mean a pair of absolutely continuous (W 1,1 ) functions (x, y) satisfying (1), (2) for almost all (a.e.) t ∈ T with respect to (w.r.t.) the usual Lebesgue measure.
As is known [Gantmacher, 1988], for any regular pair (A, B), there exist nonsingular matrices M, P such that where J has the Jordan canonical form, and N is a nilpotent matrix of index l (upper triangular matrix with l square null blocks on diagonal, such that N l = 0) also having the Jordan form. We assume that matrix N is nilpotent of index 2. Then, system (1), (2) rewrites: where z(t) . = The following assertion gives a necessary and sufficient condition for the existence of a unique solution to system (1), (2) (or rather (3)-(5)).
2) Assume that (A, B) is regular, and (6) holds. Denote x . = (z 1 , y,ẏ), f (x) . = (J z 1 ,ẏ, 0 ∈ R n ), Q(t) . = R k+n × Q(t), t ∈ T , and rewrite (3), (5) as a perturbed sweeping process Note that the set Q(t) is closed and convex for all t ∈ T . Furthermore, function f is linear. Then, the existence and uniqueness of a solution to an initial value problem for system (7) follows from the "catching-up" algorithm for perturbed sweeping processes [Adly, Haddad, and Thibault, 2014]. It remains to agree the value z 2 (t 0 ) provided (3) and (4), which is given by condition (6).
Since the addressed sweeping process is defined through a play operator, differential inclusion is equal to the following differential variational inequality: Example 1: Modeling of an Electric Circuit by a Differential-algebraic Sweeping Process Consider an electric circuit involving six resistors, three inductors, two ideal diodes, and a current source c(t) (see Fig. 1). Recall that the ideal diode's voltage/current law is described by the complementary condition where V d denotes the potential across the diode and i is the current through the diode. This complementary condition can be reformulated as an inclusion denote the load resistances and the inductors, respectively. Then the Kirchhoff's law writes: relations (8)-(11) take the form of a differentialalgebraic sweeping process (DASP): where is Lipschitz continuous, one can show that DASP (12) has a unique solution for any initial condition x(0) = x 0 ∈ C(0) satisfying the relation Indeed, equality (9) gives: expression into (8), we obtaiṅ where a = 1 x 4 ) and rewrite (10), (11) as follows: It remains to note that (14), (15) are perturbed sweeping processes, and each of them has a unique solution for given x 0 and c(·) [Adly, Haddad, and Thibault, 2014]. Note that the current source in Example 1 can be described by a discontinuous control function c(·). In this case, jumps of the moving set C(t) produce jumps of the respective state solution, which leads to a concept of impulsive DAE with hysteresis.
A typical trajectory of this DAE, in its equivalent form (3)-(5), is presented on Fig. 2.
Consider a generalization of (1), (2), where the part of control inputs is played by bounded Borel measures: µ ∈ C * (T, K).
, · 1 is the Manhattan norm in R n , and co A is the convex hull of a set A.

Further Generalizations and Concluding Remarks
In this section, we shall propose some promising generalizations of the considered DAEs, and mark out natural direction of the future work.
A natural extension of the impulsive model, addressed in the previous section, then leads to the concept of measure-driven differential variational inequality (MDVI), which is a combination of a measure differential equation, a finite-dimensional variational inequality, and/or a differential quasi-variational inequality. MDVIs are studied in concern with rate independent hysteresis processes, and sweeping processes in [Kopfova and Recupero, 2016, Recupero, 2011, Recupero, 2015, Samsonyuk and Timoshin, 2018, Samsonyuk and Timoshin, 2019.
A class of such dynamical systems, which could be a challenging object of our further study, formally looks as follows: F t, x(t), y(t) , ρ − y(t) ≥ 0 for all ρ ∈ Z and a.e. t ∈ T.
Here, Z is a given closed convex subset of R k , F is a given map of a certain appropriate type, K is a closed convex cone in R m , and x(·) and y(·) are functions of bounded variation. Note that (31), (32) can be regarded as a relaxation of the non-impulsive DVİ x = f (t, x, y) + G(t, x, y) v, v(t) ∈ K for a.e. t ∈ T, F t, x(t), y(t) , ρ − y(t) ≥ 0 for all ρ ∈ Z and t ∈ T, with absolutely continuous trajectories x(·), y(·) and "usual" control v(·) ∈ L ∞ (T, R m ), and can be treated by the standard singular space-time transformation [Karamzin, Oliveira, Pereira, and Silva, 2015, Miller, 1996, Miller and Rubinovich, 2013, Sesekin and Zavalishchin, 1997. We also remark that, for Z = R k and F : T × R n × R m → R m , DVI (34) boils down to F t, x(t), y(t) = 0, while (33), (34) turns into a DAE.
As a concluding remark, we mention that the concept of MDVI can be extended to impulsive control systems paired with another type of complementarity conditions, namely the "orthogonality" between the driving measure and an input, depending on the one-sided limits of the state. Such systems seem to be a meaningful formalization of mechanical systems with blockable degrees of freedom, and are recently studied in [Goncharova and Staritsyn, 2017, Goncharova and Staritsyn, 2018, Staritsyn and Goncharova, 2019.