BV solutions of rate independent variational inequalities

Recupero, Vincenzo

doi:10.2422/2036-2145.2011.2.02

Cited by 13 publications

(17 citation statements)

References 18 publications

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“…as shown in [25,Section A.4] and in [12,Theorem 3.2], or in [28,Lemma 4.1] in the more general setting of sweeping processes. We also mention another approach to the vector play operator with discontinuous inputs based on the papers [8,9,10,11].…”

Section: Introductionmentioning

confidence: 94%

“…Consequently we traverse these trajectories with an infinite speed in order to obtain the extension of P to BV([0, T ] ; H). The main difficulty in this procedure is the choice of the trajectories: if we consider a left continuous input and we fill in the jumps with segments, which seems the natural way, then in [25] it is proved that the resulting operator is different from the play operator defined by (1.10)-(1.12) (a complete comparison between the two operators is performed in [16,17] in the finite dimensional case). Thus another choice is in order and it seems that there is no chance to join the jumps in a canonical way that is intrinsic in the nature of the play operator and independent of the particular input.…”

Section: Introductionmentioning

confidence: 99%

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Multidimensional play operators with arbitraryBVinputs

Recupero

2020

Math. Model. Nat. Phenom.

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In this paper we provide an integral variational formulation for a vector play operator where the inputs are allowed to be arbitrary functions with (pointwise) bounded variation, not necessarily left or right continuous. We prove that this problem admits a unique solution, and we show that in the left continuous and right continuous cases it reduces to the well known existing formulations.2010 Mathematics Subject Classification. 47J20, 34C55, 26A45.

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Section: Introductionmentioning

confidence: 94%

Section: Introductionmentioning

confidence: 99%

Multidimensional play operators with arbitraryBVinputs

Recupero

2020

Math. Model. Nat. Phenom.

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“…And this is how it is introduced in Krasnosel'skiǐ-Pokrovskiǐ [23] and in Visintin [31]. Possible extensions to the case of discontinuous inputs have been analyzed in Brokate-Sprekels [11], Krejčí-Laurençot [22], Recupero [19,20]. Note that, when considering discontinuous inputs, in particular a special kind of jump/continuous functions (the so-called regulated functions), one has to decide how to fill the gap in the jumps, in order to recover a (at least approximating) continuous input.…”

Section: Hysteresis Operatorsmentioning

confidence: 99%

“…Indeed, the play operator is continuous only with respect to the topology of uniform convergence but we can grant only pointwise convergence. Moreover note that the result obtained using the extended definition of the Play operator for discontinuous inputs ( [11,22,19,20]), would require the use of measure control instead of measurable ones. More precisely the good discontinuous input v such that P [v, w 0 ] = x, can be obtained using a control which is ū1 + 2ρ number of jumps δ t i , where δ t i is the Dirac delta function centered in the jumping times.…”

Section: Lemma 31mentioning

confidence: 99%

Hysteresis and controllability of affine driftless systems: some case studie

Bagagiolo,

Zoppello

2020

Preprint

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We investigate the controllability of some kinds of driftless affine systems where hysteresis effects are taken into account, both in the realization of the control and in the state evolution. In particular we consider two cases: the one when hysteresis is represented by the socalled play operator, and the one when it is represented by a so-called delayed relay. In the first case we prove that, under some hypotheses, whenever the corresponding non-hysteretic system is controllable, then we can also, at least approximately, control the hysteretic one. This is obtained by some suitably constructed approximations for the inputs in the hysteresis operator. In the second case we prove controllability for a generic hysteretic delayed switching system. Finally, we investigate some possible connections between the two cases.

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“…Several notions of a solution are thus available for (49)-(51) for discontinuous inputs. In this regard, we mention the works [31][32][33][34], where an extension approach based on a time-reparametrization was used. For q ∈ BV r (T , H) the time-reparametrization is given by the function…”

Section: The Play Operator and Its Extensions To Discontinuous Inputsmentioning

confidence: 99%

Optimal control problems with states of bounded variation and hysteresis

Samsonyuk

Timoshin

2019

J Glob Optim

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This paper is concerned with an optimal control problem for measure-driven differential equations with rate independent hysteresis. The latter is modeled by an evolution variational inequality equivalent to the action of the play operator generalized to the case of discontinuous inputs of bounded variation. The control problem we consider is, a priori, singular in the sense that it does not have, in general, optimal solutions in the class of absolutely continuous states and Lebesgue measurable controls. Hence, we extend our problem and consider it in the class of the so-called impulsive optimal control problems. We introduce the notion of a solution for the impulsive control problem with hysteresis and establish a relationship between the solution set of the impulsive system and that of the corresponding singular system. A particular attention is paid to approximation of discontinuous solutions with bounded total variation by absolutely continuous solutions. The existence of an optimal solution to the impulsive control problem with hysteresis is proved. We also propose a variant of the socalled space-time reparametrization adapted to optimal impulsive control problems with hysteresis and discuss a method for obtaining optimality conditions for impulsive processes.The work is partially supported by the Russian Foundation for Basic Research, projects nos 17-01-00733 and 18-01-00026.

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BV solutions of rate independent variational inequalities

Abstract: We prove a theorem providing a geometric characterization of BV continuous vector rate independent operators. We apply this theorem to rate independent evolution variational inequalities and deduce new continuity properties of their solution operator: the vectorial play operator.

Cited by 13 publications

References 18 publications

Multidimensional play operators with arbitraryBVinputs

Multidimensional play operators with arbitraryBVinputs

Hysteresis and controllability of affine driftless systems: some case studie

Optimal control problems with states of bounded variation and hysteresis

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