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Colombo, G.; Goncharov, Vladimir V.

doi:10.1023/a:1008774529556

Cited by 60 publications

(4 citation statements)

References 8 publications

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“…(because it is bounded and uniformly converges to u), using relation (18), we conclude by Proposition 2.9, that for almost every t ∈ I u(t) ∈ P C(t,u(t)) u(t) − r ∆(t) , that is, −∆(t) ∈ N C(t,u(t)) (u(t)) (see the definition of the proximal normal cone), or equivalently −u(t) − z(t) ∈ N C(t,u(t)) (u(t)), a.e. t ∈ I, and by (17) we get −u(t) ∈ N C(t,u(t)) (u(t)) + F (t, u(t)), a.e. t ∈ I…”

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confidence: 97%

“…is the normal cone to C(t) in the sense of convex analysis, see also [30] and [31]. Then, some contributions in the context of nonconvex sets C(t) were given in a series of papers, see for instance [4,17,18,34,35,36].…”

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confidence: 99%

“…a lot of work has been done in the finite dimensional setting, we refer for example to [10,11,17,34] and the references therein.…”

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confidence: 99%

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On state-dependent sweeping process in Banach spaces

Azzam-Laouir¹,

Selamnia²

2018

Evolution Equations &Amp; Control Theory

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In this paper we prove, in a separable reflexive uniformly smooth Banach space, the existence of solutions of a perturbed first order differential inclusion governed by the proximal normal cone to a moving set depending on the time and on the state. The perturbation is assumed to be separately upper semicontinuous. Recently, the differential inclusion (1), where the sets in the normal cone depend on the time and the state, has been studied by many authors motivated by the applications of this type of sweeping processes to parabolic quasi variational inequalities and modelisation of 2D and 3D quasistatic evolution problems with

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confidence: 97%

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confidence: 99%

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On state-dependent sweeping process in Banach spaces

Azzam-Laouir¹,

Selamnia²

2018

Evolution Equations &Amp; Control Theory

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“…There has been a significant interest in developing the qualitative theory and control methods for sweeping processes with prox-regular constraints lately. Colombo-Goncharov [9], Benabdellah [3], Colombo and Monteiro Marques [10], and Thibault [27] studied the existence and uniqueness of solutions to non-perturbed sweeping processes with nonconvex prox-regular sets. Existence and uniqueness for perturbed sweeping processes is considered in Edmond-Thibault [11], [12].…”

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confidence: 99%

Global asymptotic stability of nonconvex sweeping processes

Wadippuli¹,

Gudoshnikov²,

Makarenkov³

2020

Discrete &Amp; Continuous Dynamical Systems - B

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Building upon the technique that we developed earlier for perturbed sweeping processes with convex moving constraints and monotone vector fields (Kamenskii et al, Nonlinear Anal. Hybrid Syst. 30, 2018), the present paper establishes the conditions for global asymptotic stability of global and periodic solutions to perturbed sweeping processes with prox-regular moving constraints. Our conclusion can be formulated as follows: closer the constraint to a convex one, weaker monotonicity is required to keep the sweeping process globally asymptotically stable. We explain why the proposed technique is not capable to prove global asymptotic stability of a periodic regime in a crowd motion model (Cao-Mordukhovich, DCDS-B 22, 2017). We introduce and analyze a toy model which clarifies the extent of applicability of our result.

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