Abstract
In this paper, we analyze and discuss the well-posedness of two new variants of the so-called sweeping process, introduced by Moreau in the early 70s (Moreau in Sem Anal Convexe Montpellier, 1971) with motivation in plasticity theory. The first new variant is concerned with the perturbation of the normal cone to the moving convex subset $$C(t)$$ C ( t ) , supposed to have a bounded variation, by a Lipschitz mapping. Under some assumptions on the data, we show that the perturbed differential measure inclusion has one and only one right continuous solution with bounded variation. The second variant, for which a large analysis is made, concerns a first order sweeping process with velocity in the moving set $$C(t)$$ C ( t ) . This class of problems subsumes as a particular case, the evolution variational inequalities [widely used in applied mathematics and unilateral mechanics (Duvaut and Lions in Inequalities in mechanics and physics. Springer, Berlin, 1976]. Assuming that the moving subset $$C(t)$$ C ( t ) has a continuous variation for every $$t\in [0,T]$$ t ? [ 0 , T ] with $$C(0)$$ C ( 0 ) bounded, we show that the problem has at least a Lipschitz continuous solution. The well-posedness of this class of sweeping process is obtained under the coercivity assumption of the involved operator. We also discuss some applications of the sweeping process to the study of vector hysteresis operators in the elastoplastic model (Kreja?i in Eur J Appl Math 2:281---292, 1991), to the planning procedure in mathematical economy (Henry in J Math Anal Appl 41:179---186, 1973 and Cornet in J. Math. Anal. Appl. 96:130---147, 1983), and to nonregular electrical circuits containing nonsmooth electronic devices like diodes (Acary et al. Nonsmooth modeling and simulation for switched circuits. Lecture notes in electrical engineering. Springer, New York 2011). The theoretical results are supported by some numerical simulations to prove the efficiency of the algorithm used in the existence proof. Our methodology is based only on tools from convex analysis. Like other papers in this collection, we show in this presentation how elegant modern convex analysis was influenced by Moreau's seminal work.
Highlights
In the seventies Moreau introduced and thoroughly studied the sweeping process, which is a particular differential inclusion
The first new variant is concerned with the case where the sweeping process (1.1) is perturbed by a Lipschitz mapping and where the moving convex set C(t) has a bounded variation
The first new variant presented above of the perturbation with a Lipschiz mapping of the sweeping process involving convex set C(t) with bounded variation is studied in great detail in Sect. 4; a theorem of existence and uniqueness is established
Summary
In the seventies Moreau introduced and thoroughly studied the sweeping process, which is a particular differential inclusion. The first new variant is concerned with the case where the sweeping process (1.1) is perturbed by a Lipschitz mapping and where the moving convex set C(t) has a bounded variation. The first new variant presented above of the perturbation with a Lipschiz mapping of the sweeping process involving convex set C(t) with bounded variation is studied in great detail in Sect. The fundamental concepts of subdifferential or normal cone, directional derivative and Legendre–Fenchel conjugate will be at the heart of our present paper From the definitions it directly follows the monotonicity property of the subdifferential of the convex function φ ∂0ψ is monotone, where ∂0 is any subdifferential with appropriate fuzzy sum rule (see [22,52]) on the Banach space X Another deep important property of the subdifferential in Convex Analysis The subdifferential of a proper lower semicontinuous convex function φ on X is maximal monotone in the sense that there is no monotone set-valued operator from X into X ∗ whose graph is larger than the graph of ∂φ
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