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Chadi Nour

Publications and source records attributed to Chadi Nour.

11 recordsLinked to original sources
arXiv2026

Optimal Control of Sweeping Processes with Nonsmooth Moving Sets: A Numerical Algorithm

This paper extends the numerical method introduced by de Pinho et al. [22] and later developed in Nour and Zeidan [33, 37] to the nonautonomous case where the nonsmooth sweeping set depends explicitly on time. The moving character of the sweeping set creates substantial additional difficulties, since several geometric constants involved in the analysis may a priori depend on time. To overcome this issue, we establish uniform geometric estimates for the moving constraint sets and their boundaries. These estimates allow us to extend the discrete approximation scheme developed in [22, 33, 37] and prove its convergence toward admissible solutions of the original problem. Several examples illustrating the applicability of the proposed method are also presented.

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arXiv2025

New Characterizations of Strong Convexity

Parallel to the main results of [13] and [14], which explore the equivalence between prox-regularity, the exterior sphere condition, and $S$-convexity, we present novel characterizations of the $r$-strong convexity property, namely, of the sets that can be expressed as the intersection of closed balls with the same radius $r>0$.

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arXiv2024

The Complement of a Closed Set Satisfying the Extended Exterior Sphere Condition

We provide a novel analytical proof of an improved version of [10, Theorem 3.1], showing that the complement of a closed set satisfying the extended exterior sphere condition is nothing but the union of closed balls with lower semicontinuous radius function. The improvement lies in the radius function, which is now larger than the one used in [10, Theorem 3.1].

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arXiv2024

The Variable Radius Form of the Extended Exterior Sphere Condition

We introduce a variable radius form of the extended exterior sphere condition of [16], and then, we prove that the complement of a closed set satisfying this new property is nothing but the union of closed balls with lower semicontinous radius function. This generalizes, to the variable radius case, the main result of [16], namely, [16, Theorem 1.2]. On the other hand, as it is shown in [14,15] for prox-regularity, the exterior sphere condition, and the union of closed balls property, we prove that the constant and the variable radius forms of the extended exterior sphere condition belong to the S-convexity regularity class.

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arXiv2023

Numerical Method for a Controlled Sweeping Process with Nonsmooth Sweeping Set

The numerical method developed in [30] for optimal control problems involving sweeping processes with smooth sweeping set C is generalized to the case where C is nonsmooth, namely, C is the intersection of a finite number of sublevel sets of smooth functions. The novelty of this extension resides in producing for the general setting a different approach, since the one used for the smooth sweeping sets is not applicable here.

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arXiv2023

Nonsmooth optimality criterion for a $\mathrm{W}^{1,2}$-controlled sweeping process: Nonautonomous perturbation

We extend to $\mathrm{C}\times \mathrm{W}^{1,2}$-local minimizers and nonautonomous perturbation function, the necessary optimality conditions derived in [1], via continuous-time approximations, for $\mathrm{W}^{1,2}\times \mathrm{W}^{1,2}$-local minimizers of an optimal control problem governed by a controlled nonconvex sweeping process with autonomous perturbation function.

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arXiv2023

The Extended Exterior Sphere Condition

We prove that the complement of a closed set S satisfying an extended exterior sphere condition is nothing but the union of closed balls with common radius. This generalizes [11, Theorem 3] where the set S is assumed to be prox-regular, a property stronger than the extended exterior sphere condition. We also provide a sufficient condition for the equivalence between prox-regularity and the extended exterior sphere condition that generalizes [13, Corollary 3.12] to the case in which S is not necessarily regular closed.

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arXiv2023

A Control Space Ensuring the Strong Convergence of Continuous Approximation for a Controlled Sweeping Process

A controlled sweeping process with prox-regular set, $W^{1,2}$-controls, and separable endpoints constraints is considered in this paper. Existence of optimal solutions is established and local optimality conditions are derived via strong converging continuous approximations that entirely reside in the interior of the prox-regular set. Consequently, these results are expressed in terms of new subdifferentials for the original data that are strictly smaller than the standard Clarke and Mordukhovich subdifferentials.

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