Le but de cette thèse est d'étudier les trois problèmes suivantes : 1) On s'intéresse à la stabilité des cônes normaux et des sous-différentiels via deux types de convergence d'ensembles et de fonctions : La convergence au sens de Mosco et celle d'Attouch-Wets. Les résultats obtenus peuvent être vus comme une extension du théorème d'Attouch aux fonctions non nécessairement convexes sur des espaces de Banach localement uniformément convexes. 2) Pour une bornologie β donnée sur un espace de Banach X, on étudie la validité de la formule suivante (…). Ici Tβ(C; x) et Tc(C; x) désignent le β -cône tangent et le cône tangent de Clarke à C en x. On montre que si, X x X est ∂β-« trusted » alors cette formule est valable pour tout ensemble fermé non vide C ⊂ X et x ∈ C. Cette classe d'espaces contient les espaces ayant une norme équivalent β-différentiable, etplus généralement les espaces possédant une fonction "bosse" lipschitzienne et β-différentiable). Comme conséquence, on obtient que pour la bornologie de Fréchet, cette formule caractérise les espaces d'Asplund. 3) On examine la convexité des ensembles de Chebyshev. Il est bien connu que, dans un espace normé réflexif ayant la propriété Kadec-Klee, tout ensemble de Chebyshev faiblement fermé est convexe. On démontre que la condition de faible fermeture peut être remplacée par la fermeture faible locale, c'est-à-dire pour tout x ∈ C il existe ∈ > 0 tel que C ∩ B(x, ε) est faiblement fermé. On montre aussi que la propriété Kadec-Klee n'est plus exigée lorsque l'ensemble de Chebyshev est représenté comme une union d'ensembles convexes fermés. / The aim of this thesis is to study the following three problems: 1) We are concerned with the behavior of normal cones and subdifferentials with respect to two types of convergence of sets and functions: Mosco and Attouch-Wets convergences. Our analysis is devoted to proximal, Fréchet, and Mordukhovich limiting normal cones and subdifferentials. The results obtained can be seen as extensions of Attouch theorem to the context of non-convex functions on locally uniformly convex Banach space. 2) For a given bornology β on a Banach space X we are interested in the validity of the following "lim inf" formula (…).Here Tβ(C; x) and Tc(C; x) denote the β-tangent cone and the Clarke tangent cone to C at x. We proved that it holds true for every closed set C ⊂ X and any x ∈ C, provided that the space X x X is ∂β-trusted. The trustworthiness includes spaces with an equivalent β-differentiable norm or more generally with a Lipschitz β-differentiable bump function. As a consequence, we show that for the Fréchet bornology, this "lim inf" formula characterizes in fact the Asplund property of X. 3) We investigate the convexity of Chebyshev sets. It is well known that in a smooth reflexive Banach space with the Kadec-Klee property every weakly closed Chebyshev subset is convex. We prove that the condition of the weak closedness can be replaced by the local weak closedness, that is, for any x ∈ C there is ∈ > 0 such that C ∩ B(x, ε) is weakly closed. We also prove that the Kadec-Klee property is not required when the Chebyshev set is represented by a finite union of closed convex sets.

Surveillance wireless sensor networks (WSNs) are highly vulnerable to the failure of the base station (BS) because attackers can easily render the network useless for relatively long periods of time by only destroying the BS. The time and effort needed to destroy the BS is much less than that needed to destroy the numerous sensing nodes. Previous works have tackled BS failure by deploying a mobile BS or by using multiple BSs, which requires extra cost. Moreover, despite using the best electronic countermeasures, intrusion tolerance systems and anti-traffic analysis strategies to protect the BSs, an adversary can still destroy them. The new BS cannot trust the deployed sensor nodes. Also, previous works lack both the procedures to ensure network reliability and security during BS failure such as storing then sending reports concerning security threats against nodes to the new BS and the procedures to verify the trustworthiness of the deployed sensing nodes. Otherwise, a new WSN must be re-deployed which involves a high cost and requires time for the deployment and setup of the new WSN. In this thesis, we address the problem of reliable recovery from a BS failure by proposing a new security architecture called Surveillance Security (SurvSec). SurvSec continuously monitors the network for security threats and stores data related to node security, detects and authenticates the new BS, and recovers the stored data at the new BS. SurvSec includes encryption for security-related information using an efficient dynamic secret sharing algorithm, where previous work has high computations for dynamic secret sharing. SurvSec includes compromised nodes detection protocol against collaborative work of attackers working at the same time where previous works have been inefficient against collaborative work of attackers working at the same time. SurvSec includes a key management scheme for homogenous WSN, where previous works assume heterogeneous WSN using High-end Sensor Nodes (HSN) which are the best target for the attackers. SurvSec includes efficient encryption architecture against quantum computers with a low time delay for encryption and decryption, where previous works have had high time delay to encrypt and decrypt large data size, where AES-256 has 14 rounds and high delay. SurvSec consists of five components, which are: 1. A Hierarchical Data Storage and Data Recovery System. 2. Security for the Stored Data using a new dynamic secret sharing algorithm. 3. A Compromised-Nodes Detection Algorithm at the first stage. 4. A Hybrid and Dynamic Key Management scheme for homogenous network. 5. Powerful Encryption Architecture for post-quantum computers with low time delay. In this thesis, we introduce six new contributions which are the followings: 1. The development of the new security architecture called Surveillance Security (SurvSec) based on distributed Security Managers (SMs) to enable distributed network security and distributed secure storage. 2. The design of a new dynamic secret sharing algorithm to secure the stored data by using distributed users tables. 3. A new algorithm to detect compromised nodes at the first stage, when a group of attackers capture many legitimate nodes after the base station destruction. This algorithm is designed to be resistant against a group of attackers working at the same time to compromise many legitimate nodes during the base station failure. 4. A hybrid and dynamic key management scheme for homogenous network which is called certificates shared verification key management. 5. A new encryption architecture which is called the spread spectrum encryption architecture SSEA to resist quantum-computers attacks. 6. Hardware implementation of reliable network recovery from BS failure. The description of the new security architecture SurvSec components is done followed by a simulation and analytical study of the proposed solutions to show its performance.