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In the paper we extend some Hardy and Littlewood type inequalities on time scales for the function of n variables. Special cases of obtained results include generalized Wirtinger, Hardy and Littlewood type inequalities.

The purpose of this paper is to develop a methodology for the stochastically asymptotic stability of fuzzy Markovian jumping neural networks with time-varying delay and continuously distributed delay in mean square.

The authors perform Briat Lemma, multiple integral approach and linear convex combination technique to investigate a class of fuzzy Markovian jumping neural networks with time-varying delay and continuously distributed delay. New sufficient criterion is established by linear matrix inequalities conditions.

It turns out that the obtained methods are easy to be verified and result in less conservative conditions than the existing literature. Two examples show the effectiveness of the proposed results.

The novelty of the proposed approach lies in establishing a new Wirtinger-based integral inequality and the use of the Lyapunov functional method, Briat Lemma, multiple integral approach and linear convex combination technique for stochastically asymptotic stability of fuzzy Markovian jumping neural networks with time-varying delay and continuously distributed delay in mean square.

This paper provides a self-contained introduction to the mathematical aspects of the topological derivative.

Full justifications are given on simple model problems following a modern approach based on the averaged adjoint state technique. Extensions are discussed in relation with the literature on the field.

Closed expressions of topological derivatives are obtained and commented.

Several cases are covered in a unified and didactic presentation. Some elements of proof are novel.

The power framework has become a vital part in the day-to-day life and exhibits a rapid development in this current era. Due to the fact of huge power utilization, the power frameworks fall under several power transmission-related concerns. Precisely, frequency deviation has generated a huge impact during power transmission; this in turn highly reduces the power system stability as well as reliability too.

To boost the system’s efficacy, this study proposes a neoteric closed loop feedback controller in which a control algorithm named correlative-elemental-curvature algorithm is introduced with a constant threshold.

With the aim of mitigating frequency deviation, a stability analysis technique called Retrofit Lyapunov’s method is deployed in the controller. This would simultaneously reduce the load disturbances along with tie-line synchronization issues faced with the prior controllers. Optimization is carried out with the aid of duelist optimization algorithm, which tunes the controller parameters thereby mitigating the complexities while designing a loop feedback controller power framework.

The efficacy of the proposed work is assessed with the aid of metrics, such as integral absolute error, accuracy and settling time. Thus, the proposed work enhances the system reliability as well as the stability by mitigating the frequency deviation related issues and guarantees reliable power transmission.