This paper addresses the multivariable gradient-based extremum seeking control (ESC) subject to saturation. Two distinct saturation scenarios are investigated here: saturation acting on the input of the function to be optimized, which is addressed using an anti-windup compensation strategy, and saturation affecting the gradient estimate. In both cases, the unknown Hessian matrix is represented using a polytopic uncertainty description, and sufficient conditions in the form of linear matrix inequalities (LMIs) are derived to design a stabilizing control gain. The proposed conditions guarantee exponential stability of the origin for the average closed-loop system under saturation constraints. With the proposed design conditions, non-diagonal control gain matrices can be obtained, generalizing conventional ESC designs that typically rely on diagonal structures. Stability and convergence are rigorously proven using the Averaging Theory for dynamical systems with Lipschitz continuous right-hand sides. Numerical simulations illustrate the effectiveness of the proposed ESC algorithms, confirming the convergence even in the presence of saturation.

Effective communication during health crises is critical, with social media serving as a key platform for public health experts (PHEs) to engage with the public. However, it also amplifies pseudo-experts promoting contrarian views. Despite its importance, the role of emotional and moral language in PHEs' communication during COVID-19 remains under explored. This study examines how PHEs and pseudo-experts communicated on Twitter during the pandemic, focusing on emotional and moral language and their engagement with political elites. Analyzing tweets from 489 PHEs and 356 pseudo-experts from January 2020 to January 2021, alongside public responses, we identified key priorities and differences in messaging strategy. PHEs prioritize masking, healthcare, education, and vaccines, using positive emotional language like optimism. In contrast, pseudo-experts discuss therapeutics and lockdowns more frequently, employing negative emotions like pessimism and disgust. Negative emotional and moral language tends to drive engagement, but positive language from PHEs fosters positivity in public responses. PHEs exhibit liberal partisanship, expressing more positivity towards liberals and negativity towards conservative elites, while pseudo-experts show conservative partisanship. These findings shed light on the polarization of COVID-19 discourse and underscore the importance of strategic use of emotional and moral language by experts to mitigate polarization and enhance public trust.