Real-world systems have to operate in uncertain and changing environments, thus the necessity to design feedback controllers which can adapt to such uncertainties. Controllers which are designed to satisfy this goal are called adaptive controllers. We can classify adaptive control into three main subclasses, for example, Reference 1: Classical model-based adaptive control, which mainly uses models of the controlled system; learning-based adaptive control, which uses both model-based and data-driven techniques to design modular adaptive controllers; and data-driven adaptive control, which is based on the interaction of the controller with the system. In this special issue, we shall focus on a specific class of data-driven adaptive controllers, namely, extremum seekers.Recently, there have been a lot of efforts in this direction of control. One of the reasons behind this increasing interest is that, following the pioneering ESC analysis paper, 2 the field of ESC has reached a certain maturity, which has led to good analysis and understanding of the main properties of the available algorithms. The idea behind ESC is to design a closed-loop dynamical system, such that its flow converges to the optimum of a given static or dynamic cost function, for example, References 3-5. This type of optimization algorithms is also referred to as continuous optimization, for example, Reference 6. The advantages of such optimization approaches comparatively to classical optimization algorithms is the fact that extremum seekers do not require a closed-form expression of the cost function. Moreover, they do not require multiple evaluation of the cost function at each point to estimate the gradient (or higher order derivatives) of the cost function, since they include some internal states which converge to an estimate of the derivatives of the cost over time. Besides, their formulation as a closed-loop control problem allows us to use numerous tools from dynamical systems theory and control theory, for example, time-scale separation, averaging theory, Lyapunov stability theory, to perform rigorous analysis of their convergence and robustness properties w.r.t. initial conditions changes and other hyperparameters tuning.In this special issue, we have included 10 papers, which could be divided into two main types. The first type, more theoretical, presents new extremum seeking methods designed for different dynamical systems, from systems with delays, and multiagent systems, to distributed systems. The second type of papers deals with real-world applications of extremum seeking controllers to challenging systems, ranging from microalgae cultivation control to heating and ventilation systems. These papers are compiled in a special virtual issue of the journal at the journal homepage. To access all of the papers, please follow the following link (https://onlinelibrary.wiley.com/toc/10991115/2021/35/7).For instance, Reference 7 presents a novel ESC for batch-to-batch optimal control of unknown time-varying systems. The authors show that the ...
