Shahshahani gradient-like extremum seeking

In this paper we present novel extremum seeking controllers designed to achieve online constrained optimization of uncertain input-to-output maps of a class of nonlinear dynamical systems. The algorithms, inspired by a class of evolutionary dynamics that emerge in the context of population games, generate on-line Shahshahani gradient-like systems able to achieve extremum seeking under simplex-like constraints on the positive orthant. The method attains semi-global practical convergence to the optimal point, even when the initial conditions are not in the feasible set, and it can be naturally adapted to address distributed extremum seeking problems in multi-agent systems where an all-to-all communication structure is not available. Potential applications include problems on distributed dynamic resource allocation, congestion and flow control in networked systems, as well as portfolio optimization in financial markets. Via simulations, we illustrate our results under different scenarios.