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Extremum seeking, an online model-free optimization algorithm with traditionally exponential convergence, was recently advanced by Poveda and coauthors to fixed-time convergence, using nonsmooth time-invariant feedback. In this article, we introduce an alternative time-varying prescribed-time extremum seeking (PT-ES) approach to reaching the optimum in a user-assignable “prescribed” time (PT), independent of the initial condition of the estimator. Instead of conventional sinusoidal probing signals, we employ “chirpy” perturbations, i.e., sinusoids with growing frequencies. In addition to providing a result for a static input–output map, we provide algorithms for a map in cascade with a delay, a wave partial differential equation (PDE), and a heat/diffusion PDE. The designs are based on the time-varying PT backstepping approach, which transforms the PDE–ordinary-differential-equation cascade into a suitable PT-stabilized target system, and on averaging-based estimates of the gradient and Hessian of the map. Classical averaging does not apply for the analysis of PT-ES algorithms for four reasons: neither is probing periodic, nor is the “average system” exponentially stable, nor is the original system finite-dimensional, nor is the interval of operation infinite. We develop an averaging technique needed for this non-periodic finite-time infinite-dimensional problem. Along with PDE Lyapunov analysis, we show that the input converges to the optimizer in PT. We present three numerical examples to illustrate the effectiveness of the proposed technique for delay-free case as well as for compensation of time-delay and diffusion PDE dynamics.

A cascade connection comprising a delay, a partial differential equation (PDE), and an ordinary differential equation (ODE) is referred to as “distributed” if an integral operator of the functional state of the delay or the PDE enters, as a scalar, the next system in the cascade, or if the ODE state acts as an input to the PDE throughout the PDE's domain. While boundary control of systems involving PDEs or delays typically calls for the PDE backstepping approach, such “distributed” interconnections call for the PDE forwarding approach. The cascaded triple consisting of a delay, a parabolic PDE, and an ODE can constitute nine (3 × 3) distinct interconnections and each of the nine structures is “doubly distributed” if both of its two connections are distributed. We focus on the most interesting three of the nine possible structures and present mixed backstepping-forwarding designs of controllers and observers. We first propose controllers to exponentially stabilize delay-PDE-ODE and PDE-delay-ODE cascades. Then, we introduce an observer to estimate the states of PDE-ODE systems with sensor delays. Besides advancing the backstepping-forwarding design for “doubly-distributed” systems, we do so with the aid of the input-to-state stability approach for parabolic PDEs. This enables proving stability without constructing Lyapunov functions, as well as proving stability in both square-integral and supremum spatial norms. Our results are novel even for the particular cases of delay-free cascades. Simulations illustrate our theory.