Extremum-seeking boundary control for Schrödinger-type PDEs

This paper addresses the design and analysis of an extremum-seeking control (ESC) methodology for scalar static maps in the context of infinite-dimensional dynamics governed by complex-valued partial differential equations (PDEs) of Schrödinger type. The system is actuated at one boundary, while the complex-valued state at the uncontrolled boundary defines the input to an unknown real-valued performance map, locally modeled by a quadratic functional. An isomorphism between the complex Hilbert space and its two-dimensional real-valued representation is established to enable the use of the standard multivariable Newton-based ESC. To compensate for the PDE actuation dynamics, a boundary control strategy based on a two-step backstepping procedure is employed. With a perturbation-based estimate of the Hessian’s inverse, the local exponential convergence to a neighborhood of the unknown extremum point is proved. A numerical example illustrates the effectiveness of the proposed extremum-seeking methodology.