Adaptive output feedback for hyperbolic PDE pairs with non-local coupling

We solve the problem of adaptive output-feedback stabilization for one-dimensional 2 × 2 hyperbolic partial integro-differential equations (PIDEs) with spatially-varying coupling coefficients and non-local (in space) terms. The functional coefficients of the system are assumed unknown. Control is applied at one boundary, and measurements are taken from both boundaries. To deal with the absence of both full-state measurement and parameter knowledge, we use a backstepping pre-transformation of the system into an observer canonical form. Whereas the original plant had products and integrals of unknown coefficients and unmeasured states, the canonical form has two unknown spatially-varying parameters multiplied by the measured output. For state estimation, we introduce an explicit state observer involving the delayed values of the input over one unit of time and the output over two units of time, which enables us to design an output-feedback controller. The parametric model is in the form of an integral equation relating delayed values of the input and output. Based on this model, we employ gradient-based parameter estimators. For the closed-loop system we establish boundedness of all signals, pointwise in space and time, and convergence of the PDE state to zero pointwise in space.