The primary concern of the work is to develop adaptive identification approach to uncertain dynamic systems governed by nonlinear partial differential equations. The well-known nonlinear sine-Gordon PDE model, which describes distributed wave dynamics (e.g., of continuum of interacting oscillators), serves as a testbed. It is shown that the sine-Gordon model with uncertain external force and viscous friction, which are distributed in space, is identifiable over the entire state measurement provided that it is excited by a specific nonzero input. Numerical simulations support the theory in the case where unknown plant parameters are spatially invariant.

As well-known, prescribed-time stabilizing design faces the need of using time-varying high gains which escape to infinity as time approaches the desired instant. In the presence of measurement noise, the corresponding state response is also significantly amplified that leads to the lack of robustness in the closed-loop implementation. In order to eliminate this drawback, the implicit Euler discretization of the closed-loop in question is recently developed in where desired robustness properties are conserved beyond the prescribed-time interval while also bounded state dynamics are ensured in the presence of measurement noise. Along this line, stabilizing prescribed-observer-based output feedback algorithms and their digital implementation are reviewed. For tutorial value, the underlying state feedback and observer designs are recalled side by side in continuous- and discrete-time perspectives, followed by the desired output feedback design. Open problems, calling for future investigation, conclude the review.