Time delays are ubiquitous in engineering, physics, and biology and are notorious for reducing performance, potentially degrading stability and robustness. The control strategies designed to counteract these issues are based on mathematical models that are never exact and are subject to hidden dynamics, imprecise parameters, and external perturbations. Hence, the designed controllers must be robust with respect to these inaccuracies, with a quantifiable robustness margin.

In a linear setting, a vast body of literature is available to guarantee this robustness. In a nonlinear context, the most appropriate tool is the notion of input-to-state stability (ISS), which guarantees reasonable transient overshoots and a steady-state error “proportional” to the disturbance magnitude. While the ISS theory benefits from a wide range of tools for ordinary differential equations (ODEs), it is still far from complete for time-delay systems (TDS). In the presence of delays, the state is no longer a point but rather a function (the history of the solution over a time interval): TDS are thus infinite-dimensional. As such, it is possible to exploit the ISS theory developed in an abstract infinite-dimensional setting. Yet, this approach fails to exploit the specific properties of TDS, thus leading to overly conservative results. More crucially, some notions related to the ISS framework are in their infancy or not developed for TDS. These include, in particular, the notions of input-to-output stability (IOS), formalizing the robust stability of the output dynamics, and input/output-to-state stability (IOSS), which can be seen as a robust detectability notion.

Another notion that is still underdeveloped for TDS is incremental stability. This property, related to contraction, essentially imposes that any two solutions eventually tend to one another. This feature can be generalized to the notion of k-contraction, which requires that any k-volume vanishes along the system’s dynamics. As such, incremental stability essentially boils down to 1-contraction. Depending on the order k, k-contraction can be used to determine the uniqueness of fixed points, the absence of limit cycles, or the absence of higher-order chaotic attractors. To date, qualitative analysis tools for k-contraction of TDS are sorely missing, even in the absence of perturbations.

A key motivation to address robust incremental properties and the IOSS notion is their fundamental role in observer design. In particular, incremental IOSS is a necessary condition for the existence of a robust observer. The project thus aims to exploit these properties not only to derive observability conditions for TDS but also to design provably robust observers for TDS.

While most outcomes of the project are of a theoretical and methodological nature, they are driven by a practical application in neuroscience: the selective disruption of targeted brain oscillations with limited impact on the dynamics at other frequencies. This application will not only validate the practical significance of the developed tools but will also serve as a guide to designing easily testable and implementable strategies. In particular, our developments could be decisive in the disruption of Parkinsonian oscillations by closed-loop deep brain stimulation without altering healthy dynamics.