Robust stabilization of delayed neural fields with partial measurement and actuation

Simulation of Closed-Loop Deep Brain Stimulation Control Schemes for Suppression of Pathological Beta Oscillations in Parkinson’s Disease

“…Due to the dynamic, non-linear nature of the parkinsonian neuromuscular system, identified models or controller parameters which were initially suitable during controller tuning may become unsuitable or provide suboptimal performance during different tasks, times throughout the day or as the disease progresses. Advanced adaptive techniques which automatically update autoregressive model coefficients or controller gains may be required to overcome this limitation (Cameron and Seborg, 1984;Santaniello et al, 2011;Chaillet et al, 2017). Nevertheless, the PI parameter rule-tuning approach presented in this study provides improved performance over currently tested closed-loop controllers, is simple to implement in a clinical setting and adheres to clinical considerations.…”

Incremental stability of spatiotemporal delayed dynamics and application to neural fields

“…Consequently, the map t → V (x t ) is also absolutely continuous, ensuring that the negativity ofV for almost all t in an interval [t 1 ; t 2 ] implies V (x t2 ) < V (x t1 ). Reasoning as in [32], it might be possible to extend the results presented here to merely measurable locally essentially bounded inputs: see [7] for a spatiotemporal version of this observation.…”

“…Qualitative behavior can also be assessed through bifurcation theory [4], [39]. Delayed neural fields have also been the subject of stability and robustness studies, using linearization techniques [3] and a spatiotemporal extension of the Lyapunov-Krasovskii approach [16], [40], [7]. This mathematical background constitutes a fertile ground for the analysis of incremental stability of delayed neural fields, and particularly their ability to be entrained by a periodic input.…”

Monotonicity Methods for Input-to-State Stability of Nonlinear Parabolic PDEs with Boundary Disturbances

“…we are in a position to study an equivalent to (21) initial boundary value problem ∂ y Therefore, using the transformation (23), we conclude that for every x 0 ∈ H 2m+1 (G) and for every input u ∈ R + × ∂ G → R being the trace of a function ν ∈ C 0 (R + × G) C 1,2 ((0, +∞) × G) and satisfying (p-i), (p-ii), also the initial boundary value problem (21) has a unique solution x ∈ CL. Hence, (21) defines a control system with • X := H 2m+1 (G) with · p -norm, for any fixed p ∈ (21). Notice that X contains the constant functions.…”