2019/05/28 by Arsalan Sharifnassab, Sharifnassab, Arsalan, S. Jamaloddin Golestani +1
Biochemistry, Genetics and Molecular Biology · Engineering · Environmental Science · Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Control and Stability of Dynamical Systems #Ecosystem dynamics and resilience #FOS: Electrical engineering #Gene Regulatory Network Analysis #Mathematical Biology Tumor Growth #Micro and Nano Robotics #Systems and Control (eess.SY) #electronic engineering #information engineering
paper · pdf · doi:10.48550/arxiv.1905.11746
openalex publication_date 2019/05/28 · openalex created_date 2022/07/29 · openalex updated_date 2026/07/28
We investigate sensitivity to cumulative perturbations for a few dynamical\nsystem classes of practical interest. A system is said to have bounded\nsensitivity to cumulative perturbations (bounded sensitivity, for short) if an\nadditive disturbance leads to a change in the state trajectory that is bounded\nby a constant multiple of the size of the cumulative disturbance. As our main\nresult, we show that there exist dynamical systems in the form of (negative)\ngradient field of a convex function that have unbounded sensitivity. We show\nthat the result holds even when the convex potential function is piecewise\nlinear. This resolves a question raised in [1], wherein it was shown that the\n(negative) (sub)gradient field of a piecewise linear and convex function has\nbounded sensitivity if the number of linear pieces is finite. Our results\nestablish that the finiteness assumption is indeed necessary.\n Among our other results, we provide a necessary and sufficient condition for\na linear dynamical system to have bounded sensitivity to cumulative\nperturbations. We also establish that the bounded sensitivity property is\npreserved, when a dynamical system with bounded sensitivity undergoes certain\ntransformations. These transformations include convolution, time\ndiscretization, and spreading of a system (a transformation that captures\napproximate solutions of a system).\n