Numerical Construction of LISS Lyapunov Functions under a Small Gain Condition

In the stability analysis of large-scale interconnected systems it is frequently desirable to be able to determine a decay point of the gain operator, i.e., a point whose image under the monotone operator is strictly smaller than the point itself. The set of such decay points plays a crucial role in checking, in a semi-global fashion, the local input-to-state stability of an interconnected system and in the numerical construction of a LISS Lyapunov function. We provide a homotopy algorithm that computes a decay point of a monotone op- erator. For this purpose we use a fixed point algorithm and provide a function whose fixed points correspond to decay points of the monotone operator. The advantage to an earlier algorithm is demonstrated. Furthermore an example is given which shows how to analyze a given perturbed interconnected system.Comment: 30 pages, 7 figures, 4 table

Existence of periodic solutions of the FitzHugh-Nagumo equations for an explicit range of the small parameter

The FitzHugh-Nagumo model describing propagation of nerve impulses in axon is given by fast-slow reaction-diffusion equations, with dependence on a parameter $\epsilon$ representing the ratio of time scales. It is well known that for all sufficiently small $\epsilon>0$ the system possesses a periodic traveling wave. With aid of computer-assisted rigorous computations, we prove the existence of this periodic orbit in the traveling wave equation for an explicit range $\epsilon \in (0, 0.0015]$. Our approach is based on a novel method of combination of topological techniques of covering relations and isolating segments, for which we provide a self-contained theory. We show that the range of existence is wide enough, so the upper bound can be reached by standard validated continuation procedures. In particular, for the range $\epsilon \in [1.5 \times 10^{-4}, 0.0015]$ we perform a rigorous continuation based on covering relations and not specifically tailored to the fast-slow setting. Moreover, we confirm that for $\epsilon=0.0015$ the classical interval Newton-Moore method applied to a sequence of Poincar\'e maps already succeeds. Techniques described in this paper can be adapted to other fast-slow systems of similar structure

Continuation-Based Pull-In and Lift-Off Simulation Algorithms for Microelectromechanical Devices

The voltages at which microelectromechanical actuators and sensors become unstable, known as pull-in and lift-off voltages, are critical parameters in microelectromechanical systems (MEMS) design. The state-of-the-art MEMS simulators compute these parameters by simply sweeping the voltage, leading to either excessively large computational cost or to convergence failure near the pull-in or lift-off points. This paper proposes to simulate the behavior at pull-in and lift-off employing two continuation-based algorithms. The first algorithm appropriately adapts standard continuation methods, providing a complete set of static solutions. The second algorithm uses continuation to trace two kinds of curves and generates the sweep-up or sweep-down curves, which can provide more intuition for MEMS designers. The algorithms presented in this paper are robust and suitable for general-purpose industrial MEMS designs. Our algorithms have been implemented in a commercial MEMS/integrated circuits codesign tool, and their effectiveness is validated by comparisons against measurement data and the commercial finite-element/boundary-element (FEM/BEM) solver CoventorWare