5 research outputs found
An Overview of Integral Quadratic Constraints for Delayed Nonlinear and Parameter-Varying Systems
A general framework is presented for analyzing the stability and performance
of nonlinear and linear parameter varying (LPV) time delayed systems. First,
the input/output behavior of the time delay operator is bounded in the
frequency domain by integral quadratic constraints (IQCs). A constant delay is
a linear, time-invariant system and this leads to a simple, intuitive
interpretation for these frequency domain constraints. This simple
interpretation is used to derive new IQCs for both constant and varying delays.
Second, the performance of nonlinear and LPV delayed systems is bounded using
dissipation inequalities that incorporate IQCs. This step makes use of recent
results that show, under mild technical conditions, that an IQC has an
equivalent representation as a finite-horizon time-domain constraint. Numerical
examples are provided to demonstrate the effectiveness of the method for both
class of systems
Exit time asymptotics for small noise stochastic delay differential equations
Dynamical system models with delayed dynamics and small noise arise in a
variety of applications in science and engineering. In many applications,
stable equilibrium or periodic behavior is critical to a well functioning
system. Sufficient conditions for the stability of equilibrium points or
periodic orbits of certain deterministic dynamical systems with delayed
dynamics are known and it is of interest to understand the sample path behavior
of such systems under the addition of small noise. We consider a small noise
stochastic delay differential equation (SDDE) with coefficients that depend on
the history of the process over a finite delay interval. We obtain asymptotic
estimates, as the noise vanishes, on the time it takes a solution of the
stochastic equation to exit a bounded domain that is attracted to a stable
equilibrium point or periodic orbit of the corresponding deterministic
equation. To obtain these asymptotics, we prove a sample path large deviation
principle (LDP) for the SDDE that is uniform over initial conditions in bounded
sets. The proof of the uniform sample path LDP uses a variational
representation for exponential functionals of strong solutions of the SDDE. We
anticipate that the overall approach may be useful in proving uniform sample
path LDPs for a broad class of infinite-dimensional small noise stochastic
equations.Comment: 39 page