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Conditions for asymptotic stability of first order scalar differential-difference equation with complex coefficients

Kapica Rafał, Zawiski Radosław

Archives of Control Sciences

2023

Vol. 33, No. 3

607--629

We investigate a scalar characteristic exponential polynomial with complex coefficients associated with a first order scalar differential-difference equation. Our analysis provides necessary and sufficient conditions for allocation of the roots in the complex open left half-plane what guarantees asymptotic stability of the differential-difference equation. The conditions are expressed explicitly in terms of complex coefficients of the characteristic exponential polynomial, what makes them easy to use in applications. We show examples including those for retarded PDEs in an abstract formulation.

Random Iteration with Place Dependent Probabilities

Kapica Rafał, Ślęczka Maciej

Probability and Mathematical Statistics

2020

Vol. 40, Fasc. 1

119--137

We consider Markov chains arising from random iteration of functions Sθ : X → X, θ ϵ Θ, where X is a Polish space and Θ is an arbitrary set of indices. At x ϵ X, θ is sampled from a distribution ϑx on Θ, and the ϑx are different for different x. Exponential convergence to a unique invariant measure is proved. This result is applied to the case of random affine transformations on Rd, giving the existence of exponentially attractive perpetuities with place dependent probabilities.