Generalized Markov-Bernstein inequalities and stability of dynamical systems

TL;DR

This paper explores inequalities relating functions and their derivatives for exponential polynomials, linking these to the stability of linear switching systems and providing bounds on discretization steps.

Contribution

It establishes a connection between Markov-Bernstein inequalities and system stability, deriving bounds and monotonicity results for real exponents, and identifying open problems for complex exponents.

Findings

Derived bounds for discretization steps in stability analysis.

Proved monotonicity of sharp constants for real exponents.

Identified open problems for complex exponents.

Abstract

The Markov-Bernstein type inequalities between the norms of functions and of their derivatives are analysed for complex exponential polynomials. We establish a relation between the sharp constants in those inequalities and the stability problem for linear switching systems. In particular, the maximal discretization step is estimated. We prove the monotonicity of the sharp constants with respect to the exponents, provided those exponents are real. This gives asymptotically tight uniform bounds and the general form of the extremal polynomial. The case of complex exponent is left as an open problem.