Well-posedness and properties of the flow for semilinear evolution equations

TL;DR

This paper establishes conditions for the well-posedness and key properties of the flow in semilinear evolution equations, aiding stability and robustness analysis of boundary control systems with disturbances.

Contribution

It provides new sufficient conditions for well-posedness and flow properties in semilinear evolution equations with unbounded input operators, including boundary control systems.

Findings

Conditions for well-posedness of semilinear evolution equations.

Sufficient criteria for flow map properties like Lipschitz continuity.

Application to Burgers' equation with disturbances.

Abstract

We derive conditions for well-posedness of semilinear evolution equations with unbounded input operators. Based on this, we provide sufficient conditions for such properties of the flow map as Lipschitz continuity, bounded-implies-continuation property, boundedness of reachability sets, etc. These properties represent a basic toolbox for stability and robustness analysis of semilinear boundary control systems. We cover systems governed by general $C_0$-semigroups, and analytic semigroups that may have both boundary and distributed disturbances. We illustrate our findings on an example of a Burgers' equation with nonlinear local dynamics and both distributed and boundary disturbances.