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

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Mironchenko, Andrii:
Well-posedness and properties of the flow for semilinear evolution equations.
In: Mathematics of Control, Signals, and Systems. Bd. 36 (2024) Heft 3 . - S. 483-523.
ISSN 0932-4194
DOI: https://doi.org/10.1007/s00498-023-00378-x

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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 C₀-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.