Positivity and monotonicity methods in infinite-dimensional systems theory
Final Report Abstract
The project aimed to explore how structural properties such as positivity and monotonicity – wherever applicable – influence the behaviour and analysis of infinite-dimensional systems – systems that arise in models involving partial differential equations, delay equations, and other functional-analytic formulations. These systems are often used to describe processes where key quantities, such as temperature, population density, or chemical concentrations, must remain non-negative over time. While positivity and monotonicity have been well studied in finitedimensional systems, their roles in infinite-dimensional settings, particularly in the presence of inputs and outputs, are much less understood or amenable to positivity-based methods. The proposed research focused on three central themes, namely, the admissibility of control and observation operators, the use of monotonicity techniques in input-to-state stability (ISS) analysis, the stability, and the spectral and asymptotic behaviour of systems exhibiting positivity. The project led to several developments that extended and, in some cases, shifted the original research focus: • An initial study on the limiting case admissibility of observation and control operators associated with positive systems revealed the need for a more refined understanding of the order structure in extrapolation spaces – mathematical constructions that extend the system’s state space to accommodate boundary control. This led to new results on the lattice structure of negatively ordered Sobolev spaces, contributing to the broader theory of ordered Banach spaces. • In the course of analysing admissibility, fundamental limitations were discovered in existing duality results for semigroups whose duals lack strong continuity. As a result, a generalised form of duality theory was developed beyond the context of positive systems. • New insights were gained into energy-based models associated with hyperbolic PDEs, through an investigation of their semi-uniform stability. This work provided a quantitative framework for describing how energy dissipates over time, further enriching the stability theory for structured infinite-dimensional systems. • The concept of input-to-state stability in integral norms was extended to infinite-dimensional systems. Originally studied in finite-dimensional non-linear control theory, this integral form of stability allows for a more flexible analysis of how external disturbances affect the system over time. • Another line of research focused on the robustness of semigroup properties under perturbations. A class of relatively bounded perturbations was shown to preserve certain smoothing effects (ultracontractivity), broadening the scope of known perturbation results. The project led to a number of new mathematical findings and initiated several lines of research that extend beyond the original project scope. These results deepen the theoretical understanding of infinite-dimensional systems and lay the groundwork for future developments in control, stability, and functional analytic methods.
Publications
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The lattice structure of negative Sobolev and extrapolation spaces.
S. Arora, J. Gluck & F. Schwenninger
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Admissible operators for sun-dual semigroups. Mathematics of Control, Signals, and Systems, 38(1), 103-125.
Arora, Sahiba & Schwenninger, Felix L.
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Input-to-state stability in integral norms for linear infinitedimensional systems
S. Arora & A. Mironchenko
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Limit-case admissibility for positive infinite-dimensional systems. Journal of Differential Equations, 440, 113435.
Arora, Sahiba; Glück, Jochen; Paunonen, Lassi & Schwenninger, Felix L.
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A universal example for quantitative semi‐uniform stability. Journal of the London Mathematical Society, 113(2).
Arora, Sahiba; Schwenninger, Felix L.; Vukusic, Ingrid & Waurick, Marcus
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Smoothing of operator semigroups under relatively bounded perturbations. Journal of Mathematical Analysis and Applications, 130640.
Arora, Sahiba & Mui, Jonathan
