[Paper Review] Banach function spaces and Datko-type conditions for nonuniform exponential stability of evolution families

Nicolae Lupa, Liviu Horia Popescu|arXiv (Cornell University)|Jul 5, 2016

Stability and Controllability of Differential Equations16 references3 citations

TL;DR

This paper introduces a new class of Banach function spaces tailored to capture the nonuniform exponential behavior of evolution families on Banach spaces. By generalizing Datko's classical theorem within this framework, it establishes necessary and sufficient conditions for nonuniform exponential stability using norms intrinsically linked to the family's nonuniform decay properties.

ABSTRACT

For nonuniform exponentially bounded evolution families defined on Banach spaces, we introduce a class of Banach function spaces, whose norms are uniquely determined by the nonuniform behavior of the corresponding evolution family. We generalize the classical theorem of Datko on these spaces.

Motivation & Objective

• To develop a class of Banach function spaces whose norms reflect the nonuniform behavior of evolution families on Banach spaces.
• To extend the classical Datko theorem to nonuniform exponential stability settings.
• To characterize nonuniform exponential stability through functional analytic conditions on these newly defined spaces.
• To establish necessary and sufficient conditions for nonuniform exponential stability using the structure of the introduced function spaces.

Proposed method

• Define a new class of Banach function spaces where the norm is constructed from the nonuniform decay rates of the evolution family.
• Use the evolution family's nonuniform exponential boundedness to induce a norm that encodes its long-term behavior.
• Generalize Datko's theorem by replacing the $ L^2 $-integrability condition with a condition in the newly defined Banach function space.
• Establish equivalence between nonuniform exponential stability and the boundedness of a specific integral operator in the function space.
• Employ spectral and functional analytic techniques to relate the stability properties to the geometry of the function space.
• Demonstrate that the norm of the function space is uniquely determined by the nonuniform behavior of the evolution family.

Experimental results

Research questions

• RQ1 How can Banach function spaces be constructed such that their norms reflect the nonuniform decay behavior of evolution families?
• RQ2 What is the appropriate generalization of Datko's theorem for nonuniform exponential stability?
• RQ3 Under what conditions is nonuniform exponential stability equivalent to boundedness in the introduced function space?
• RQ4 Can the new function space framework yield necessary and sufficient conditions for nonuniform exponential stability?
• RQ5 How does the structure of the function space relate to the underlying evolution family's nonuniform exponential boundedness?

Key findings

• The proposed Banach function spaces are uniquely determined by the nonuniform behavior of the evolution family, ensuring a direct link between the space's norm and the family's decay properties.
• A generalized Datko-type theorem is established, showing that nonuniform exponential stability is equivalent to the boundedness of a specific integral operator in the new function space.
• The new framework provides necessary and sufficient conditions for nonuniform exponential stability, extending classical results beyond the uniform case.
• The function space norm is explicitly constructed from the evolution family's nonuniform exponential bounds, ensuring intrinsic relevance to stability analysis.
• The method successfully generalizes the classical Datko theorem to nonuniform settings, offering a functional analytic tool for stability analysis in nonuniformly bounded systems.