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IMA Journal of Mathematical Control and Information 2009 26(2):213-229; doi:10.1093/imamci/dnp010
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© The author 2009. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Stability of abstract non-linear non-autonomous difference-delay equations

Rigoberto Medina{dagger}

Departamento de Ciencias Exactas, Universidad de Los Lagos, Casilla 933, Osorno, Chile

{dagger} Email: rmedina{at}ulagos.cl

Received on November 1, 2007; Revision received March 20, 2009. Accepted on April 19, 2009

We consider a class of functional–difference equations in a Banach space with causal operators, which have the local Lipschitz property. Sufficient conditions for the boundedness of solutions and the Lyapunov stability are given. We offer some explicit stability criteria when the equations are defined on separable Hilbert spaces. Our approach is based on the representation of solutions combined with new norm estimates for operator-valued functions. Applications of the main results of the paper to the stability of abstract discrete-time control systems are especially investigated.

Keywords: abstract discrete-time systems; uniform stability; input-to-state stability; robust Lyapunov stability; causal operators; discrete evolution family; small-gain theorems.


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