Stability of abstract dynamic equations on time scales

By Alaa E. Hamza, K. Oraby. 2012

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AbstractIn this paper, we investigate many types of stability, like uniform stability, asymptotic stability, uniform asymptotic stability, global stability, global asymptotic stability, exponential stability, uniform exponential stability, of the homogeneous first-order linear dynamic equations of the form xΔ(t)=Ax(t),t>t0,t,t0∈Tx(t0)=x0∈D(A), where A is the generator of a C0-semigroup {T(t):t∈T}⊂L(X), the space of all bounded linear operators from a Banach space X into itself. Here, T⊆R≥0 is a time scale which is an additive semigroup with the property that a−b∈T for any a,b∈T such that a>b. Finally, we give an illustrative example for a nonregressive homogeneous first-order linear dynamic equation and we investigate its stability.

Subfields: Applied Mathematics, Control and Systems Engineering

View Paper: https://www.semanticscholar.org/paper/7b49055b9bfdb1a4e40aacb2d7539022a778c604