Stability analysis of quasilinear systems on time scale based on a new estimation of the upper bound of the time scale matrix exponential function
Résumé
In this paper, we analyze the stability of quasilinear systems on time scale based on a new estimation of the upper bound of the time scale matrix exponential function. First, some upper bounds for the norm of the matrix
exponential function eA(t, t0) are derived, where A is a regressive square matrix, t, t0 ∈ T, T being an arbitrary nonempty closed subset of the set of real numbers R, which is called a time scale. The matrix exponential
function generalizes the usual matrix exponential as well as the integer power of a matrix: if T = R and A is a constant matrix, then eA(t, t0) = eA(t−t0), while if T = Z and I + A is invertible, then eA(t, t0) = (I + A)(t−t0). Using
Putzer algorithm and estimating the maximum value of a function defined and bounded on a time scale T, we derive an upper bound for eA(t, t0) in the form: ∥eA(t, t0)∥ ≤ Keα(t, t0), where t, t0 ∈ T, K is a positive constant, α(t)
is a function defined on T and eα(t, t0) is a scalar exponential function. One of the upper bounds is used for stability investigation of quasilinear systems on a time scale.
Domaines
Automatique
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