Stability and stabilisability of the singularly perturbed system with delay on time scales: a decomposition approach

Asymptotic Analysis of Spectrum and Stability for One Class of Singularly Perturbed Neutral-Type Time-Delay Systems

“…Applying the formula for the determinant of a block matrix (see [37]) to (31), and taking into account the inequality (12) and Equation ( 14), we obtain for any complex λ:…”

“…In [11], the exact slow-fast decomposition was proposed for a linear singularly perturbed time-invariant system having multiple point-wise commensurate non-small delays only in the slow state variable, while the fast state variable is delay free. In [12], a singularly perturbed linear time-invariant system with delay on time scales was considered. The delay in the form of Stieltjes integral appears only in the slow state variable.…”

“…The exact slow-fast decomposition of this system was proposed in this work. For the asymptotic slow-fast decomposition of singularly perturbed time-delay differential systems of both, non neutral and neutral, types one can see e.g., [1,3,[12][13][14][15][16][17][18][19][20] and references therein.…”

“…In [11], the spectrum analysis of a singularly perturbed linear time-invariant system with multiple point-wise commensurate non-small delays only in the slow state variable was carried out by using the exact slow-fast decomposition of this system. In [12], the spectrum of a singularly perturbed linear time-invariant system with the delay (in the form of Stieltjes integral) only in the slow state variable was analyzed based on the exact slow-fast decomposition of this system.…”

“…A similar stabilization problem by a linear state-feedback control with both, small and non-small, state delays was studied in [33] using the frequency domain approach. The exponential stability and stabilization of a singularly perturbed system with the delay (in the form of Stieltjes integral) only in the slow state variable were studied in [12] by application of the exact slow-fast decomposition of this system. In [16,30,[34][35][36], the exponential/asymptotic stability and stabilization problems for various singularly perturbed time-delay systems were studied in the framework of the Linear Matrix Inequalities method.…”