Practical stability in terms of two measures for fractional order dynamic systems in Caputo's sense with initial time difference

Impulsive Fractional-Like Differential Equations: Practical Stability and Boundedness with Respect to h-Manifolds

“…In this section, we will state our main practical stability and boundedness criteria for impulsive systems with FLDs. These results extend and generalize the results in [29,[31][32][33][34]36,40,41] for different classes of differential, functional differential and fractional differential equations, and are first contributions to the stability theory of impulsive equations with FLDs.…”

“…Impulsive systems with conformable derivatives have been considered only in [38,39], where some oscillation criteria and inequalities are proposed. However, the above papers did not offer stability results.More presently, some results on practical stability theory for impulsive fractional differential systems with Caputo fractional derivatives have been presented in [40,41]. However, the mentioned studies do not consider FLDs.On the other hand, the stability with respect to sets or the manifold concept generalizes the idea of stability of a system [42][43][44][45].…”

“…More presently, some results on practical stability theory for impulsive fractional differential systems with Caputo fractional derivatives have been presented in [40,41]. However, the mentioned studies do not consider FLDs.…”

Fractional Lotka-Volterra-Type Cooperation Models: Impulsive Control on Their Stability Behavior

“…In Theorem 2 we proved that even in some concrete cases, when the system (5) is not globally Mittag-Leffler stable, by the impulsive controller (8) we can made the vicinity of a state arbitrarily small so that all other states to remain arbitrarily close to it provided that their initial functions were sufficiently near the initial data of the particular state . The property of practical stability is very important in many applications when the Mittag-Leffler stability is conservative [ 58 , 60 , 61 ]. For type (2) models from the population biology this can compensate a not stable initial population by keeping the population densities between particular bounds for .…”

“…The concepts of practical stability [ 55 , 56 ] and stability of sets [ 57 ] have been applied to fractional-order systems [ 58 , 59 , 60 , 61 ]. Using a practical stability approaches is beneficial in the cases when the equilibrium of the considered system is unstable in the classical Lyapunov or Mittag-Leffler sense, but its performance may be sufficient for a particular application [ 56 ].…”