Boundedness of Solutions of Conformable Fractional Equations of Perturbed Motion*
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International Applied Mechanics, Vol. 56, No. 5, September, 2020
BOUNDEDNESS OF SOLUTIONS OF CONFORMABLE FRACTIONAL EQUATIONS OF PERTURBED MOTION*
À. À. Martynyuk and Yu. A. Martynyuk-Chernienko
The results of analyzing the boundedness of the solutions of nonlinear systems with conformable fractional derivative of the state vector are discussed. The solutions are estimated and their boundedness conditions are established using the method of integral inequalities. Systems subject to constant perturbations are considered as an example. Keywords: nonlinear conformable fractional system of equations, method of integral inequalities, boundedness of solutions Introduction. The concept of fractional derivative a continuous function stemmed from the following question raised in 1695 by de L’Hopital to Leibniz: What if n = 1/2 in d n x / dt n ? Leibniz’s response was “This is an apparent paradox from which, one day, useful consequences will be drawn” (see [9] and the references therein). The interest in equations with fractional derivatives [4, 8, 10, 14, 16, 23] increased in the last two decades is due to the capability of more accurate description of processes in some models of real phenomena. Namely, it was proposed to use control laws containing fractional-order derivatives in the theory of automatic control [5, 7]. It was shown that the best way to effectively control fractional systems is to use fractional controllers. Another example of fractional systems is the three-dimensional equation of heat transfer with edge control [3]. The experimental data obtained by Schmidt and Drumheller [24] indicate that the current flowing through a capacitor is proportional to a noninteger derivative of the voltage. In rheology, when viscoelastic substances are used as insulators or vibration dampers, fractional derivatives are suitable for more accurate description of damping in the system, reducing the number of the parameters of the model [25, 26]. The most popular definitions of fractional derivative are Riemann–Liouville, Hadamard, Grünwald–Letnikov [23, 10]. In 1969, Caputo proposed a new definition of fractional derivative [6], which expanded the capabilities of the analysis of the equations of perturbed motion with Caputo fractional derivative of the state vector. The results on equations of this type obtained before 2009 are reported in the monograph [12]. In [2, 11], the term “conformable fractional derivative” was proposed. We will use it below. The present paper continues the studies on conformable fractional equations of perturbed motion started in [1, 15, 17–20, 27], where Lyapunov’s second method was generalized to this class of equations of perturbed motion, the principle of comparison with scalar and vector Lyapunov functions was formulated, the conditions of practical stability with respect to manifolds of solutions of conformable fractional equations and the conditions of practical stability for impulsive systems with
S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, 3 Nesterov S
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