Cone-type constrained relative controllability of semilinear fractional systems with delays

Abstract

Cone-type constrained relative controllability of semilinear fractional systems with delays Kybernetika, Vol. 53 (2017) CONE-TYPE CONSTRAINED RELATIVE CONTROLLABILITY OF SEMILINEAR FRACTIONAL SYSTEMS WITH DELAYS Beata Sikora and Jerzy KlamkaThe paper presents fractional-order semilinear, continuous, finite-dimensional dynamical systems with multiple delays both in controls and nonlinear function f . The constrained relative controllability of the presented semilinear system and corresponding linear one are discussed. New criteria of constrained relative controllability for the fractional semilinear systems with delays under assumptions put on the control values are established and proved. The conical type constraints are considered. The results are illustrated by an example.

On the constrained and unconstrained controllability of semilinear Hilfer fractional systems

“…The two-parameter family of the Hilfer fractional derivatives is a generalization of both the Caputo derivative and the Riemann-Liouvile derivative (definitions of both the derivatives can be found in [30]; applications to control theory can be found in [18,20,33,38,40,44], among others). The Hilfer derivative allows us to interpolate between the fractional derivatives mentione above.…”

On application of Rothe's fixed point theorem to study the controllability of fractional semilinear systems with delays

“…For time-delay systems, only the complete state z(t) = (x(t), u t (s)), where u t (s) = u(s) for s ∈ [t − h M (t), t), completely describes the behavior of the system at time t. Theorem 3.1. (Sikora and Klamka [38]) For the given initial conditions z(0…”

“…Controllability problems for linear continuoustime fractional systems with delayed control are analyzed in [4,7,22,35,36,37,42]. Semilinear and nonlinear fractional-order systems with delays are discussed, among others, in [38] where the Frechet derivative method is applied, in [5,8] and [27] the Schauder fixed point theorem is used, in [33] the Schaefer fixed point theorem is considered.…”

Approximate controllability of impulsive system involving state-dependent delay and variable delay in control via fundamental solution

“…Klamka [12,13] discussed controllability of linear system involving delays in control. Sikora and Klamka [29,30] developed some interesting results by assuming constrained controls (that is the control functions are restricted to take their values in a prescribed admissible set) for linear and semilinear fractional systems with multiple delays in control in finite dimensional spaces. Balachandran [3] concerned with relative controllability of dynamical control system involving delay in control.…”