On the dynamics of two-dimensional fractional linear control systems

In this paper, we examine the behavior of phase trajectories of fractional two-dimensional linear systems with control. We focus on the double fractional integrator. Fractional differentiation operators are understood in the sense of Hilfer or Hadamard. Admissible controls are assumed to be norm-bounded; we search for them in the functional class $L_\infty[0,T]$, $T>0$. Based on explicitly specified constraints on the norm of a control, we calculate boundary trajectories of the system, which determine on the phase plane a domain in which all admissible trajectories of the system are localized. We show that the solution of the optimal control problem by the method of moments leads to some minimization problem that does not have an analytical solution in the general case (for arbitrary values of the exponents of fractional differentiation in the equations governing the system). We establish conditions under which the minimization problem considered has a solution and determine subdomains of possible localization of this solution. Exact and approximate analytical solutions of the minimization problem are constructed in some particular cases and the results of numerical computation of the minimum are given. The corresponding solutions of the optimal control problem are obtained and phase trajectories of the system are found. All results obtained are analyzed.