FIRST ONLINE CONFERENCE ON MODERN FRACTIONAL CALCULUS AND ITS APPLICATIONS (OCMFCA-2020), İstanbul, Turkey, 5 - 06 December 2020, pp.1
We discuss here the stabilization problem for the 'toy' model dv(t)/dt = u(t) - bv(t), where b is a positive constant, with an arbitrary initial condition v(0). The model represents the second Newtonian law for the velocity v accelerated with the engine force u (the control parameter) together with the viscous friction force bv. The task is to stabilize the velocity at the constant level v*. To do it, one can form a Kolesnikov's subset attracting the phase trajectories to the neighbourhood of the function psi = v(t)-v* via the construction of the appropriate feedback signal u. Kolesnikov's algorithm provides the exponential convergence, with the positive constant T (the typical time scale for the control), but in the same time it demands the permanent power support P(t)=v(t)u(t) pumping the energy to the system even when the control goal v* is achieved.
To decrease the power cost of Kolesnikov's control, let's re-formulate the feedback in the form of Caputo's fractional derivative. In this case the solution to the ODE together with the feedback control signal u could be found with Rida-Arafa method based on the generalized Mittag-Leffler function.
To minimize the asymptotic of the power P(t) as tends to infinity, one can evaluate the parameter alpha of the fractional dimension. Thus, for the certain constrain over the initial condition and the target stabilization level, the integer-dimensional Kolesnikov algorithm can be replace with the fractional target attractor feedback to provide the minimal power cost.