On control problems for fractional systems with aftereffect

Abstract

We consider dynamical models with aftereffect described by functional--differential equations with fractional derivatives. These models encompass processes, in which the system state may change abruptly at certain points in time, which is interpreted as the result of impulse effects (shocks). The trajectories of such systems may have discontinuities at certain points in time, and between these points the behavior of system is described by differentiable functions, which satisfy the equation in the usual sense. We pose a general control problem for a given system. We formulate solvability conditions for this problem in the class of impulse controls, $L_2$–controls, and their hybrids. The proposed approach to studying systems with fractional derivatives is based on the systematic use of abstract functional–differential equation theory and offers certain advantages for studying systems and processes with aftereffects.