Positional parring impulse disturbances in a control problem for linear systems with aftereffect

Dynamic models under consideration cover a wide class of models in mathematical Economics and Ecology taking into account some aftereffects. For such models, control problems are considered in a general case that the purpose of controlling is prescribed with the use of a finite set of linear functionals. The number of these functionals is independent of the dimension of the system. The system is subject to impulse disturbances which result in trajectory jumps with unknown previously time moments and values. A construction of regular (not impulsive) control is proposed wich solves the control problem despite impulse disturbances. It is assumed that information on the impulses occurred becomes known by the time of the feedback correction control that depends on the jumps values. To solve the control problem, a class of controls is described that contains both program and jumps-positional components. The proposed approach to the positional parring of impulse disturbances is essentially based on the fundamental results of contemporary theory of functional differential equations, among which are the theorems on representation of solutions to linear systems with aftereffect, the properties of Cauchy''s matrix, conditions of the solvability for control problems with general target functionals in various classes of control actions (N.V. Azbelev, V.P. Maksimov, and L.F. Rakhmatullina, Introduction to the Theory of Functional Differential Equations: Methods and Applications, www.hindawi.com/books). An example is given that illustrates the utility of feedback control to parry impulse disturbances. Without the parring, the solution of the control problem takes a larger resource in control.