Approximate construction of the set of trajectories of the control system described by a Volterra integral equation

Abstract

In this paper the set of trajectories of the control system is investigated. It is assumed that the behavior of the control system is described by a Volterra integral equation which is nonlinear with respect to the state vector and is affine with respect to the control vector, and the control functions have an integral constraint. An approximation of the set of trajectories by the set which consists of a finite number of trajectories is given. The Hausdorff distance between the set of trajectories of the system and the set, consisting of a finite number of trajectories is evaluated.

Approximation of the integral funnel of the Urysohn type integral operator

“…(see, e.g., [2], [4], [11], [13]). Approximation of the set of outputs (or trajectories) and integral funnel of the control systems described different type integral operators and integral equations, where the input functions satisfy an integral constraint, are considered in [5], [6], [7], [8], [9]. In papers [5], [6], [7], [8] the systems with scalar variable and continuous outputs are investigated, while in [9], the system with multivariable outputs is studied.…”

“…Approximation of the set of outputs (or trajectories) and integral funnel of the control systems described different type integral operators and integral equations, where the input functions satisfy an integral constraint, are considered in [5], [6], [7], [8], [9]. In papers [5], [6], [7], [8] the systems with scalar variable and continuous outputs are investigated, while in [9], the system with multivariable outputs is studied. Moreover, in papers [5], [9], it is assumed that the system is affine with respect to the input function, but in [6], [7] [8] it is supposed that the system is nonlinear with respect to both input and output functions.…”

On the compactness of the set of L-2 trajectories of the control system

“…The properties of the set of trajectories of the control systems with integral constraints on the control functions and described by different type integral equations are considered in [13,14]. In [12,15] and [16], the methods for approximate construction of the set of trajectories are discussed. Note that in aforementioned papers, it is accepted that the trajectories of the considered equations are continuous function.…”

Continuity of the $L_{p}$ Balls and an Application to Input-Output System Described by the Urysohn Type Integral Operator

“…It is obvious that the set F p (r) is the image of the closed ball B Ω,X ,p (r) ⊂ L p (Ω, Σ, µ; X ) under operator F (•) given by (6.1). The approximate construction of the set of outputs of the input-output systems described by different type integral operators are considered in papers [5], [6], [10] (see the references also therein). The set F p (r) is a bounded subset of the space L 1 (Ω 0 , Σ 0 , µ 0 ; Y) for every p > 1.…”