Approximation of the Set of Trajectories of the Nonlinear Control System with Limited Control Resources

Abstract

Abstract. In this paper the control system described by a Urysohn type integral equation is studied. It is assumed that the control functions have integral constraint. Approximation of the set of trajectories generated by all admissible control functions is considered.Step by step way, the set of admissible control functions is replaced by a set consisting of a finite number of control functions which generates a finite number of trajectories. An evaluation of the Hausdorff distance between the set of trajectories of the system and the set, consisting of a finite number of trajectories is obtained.

Approximation of the Image of the Hilbert-Schmidt Integral Operator

“…Note that integrally constrained inputs are usually applied when the input resources of the system are exhausted by consumption such as energy, fuel, finance, etc. (see, [4], [5], [8] and references therein). An evaluation of error estimation for Hausdorff distance between the image and its approximation, which consists of a finite number of functions, is given.…”

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.…”

“…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. Note that in papers [6], [7], [8] only approximations of the sections of the set of trajectories and integral funnels are obtained.…”

“…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. Note that in papers [6], [7], [8] only approximations of the sections of the set of trajectories and integral funnels are obtained. In the presented paper an approximation of the image of the closed L p ball in the space of continuous functions is given.…”