On the approximation of the set of trajectories of control system described by a Volterra integral equation

Abstract

In this paper, the set of trajectories of the control system described by a nonlinear Volterra integral equation is studied. It is assumed that the set of admissible control functions is the closed ball of the space Lp, p > 1, with radius µ and centered at the origin. It is shown that the sections of the set of trajectories can be approximated by the sections of trajectories, generated by the mixed constrained and Lipschitz continuous control functions, the Lipschitz constant of which is bounded.

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.…”

On the existence of ε-optimal trajectories of the control systems with constrained control resources

“…In papers [8], [9] various topological properties of the sets of trajectories of the control systems described by the nonlinear Volterra type integral equations with integral constraint on the control functions are studied. In [7] the approximation of the sets of trajectories of the aforementioned systems is discussed. A similar problem for the systems described by the ordinary di¤erential equations is considered in [5].…”

Approximation of the Sections of the Set of Trajectories of the Control System Described by a Nonlinear Volterra Integral Equation

“…[4,8,15,22,27,28] and references therein). Some properties of the set of trajectories of the control systems described by a nonlinear Volterra integral equation with integral constraint on the controls are discussed in [13] and [14]. Precompactness of the set of trajectories of the control systems described by a nonlinear Volterra integral equation with integral constraint on the control functions is considered in [14].…”

“…In [13] an approximation of the sections of the set of trajectories of the control systems described by a nonlinear Volterra integral equation with integral constraint on the controls is investigated, where the sections of the set of trajectories is approximated by the sections of the set of trajectories generated by the compact set of control functions. In this paper, developing the result obtained in [13], the sections of the set of trajectories are approximated by the sections of the set of trajectories which consists of a finite number of trajectories.…”

“…In Section 2 the basic conditions are formulated which satisfy the system (Conditions 2.A, 2.B and 2.C). The known result from [13] is given which is used in the following arguments (Theorem 1).…”