Precompactness of the Set of Trajectories of the Controllable System Described by a Nonlinear Volterra Integral Equation

Abstract

In this paper the controllable system whose behaviour is described by a nonlinear Volterra integral equation, is studied. The set of admissible control functions is the closed ball of the space Lp (p > 1) with radius µ0 and centered at the origin. It is shown that the set of trajectories of the system is a bounded and precompact subset of the space of continuous functions.

Compactness of the Set of Trajectories of the Control System Described by a Urysohn Type Integral Equation

“…where l(λ) is defined by (3), γ 0 ≥ 0, γ 1 ≥ 0 and γ 2 ≥ 0 are defined by (12), (13) and (14) respectively.…”

“…Note that compactness of the set of trajectories guaranties the existence of the optimal trajectories in the optimal control problem with continuous payoff functional. Compactness of the set of trajectories of control systems described by the Volterra type integral equations is studied in [12,13].…”

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 76 ANAR HUSEYIN the sets of trajectories of the aforementioned systems is discussed.…”

Dependence on the parameters of the set of trajectories of the control system described by a nonlinear Volterra integral equation

“…The control systems with integral constraint on controls are generally needed in modelling the systems having limited energy resources which are exhausted by consumption, such as fuel or finance (see, e.g. [2], [3], [4], [5], [7], [9]). For example, the motion of a flying apparatus with variable mass is described in the form of a control system, where the control functions have integral constraint (see e.g.…”

“…[1], [6], [7], [8], [10], [11], [12], [13], [14], [15]), and many problems of nonlinear mechanics lead to nonlinear integral equations (see, e.g. [8], [12], [15]).…”