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

Abstract

The control system with integral constraint on the controls is studied, where the behavior of the system by a Urysohn type integral equation is described. It is assumed that the system is nonlinear with respect to the state vector, affine with respect to the control vector. The closed ball of the space Lp(E; R m ) (p > 1) with radius r and centered at the origin, is chosen as the set of admissible control functions, where E ⊂ R k is a compact set. It is proved that the set of trajectories generated by all admissible control functions is a compact subset of the space of continuous functions.

Urysohn Tür İntegral Denklem ile Verilen Kontrol Sistemin Yörüngeler Kümesinin Özellikleri Üzerine

“…Urysohn tür integral denklem ile verilen kontrol sistemlerin farklı özellikleri [16]- [21] 'de incelenmektedir. [18]- [20] çalışmalarında yörüngeler kümesinin yaklaşımı, [21], [22] 'de kompaktlığı tartışılmaktadır.…”

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.…”

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

“…The conditions A-C guarantee that every admissible control function u(·) ∈ U p,r generates a unique trajectory of the system (2.1) (see, [16]). Now let us give some auxiliary propositions which will be used in following arguments.…”

“…In [14], [16], [15], [17] approximation and topological properties of the set of trajectories of the control systems described by nonlinear integral equations with integral constraints on the control functions are considered. In this paper the control system described by a Urysohn type integral equation with integral constraint on the control functions is studied, where the trajectory of the system is multivariable continuous function.…”