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

Abstract

In this paper, the compactness of the set of L2 trajectories of the control system described by the Urysohn-type integral equation is studied. The control functions are chosen from the closed ball of the space L2 with radius r and centered at the origin. Existence of an optimal trajectory of the optimal control problem with lower semicontinuous payoff functional is discussed.

Approximation of the set of integrable trajectories of the control system with $ L_2 $ norm constraints on control functions

“…Existence of the optimal trajectories and controllability of the system, necessary and sufficient conditions for optimality of given processes, approximation of the set of trajectories of the control systems described by integral equations are considered in [1,3,5,14,15,16,19] (see also references therein). In [14,15,16] the various properties and approximation of the set of trajectories and integral funnel of the control systems described by Urysohn type integral equations and integral constraints on the control functions are considered. In paper [16] it is assumed that the system is nonlinear with respect to the state vector and is affine with respect to the control vector.…”

“…Continuity of the trajectories permits to prove convergence approximating sets of trajectories to the set of system's trajectories. In paper [15] compactness of the set of L 2 integrable trajectories of the control systems described by the Urysohn type integral equation is investigated. It is assumed that the system is nonlinear with respect to the state vector and is affine with respect to the control vector.…”

“…It is assumed that the system is nonlinear with respect to the state vector and is affine with respect to the control vector. The results obtained in [15] are widely used in studies presented here.…”

“…Proposition 2.1. [15] Every admissible control function u(•) ∈ V ρ generates unique trajectory of the system (1). Proposition 2.2.…”

“…Proposition 2.2. [15] The set of trajectories Z ρ of the system (1) is a bounded subset of the space L 2 (E; R n ), i.e. there exists…”