On the properties of the set of p-integrable trajectories of the control system with limited control resources

ABSTRACT

The properties of the set of trajectories of the control system described by Urysohn type integral equation are considered. It is assumed that the system is affine with respect to the control vector and control functions are chosen from the bounded and closed ball of the space $L_q(E;{\mathbb{R}}^m)$, q>1. The trajectory of the system is the function from the space $L_p(E;{\mathbb{R}}^n)$, $1/q+1/p=1$, which satisfies the system's equation almost everywhere. The path-connectedness and compactness of the set of trajectories are proved. The existence of the optimal trajectories in the optimal control problem with lower semicontinuous payoff functional is discussed.

KEYWORDS:

• Control system
• integral equation
• p-integrable trajectory
• connectedness
• compactness

Acknowledgements

The author thanks anonymous referee for valuable remarks which improved the presentation of the paper.

Disclosure statement

No potential conflict of interest was reported by the authors.