Compactness of the set of trajectories of the control system described by a Urysohn type integral equation with quadratic integral constraints on the control functions

Abstract

In this paper the control system is considered described by a Urysohn type integral equation which is nonlinear with respect to the state vector and is affine with respect to the control vector. The functions from the space L 2 ([t 0 , θ ]; R m ) satisfying a quadratic integral constraint are chosen as admissible control functions. The set of trajectories generated by all admissible control functions is studied. The boundedness, closedness, precompactness, and hence the compactness of the set of trajectories in the space of continuous functions is proved. MSC: 45G15; 93C10

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. Ele alınan çalışmada davranışı Urysohn tür integral denklem ile verilen ve kontrol fonksiyonları integral kısıtlı olan kontrol sistemler araştırılmaktadır.…”

Modelling non-linear control systems using the discrete Urysohn operator

“…There is a number of papers dedicated to different aspects of solving the Urysohn integral equation for a given kernel, e.g. [2][3][4][5]. However, literature on the identification of the kernel based on known input and output functions is very limited.…”