On the continuity property of Lp balls and an application

Abstract

In this paper continuity properties of the set-valued map p → B p (μ 0 ), p ∈ (1, +∞), are considered where B p (μ 0 ) is the closed ball of the space L p ([t 0 , θ]; R m ) centered at the origin with radius μ 0 . It is proved that the set-valued map p → B p (μ 0 ), p ∈ (1, +∞), is continuous. Applying obtained results, the attainable set of the nonlinear control system with integral constraint on the control is studied. The admissible control functions are chosen from B p (μ 0 ). It is shown that the attainable set of the system is continuous with respect to p.

On the Properties of the Set of Trajectories of the Control System with Integrable Trajectories and Limited Control Resources

“…Kontrol fonksiyonları geometrik kısıtlı kontrol sistemler, kontrol sistemler teorisinin geniş biçimde incelenmiş dallarından birdir (Deimling, 1992;Kalman, 1963;Krasovskii & Subbotin, 1988;Pontryagin ve ark., 1962). Kontrol fonksiyonları üzerinde integral kısıtlamalar, genelde kullanırken tükenen kontrol etkilerde, örneğin enerji, yakıt, finans gibi kontrol etkilerde ortaya çıkmaktadır (Conti, 1974;Guseinov & Nazlipinar, 2007;Gusev & Zykov, 2017;Ibragimov ve ark., 2021;Krasovskii, 1968;Kostousova, 2020;Subbotin & Ushakov, 1975;Subbotin & Subbotina, 1975;Ukhobotov & Izmestyev, 2018). Kontrol fonksiyonu integral kısıtlı iken, bu fonksiyon geometrik kısıtlı olmayabilir.…”

Sobre la convergencia de los conjuntos de alcanzabilidad de un sistema lineal de segundo orden

“…En este contexto, el problema sobre la búsqueda de la mejor estimación es ampliamente estudiado en la literatura, ver por ejemplo los trabajos (Sabin and Summers, 1990;Shen and Zhong, 2011;Pitarch et al, 2015). El problema sobre la búsqueda de estimaciones numéricas de los conjuntos de alcanzabilidad de sistemas no lineales se puede consultar en (Kumkov and Zharinov, 2004;Vinnikov, 2015).…”