Controllability of Nonlinear Boundary Value Control Systems in Banach Spaces Using Schauder Fixed Point Theorem

Abstract

In this paper, sufficient conditions for controllability of nonlinear boundary value control system in Banach spaces are established. The results are obtained by using semigroup theory "compact semigroup"and some techniques of nonlinear functional analysis, such as, Schauder fixed point theorem. Moreover, example is provided to illustrate the theory. Keywords: Controllability, Banach space, Semigroup theory, Schauder fixed point theorem. ‫ﺑﻨﺎﺥ‬ ‫ﻓﻀﺎءﺍﺕ‬ ‫ﻓﻲ‬ ‫ﺣﺪﻭﺩﻱ‬ ‫ﺷﺮﻁ‬ ‫ﺫﺍﺕ‬ ‫ﺧﻄﻴﺔ‬ ‫ﻏﻴﺮ‬ ‫ﻷﻧﻈﻤﺔ‬ ‫ﺍﻟﺴﻴﻄﺮﺓ‬ ‫ﻗﺎﺑﻠﻴﺔ‬ ‫ﻟﺸﻮﻳﺪﺭ‬ ‫ﺍﻟﺼﺎﻣﺪﺓ‬ ‫ﺍﻟﻨﻘﻄﺔ‬ ‫ﻧﻈﺮﻳﺔ‬ ‫ﺑﺎﺳﺘﺨﺪﺍﻡ‬ ‫ﺍﻟﺨﻼﺻﺔ‬ ‫ﺍﻟﺴﻴﻄﺮﺓ‬ ‫ﻋﻠﻰ‬ ‫ﻟﻠﻘﺎﺑﻠﻴﺔ‬ ‫ﺍﻟﻜﺎﻓﻴﺔ‬ ‫ﺍﻟﺸﺮﻭﻁ‬ ‫ﻣﻊ‬ ‫ﺗﺘﻌﺎﻣﻞ‬ ‫ﺍﻟﺘﻲ‬ ‫ﺍﻟﻤﺒﺮﻫﻨﺔ‬ ‫ﺇﺛﺒﺎﺕ‬ ‫ﺗﻢ‬ ، ‫ﺍﻟﺒﺤﺚ‬ ‫ﻫﺬﺍ‬ ‫ﻓﻲ‬ ‫ﻟﻨﻈﺎﻡ‬ ‫ﻏﻴﺮ‬ ‫ﺫﺍﺕ‬ ‫ﺧﻄﻲ‬ ‫ﺣﺪﻭﺩﻱ‬ ‫ﺷﺮﻁ‬ ‫ﻓﻀﺎء‬ ‫ﻓﻲ‬ ‫ﺑﻨﺎﺥ‬ ‫ﺍﺕ‬ ‫ﺑﺎﺳﺘﺨﺪﺍﻡ‬ ‫ﻭﺫﻟﻚ‬ ‫ﻧﻈﺮﻳﺔ‬ ‫ﺍﻟﺘﻘﻨﻴﺔ‬ ‫ﺍﻟﻄﺮﺍﺋﻖ‬ ‫ﻭﺑﻌﺾ‬ (‫ﺍﻟﻤﺘﺮﺍﺻﺔ‬ ‫ﺍﻟﺰﻣﺮﺓ‬ ‫ﺍﻟﺰﻣﺮﺓ)ﺷﺒﻪ‬ ‫ﺷﺒﻪ‬ ‫ﻟﺸﻮﻳﺪﺭ‬ ‫ﺍﻟﺼﺎﻣﺪﺓ‬ ‫ﺍﻟﻨﻘﻄﺔ‬ ‫ﻧﻈﺮﻳﺔ‬ ‫ﻣﺜﻞ‬ ‫ﺍﻟﺨﻄﻲ‬ ‫ﻏﻴﺮ‬ ‫ﺍﻟﺪﺍﻟﻲ‬ ‫ﺍﻟﺘﺤﻠﻴﻞ‬ ‫ﺿﻤﻦ‬ ‫ﻛﺬﻟﻚ‬ ، ‫ﺍﻟﻨﻈﺮﻳﺔ‬ ‫ﻗﻴﻤﺔ‬ ‫ﻳﻮﺿﺢ‬ ‫ﻣﺜﺎﻝ‬ ‫ﺍﻋﻄﺎء‬ ‫ﺗﻢ‬ ‫ﺃﻋﻼﻩ‬ .