Existence of solutions and controllability of fractional semilinear neutral equations with impulses and nonlocal conditions

Résumé

This thesis investigates the controllability and solution properties of fractional-order dynam ical systems, with a particular focus on neutral systems involving impulses and nonlocal conditions. Motivated by the need to model real-world processes with memory and heredi tary effects, we study systems governed by Caputo and Riemann–Liouville fractional derivatives in the framework of Banach spaces. The core contributions are organized across four main directions. First, we study the exact controllability of a fractional neutral evolution system with Caputo derivatives and history dependent terms, addressing a minimum energy control problem. We then examine the existence of mild solutions and the approximate controllability of nonlinear fractional neutral evolution systems governed by the Riemann–Liouville derivative. These results deepen our understanding of the role of nonlinearity and memory in fractional systems. Next, we establish existence and uniqueness results for mild solutions of impulsive fractional neutral systems with nonlocal initial conditions. This part includes the analysis of discontinuities introduced by impulses and their interaction with the nonlocal structure of the system. Further more, we study the approximate controllability of nonlinear fractional systems of order α∈(1,2) with nonlocal conditions, utilizing advanced tools such as resolvent operators and fractional cosine families. Finally, we extend our analysis to fractional networked systems with impulses, contributing to the development of control methods for interconnected systems with memory effects. Overall, this thesis offers a comprehensive theoretical foundation for the controllability and analysis of fractional neutral and nonlinear systems, with implications for the modeling and control of complex dynamical systems in physics, engineering, and other applied sciences

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