The concepts of non-integer order differentiation and integration constitute an effective tool to characterize the behavior of a large category of dynamic systems of finite or infinite dimension. Applications are numerous, whether in medicine, hydrologyc [1] Mechanical systems [5], [3] viscosity [6], electronics [7], and neutron transport [9]. Since the emergence of methods for designing direct multivariate approaches to the synthesis of linear or nonlinear control laws, numerous methods currently exist in the literature to improve certain criteria such as system stability and robustness. Control laws are often functions of the system states. The entirety of the physical state components (or pseudo-states) of a system cannot be determined by direct measurements. However, in general, directly accessible pseudo-states do not pro- vide all the information necessary to detect and locate internal process faults. This makes the problem of constructing the state vector and state vector functions one of the most important topics in control engineering; this is the role of the observer, or state estimator. In recent years, the extension of control theory [19] conformal fractional differential equations [2], [4] to incorporate the notion of conformal fractional differential equations composed by another function (φ CFD) has increased. Much literature has attempted to solve problems of stability or stabilization, con- troll ability, and observability, [10, 12, 17, 14, 13, 15, 16] for the modified conformal Cauchy problem, the conformal φ Cauchy problem [24, 23] . This work follows the path of semi group operators in Banach spaces to prove the uniqueness and existence of the solution, then stability through feedback control of the system. This work is organized as follows:

Section 2 provides an introduction to the preliminaries, focusing on the Con formable fractional derivative and phi-semi group.

• In Section 3, we delve into the study of the existence of phi semi group and the existence of mild solutions for phi Conformable fractional systems of order α in the interval [0, 1], with control in Banach spaces.
• Section 4 introduces a fractional control system, which serves to characterize the stability of phi fractional control systems.
• Next, we can then apply the abstract results to the RLC electrical circuit.
• Finally, within the concluding, we can then apply the abstract results to a particular class of quantum mechanics fields equations.

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