This chapter presents several important definitions and lemmas needed in the study of fractional calculus theory and system stability theory, providing essential theoretical support for subsequent research. Fractional calculus extends the concept of integer-order differentiation and integration to noninteger orders, offering a powerful tool for modeling various physical and engineering processes. System stability theory, on the other hand, addresses the behavior of dynamic systems over time, ensuring that they remain within desired bounds and do not exhibit undesirable behavior. By laying down the foundational concepts and theoretical constructs in these areas, this chapter serves as a critical reference point for further exploration and application in advanced research. The definitions and lemmas provided here are crucial for developing a deeper understanding and for solving complex problems in these fields.
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