his paper develops the fundamentals of optimaltuning periodic-gain fractional delayed state feedback control for a class of linear fractional order periodic time-delayed systems.his paper develops the fundamentals of optimal-tuning periodic-gain fractional delayed state feedback control for a class of linear fractional order periodic time-delayed systems.T Although there exist techniques for the state feedback control of linear periodic timedelayed systems by discretization of the monodromy operator, there is no systematic method to design state feedback control for linear fractional periodic time-delayed (FPTD) systems. This paper is devoted to defining and approximating the monodromy operator for steady state solution of FPTD systems. It is shown that the monodromy operator cannot be achieved in a close form for FPTD systems, and hence the short memory principle along with the fractional Chebyshev collocation (FCC) method are used to approximate the monodromy operator. The proposed method guarantees a near-optimal solution for FPTD systems with fractional orders close to unity. The proposed technique is illustrated in examples, specifically in finding optimal linear periodic-gain fractional delayed state feedback control laws for the fractional damped Mathieu equation and a double inverted pendulum subjected to a periodic retarded follower force with fractional dampers in which it is demonstrated that the use of time-periodic control gains in the fractional feedback control generally leads to a faster response.

This article aims to provide a comprehensive review of the latest advancements in numerical methods and practical implementations in the field of fractional stochastic partial differential equations (FSPDEs). This type of equation integrates fractional calculus, stochastic processes, and differential equations to model complex dynamical systems characterized by memory and randomness. It introduces the foundational concepts and definitions essential for understanding FSPDEs, followed by a comprehensive review of the diverse numerical methods and analytical techniques developed to tackle these equations. Then, this article highlights the significant expansion in numerical methods, such as spectral and finite element methods, aimed at solving FSPDEs, underscoring their potential for innovative applications across various disciplines.

This paper proposes a new technique to design an optimal observer-based feedback control for linear fractional-order systems with constant or periodic coefficients. The proposed observer-based feedback control assures the fastest convergence of the closed-loop system’s states. For this purpose, a state-transition operator is defined in a Banach space and approximated using the fractional Chebyshev collocation method. It is shown that periodic gains of the controller and observer can be independently tuned by minimizing the spectral radius of their associated state-transition operators. The validity and efficiency of the proposed method are demonstrated through two illustrative examples.