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.

Presented herein are a class of methodologies for conducting constrained motion analysis of rigid bodies within the Udwadia-Kalaba (U-K) formulation. The U-K formulation, primarily devised for systems of particles, is advanced to rigid body dynamics in the geometric mechanics framework and a novel development of U-K formulation for use on nonlinear manifolds, namely the special Euclidean group \begin{document}$ {\mathsf{SE}(3)}$\end{document} and its second order tangent bundle \begin{document}${\mathsf{T}^2\mathsf{SE}(3)} $\end{document}, is proposed in addition to the formulation development on Euclidean spaces. Then, a Morse-Lyapunov based tracking controller using backstepping is applied to capture disturbed initial conditions that the U-K formulation cannot account for. This theoretical development is then applied to fully-constrained and underconstrained scenarios of rigid-body spacecraft motion in a lunar orbit, and the translational and rotational motions of the spacecraft and the control inputs obtained using the proposed methodologies to achieve and maintain those constrained motions are studied.