Optimal control of linear fractional-order delay systems with a piecewise constant order based on a generalized fractional Chebyshev basis

This research studies optimal control of a new subclass of variable-order fractional delay systems whose its order is a piecewise constant function. This category of systems has not been discussed in the literature yet. An effective methodology based on a generalization of the fractional-order Chebyshev functions is offered for providing a solution with high level of precision. A detailed consideration regarding the convergence of the new framework is furnished. Moreover, two important estimates connected to the best approximation of the mentioned fractional basis in the Sobolev space and Hilbert space are achieved. Because direct implementation of the Riemann-Liouville integral operator (RLIP) leads to probably some serious drawbacks, such as numerical challenges, instability and unexpected oscillatory behaviour of the system under examination, a key integral operator connected to the basis under consideration is attained. The capacity and capability of the suggested numerical scheme are illustrated and verified through our numerical findings.