| With the development of economy,industry and society,people find that many prac-tical systems show the characteristics of fractional order.Therefore,fractional calculus was introduced into system sciences,and the models in engineering were described by fractional differential equations.Switched systems are a class of important hybrid sys-tems consisting of several continuous or discrete dynamic subsystems and a switching rule indicating the active subsystem at each instant of time.In this paper,we consider the optimal control problems for switching fractional-order systems and switching fractional-order stochastic systems by combining the fractional calculus with the switching systems.For solving this problem we do not make any assumptions about the number of switches nor about the mode sequence,they are determined by the solution of the problem.First,for fractional-order switched optimal control problem,the fractional-order switching system is embedded into a large family of systems and it is shown that the set trajectories of the fractional-order switching system is dense in the set of trajectories of the embedded system.By using the embedding,we provide a rigorous developmen-t of sufficient and necessary conditions for optimality.Also,we provide an example to demonstrate the effectiveness of our discussed techniques.Moreover,for fractional-order stochastic switched optimal control problem,to ensure the existence and uniqueness of mild solution of a class of stochastic system of fractional order,a set of sufficient conditions is constructed via contraction mapping principle.The fractional-order stochastic switching system is embedded into a large family of systems and it is shown that the set trajectories of the fractional-order switching stochastic system is dense in the set of trajectories of the embedded system.The existence of fractional optimal controls for the switched optimal control problem is investigated if we limit the number of switches. |
PDF Full Text Request