Noether Theories Of Variational Problems And Optimal Control Problems For Fractional Birkhoff Systems

This thesis,taking the non-integer derivatives,non-standard Lagrange functions and fractional Birkhoff systems as its main thread,considers the Noether theories of the variational problems and the optimal control problems for fractional Birkhoff systems,fractional forced Birkhoff systems and fractional generalized Birkhoff systems,respectively.This thesis consists of six sections which are introduced in the following way.Chapter 1is an introduction in which the significance of the selected topic,the survey and development trend of domestic and foreign and the main content of this article are introduced.Chapter 2 is devoted to the Noether theorems of variational problem for fractional Birkhoff systems.Firstly,from variational problem for fractional Birkhoff systems in the Riemann–Liouville sense,the fractional Birkhoff equations are derived.Secondly,the invariance conditions about the fractional Birkhoff equations are discussed.Finally,under the conditions that the lower bound of fractional left derivative is not transformed and is transformed respectively,the Noether theorems of variational problem for fractional Birkhoff systems are obtained.The Noether theorems for fractional forced Birkhoff systems are considered in Chapter3.Firstly,from fractional Pfaff-Birkhoff variational principle,the fractional forced Birkhoff equations are derived in the Riemann–Liouville sense.Secondly,the invariance conditions of the fractional forced Birkhoff equations in the Riemann-Liouville sense are discussed.Finally,under the infinitesimal transformation groups with time not changing and time changing respectively,the Noether theorems of the fractional forced Birkhoff equations in the Riemann–Liouville sense are obtained.Chapter 4 is devoted to the Noether theorems for fractional generalized Birkhoff systems.Firstly,from fractional Pfaff-Birkhoff variational principle,the fractional generalized Birkhoff equations are derived in the Caputo sense.Secondly,the invariance conditions of the fractional generalized Birkhoff equations in the Caputo sense are discussed.Finally,under the infinitesimal transformation groups with time not changing and time changing respectively,the Noether theorems for the fractional generalized Birkhoff equations in the Caputo sense are obtained.In Chapter 5,the Noether theorems of the optimal control problem for fractional Birkhoff systems are studied.Firstly,the optimal control problem for fractional Birkhoff systems is presented.Secondly,the extreme conditions of the optimal control problem for fractional Birkhoff systems are obtained.Finally,under the infinitesimal transformation groups with time not changing and time changing respectively,the Noether theorems of the optimal control problem for fractional Birkhoff systems are obtained.Chapter 6 is about the Noether theorems of the optimal control problem for fractional generalized Birkhoff systems.Firstly,the optimal control problem for fractional generalized Birkhoff systems is presented.Secondly,the extreme conditions about the optimal control problem for fractional generalized Birkhoff systems are obtained.Finally,under the infinitesimal transformation groups with time not changing and time changing respectively,the Noether theorems of the optimal control problem for fractional generalized Birkhoff systems are obtained.