The Research Of Several Control Theories For Riemann-liouville Fractional Evolution Diferential Systems

In recent years, the Caputo fractional evolution diferential control systems have beenresearched by many authors, and a lot of promising results have obtain. Since the Riemann-Liouville fractional diferential control systems are more complex than the Caputo’s, we cannot use the old method to discuss the Riemann-Liouville fractional diferential control sys-tems. In order to study the Riemann-Liouville fractional diferential control systems, we needto search new methods and technique. It is difcult to search new methods and technique, sothe theories of the Riemann-Liouville fractional evolution diferential control systems are stilluntreated in recent years. For this reason, we concern some control theories for Riemann-Liouville fractional evolution diferential control systems in this paper. The paper is arrangedas follows:In Chapter1, we give the background, the present development situation and the devel-opment trend of Riemann-Liouville fractional evolution diferential control systems and themain work of this thesis.In Chapter2, we introduce the necessary preliminaries such as definitions, properties offractional calculus and some related lemmas and so on.In Chapter3, the approximate controllability for Riemann-Liouville fractional evolutiondiferential control systems are investigated. Under the assumption that the correspondinglinear systems is approximately controllable, and using the reachable set, we obtain the ap-proximate controllability of the systems.In Chapter4, the approximate controllability for impulsive Riemann-Liouville fractionalevolution diferential inclusions are researched. Firstly, a new Banach space PC1α(J, X) isdefined, and then the PC1α-mild solution is also given. Finally, by applying the fixed pointtheorems, semigroup theory and multivalued maps, the approximate controllability for thenonlinear systems are proved under condition that the corresponding linear system is approx-imate controllability.In Chapter5, the “Bang-Bang” principle for a class of Riemann-Liouville fractionalsemilinear evolution inclusions be investigate. At first, the existence of the mild solution forRiemann-Liouville fractional semilinear inclusions with a nonconvex valued multifunctionF(t, x(t)) is considered. Next, the “Bang-Bang” principle between the solution sets of the original inclusion F(t, x(t)) and the inclusions with the forms of co F(t, x(t)) and extco F(t,x(t)) are discussed.In Chapter6, the relaxation property of the control systems described by a class ofRiemann-Liouville fractional semilinear evolution hemivariational inequalities is studied. Un-der some suitable conditions, we get the existence of the mild solution for Riemann-Liouvillefractional semilinear evolution hemivariational inequalities. Finally, the relaxation propertyis discussed between the nonconvex values constraint function U(t, x(t)) and the upper semi-continuous convex valued regularization of the constraint V(t, x(t)).In Chapter7, based on the research at present, we propose some ideas for our futurework. Finally, in view of the study of Riemann-Liouville fractional evolution diferentialcontrol systems, we will try to find new idea and methods to investigate the permanence ofthe Riemann-Liouville fractional evolution diferential stochastic control systems.