The Research Of Controllability For Nonlinear Fractional Evolution Systems

Recently, the controllability of the dynamic systems has received comprehensive appli-cations on the area of engineering, biology, finance, etc. For this reason, we are concernedwith the controllability of the nonlinear fractional evolution systems in this thesis. The ar-rangement of the thesis is as follows:Chapter one introduces the background, the domestic and foreign research situation ofour problems and presents the main work of this thesis.Chapter two recalls the preliminaries necessary for the study of this thesis, includingthe definitions, properties and conclusions of function spaces, Bochner integral, fractionalcalculus, operators semigroup, multivalued maps and generalized Clarke subdiferential.Chapter three deals with the exact controllability of impulsive fractional evolution inclu-sions in Banach spaces. At present, most existing results of controllability for this problemsare mainly obtained based on the impulsive function is single value. In this dissertation, weshow that the controllability of such systems under the case that the impulsive function ismultivalued. Our results extend the existing ones and represent the new continuation and de-velopment of the resent work on this issue. At the end of the chapter, we give a concreteexample to illustrate the applications of the abstract results.Chapter four discussed the exact controllability of the fractional evolution hemivaria-tional inequalities in Hilbert spaces. In recent years, much concern is devoted to controllabilityof fractional evolution equations and inclusions and the theory result is fruitful. However, theexact controllability of fractional evolution hemivariational inequality is still untreated top-ics in the literature. Therefore, we firstly present the existence of mild solutions by utilizingdefinitions of fractional calculus, fixed points theorem and properties of generalized Clarkesubdiferential in this thesis. Next, unlike the assumptions of chapter3, the exact controllabil-ity of the fractional evolution hemivariational inequalities is also formulated and proved underthe corresponding linear system is exactly controllable.In fact, not all of control systems are exactly controllable (e.g., the control system de-scribed by heat conduction equation is not exactly controllable). Therefore, it is necessary toconsider some weaker controllability (such as approximate controllability). In the followingfifth and sixth chapter, the thesis pays more attention to approximate controllability for thenonlinear control systems. Chapter five researchs the existence of mild solutions for the fractional Clarke subdif-ferential evolution inclusions with nonlocal conditions by applying the theorem of operatorsemigroup and fixed points theorem. Subsequently, this chapter studies the approximate con-trollability for the corresponding linear systems and the approximate controllability for thefractional Clarke subdiferential evolution inclusions is also obtained based on the linear sys-tem is approximately controllable. Finally, a concrete example is given to explain the maintheorem.Fractional diferential equation is an expanding and vibrant branch of applied mathe-matics that has found numerous applications. In resent years, gratifying achievement anddevelopment has made on the Caputo fractional equation. However, since the initial con-ditions of the Riemann Liouville fractional equation maybe tend to∞or even not exist, itbrings great difculties for us and is also the main reason why there is still little informationknown for the control systems governed by fractional evolution diferential equations involv-ing Riemann-Liouville fractional derivatives at present. In order to fill this gap, the purposeof chapter6is to investigate the existence of mild solutions and approximate controllabilityfor Riemann-Liouville fractional evolution equations.Chapter seven gives a conclusion of our present work and introduces some further studyideas in the future.