Optimal Control Problems For Space-fractional Wave And Poisson White Noise Driven Diffusion Equations Khattak Shahzad

In the last 30 years,the theory and applications of fractional calculus(that is,any order integrals and derivatives)have attracted much research interest and been greatly developed.Fractional partial differential equations also have been widely applied in many fields.For example,biotechnology,viscoelastic material,time-delay system,image processing,fluid dynamics and so on.Compared to integer partial differential equations,it is better to use fractional partial differential equations to model some physical processes.For example,diffusion and thermal conduction in materials with long-time memory and long-range interaction effects,fractal porous media and so on.Besides,optimal control problems for fractional partial differential equations have been developed into important research fields.Compared to integer differential systems,it is more reasonable and accurate to describe the dynamic process by using fractional differential systems.Therefore,this class of nonlinear control systems can be used more and more extensively.In this thesis,we study the existence and uniqueness of solutions for two classes of space-fractional evolution equations.Furthermore,by using the theory of Lions,we discuss quadratic optimal control problems for these two classes of evolution equations.The main contents are as follows:Based on the properties of the fractional Laplacian operator and the fractional Sobolev space,we study a class of space-fractional wave equations.By using Gal(?)rkin approximate method,we show the existence and uniqueness of weak solution.We generalize the results of a class of Poisson white noise driven diffusion equa-tions to space-fractional diffusion equations.In higher dimensions,we obtain the existence and uniqueness of solution.By using the the existence and uniqueness of solution for the above space frac-tional evolution equations,we study two classes of fractional nonlinear control systems.By showing the nature of solution mappings and the existence of mini-mum for the quadratic cost function,we obtain the existence of optimal control.