Null-Controllability Of Two Types Of Fractional Partial Differential Equations On Specified Domain

Fractional calculus operators are more nonlocality than integer order one,therefore,it can be used to describe materials with memory and genetic characteristics in life,such as electrolytic chemistry,material mechanics,signal processing,neural networks,condensed matter physics,etc.Fractional partial differential equations can describe anomalous diffu-sion phenomena which is characterized by historical dependence and global correlation,so its application more universal in the mathematical model of abnormal diffusion.This thesis mainly studies the null-controllability of the fractional order diffusion equation on(0,?)and null-controllability of one-dimensional compressible fractional order Navier-Stokes equation on I2?:=(0,2?).The research contents and results are as follows:1.We consider null-controllability of the fractional diffusion equation with Q(x)that satisfies a certain assumption as a general case on(0,?):#12 where T>0,1/2<?<1,?=R(0,?),v(t)? L2(0,T)is a control function which depends on time t.In this chapter,the existing spectral conclusions in Sturm-Liouville are used to transform the control problem into a moment one,then the orthogo-nal sequence is constructed using the fractional order Paley-Wiener type theorem,Malli-avin multiplier theory proves that the orthogonal sequence converges,which leads to the conclusion that the diffusion equation is null-controllable.Finally,we take Q(x)=x2 as a typical inference of this type of problem.2.We consider null-controllability of one-dimensional compressible fractional order Navier-Stokes equation with constant steady state(Q0,V0),Q0>0,V0>0 onI2?:=(0,2?):#12 Where ??(1/2,1),XOdenotes characteristic function of any open set O within I2?.In order to transform the control problem into a moment one,we use the Mittag-Leffler function to replace the exponential function.The main technique in the proof is apply-ing Paley-Wiener type theorem to construct an orthogonal sequence to a family of com-plex Mittag-Leffler functions.It is essential that the typical product corresponding to the eigenvalue of the operator A and the Malliavin multiplier theory are used to prove the sequence convergence,finally we show that the fractional order Navier-Stokes equation is null-controllable.In this thesis,the method of transforming the control problem into a moment one provides a theoretical reference for studying the null-controllability of differential equa-tions.