Types Of Distributed Parameter Systems With Qualitative Analysis And Parameter Identification

This paper includes three chapters.In chapter one, an initial value problem for pseudo-parabolic equation is considered. The pseudo-parabolic models appear wildly in various aspects. The initial value problem for pseudo-parabolic equation can describe the seepage of fluids through a fissured rock, nonsteady flows of second-order fluids, and some kinds of heat flow cryptosystems in cryptography etc. Since the theory and results of parabolic equation cannot apply directly to pseudo-parabolic equation, it is difficult to study pseudo-parabolic equations. In chapter 1, firstly, by using classical analysis methods, such as the introduction of Bessel function and semigroup theory, we transform the pseudo-parabolic equation into an abstract operator equation and achieve the existence and uniqueness of the local solution. Secondly, due to the integral transform, we obtain the prior estimate of the solution to prove the existence of global solution. The concept of stability shows asymptotic behaviors of a system. Finally, we study the stability under disturbance of the initial value, i.e., the Lyapunov stability problem. By constructing the Lyapunov function with a special structure, prior estimate and the solution of ordinary different equation, we prove the main result, i.e., the trivial solution has global asymptotic stable behavior.Two types of boundary problems for shape memory alloys (SMA) model are discussed in chapter 2 and chapter 3. The SMA model arises from the conservation laws of momentum and energy, and describes martensitic phase transitions in one-dimensional shape memory alloys. In Chapter 2, an initial-boundary problem for SMA model is considered. Firstly, by constructing an abstract operator, the original problem can be transformed into a Cauchy problem for an abstract operator equation. Secondly, the characters of abstract operator are discussed and the operator is proved to be an infinitesimal generator of analytic semigroup. Finally, by applying the semigroup theory, the existence and uniqueness of local solution are proved.The unknown parameter identification of distributed parameter systems often part of ill-posed problems. The theoretical analysis and numerical method for ill-posed problems need to be improved. In chapter 3, the problem of identifying unknown parameter of SMA model is considered. By using Tikhonov regularization function and Gauss-Newton method, a modified algorithm is presented and the convergence of algorithm is proved under different observation. Those works provide theoretical foundation to the unknown parameter identification of SMA model.