Distributed Parameter Systems With Qualitative Analysis And Parameter Identification

Distributed parameter systems are often described by partial differential equations. The problems, such as, reasonability of the model, asymptotic behaviors and stability under the disturbance of initial or boundary values, are always concerned.The existence of solutions is primary to check the reasonability. In Chapter 2, the 4-order elliptic equation can describe the structures of airplanes, ships and buildings. By using classical methods, such as, upper and lower solutions, an auxiliary function and Shauder fixed point theory, we achieve the existence under the presence of a couple supper and lower solutions. Then, due to the first eigenfunction of 2-order homogeneous boundary value problems, we construct a pair of upper and lower solutions to prove the existence of positive solutions under general assumptions. When the nonlinear term on the right side is singular, and the singular point is a natural upper or lower solution, this method doesn't work. We consider a series of boundary problems, and make their solution sequence converge to the solution of the original problem. In this way, we conquer the difficulty and get the conclusion of existence.Stability shows asymptotic behaviors of a system. Chapter 3 discusses Sobolev-Galpern equation with pseudoparabolic characteristic, studies the stability under disturbance of the initial value, i.e., the Lyapunov stability problem, and shows the global asymptotic behavior of the trivial solution. Different from others, we apply the known Lyapunov method to discuss it. Firstly, by constructing the Lyapunov function with a special structure, a prior estimate is obtained. Secondly, with the help of Green's function, the original problem is transformed into an equivalent operator equation, and the fixed point theory of contraction mapping in a Banach space guarantees the existence and uniqueness of the global solution. Finally, the main conclusion is presented, i.e., the trivial solution has global asymptotic stable behavior.In Chapter 4, we discuss the boundary control problem of the shape memory alloys(SMA). The nonlinear SMA model arises from the conservation laws of momentum and energy, and describes martensitic phase transitions in one-dimensional shape memory alloys. By using a state-space approach, we show that the boundary parameters arc identifiable, i.e., the solution is dependent continuously on them, which is the basis of establishing the convergence of the optimization algorithm. And sufficient conditions for the boundary control problem are derived.