The Controllability Of Quasi-linear Parabolic System By One Control Force

The aim of this article is to proper the controllability of the following quasi-linear parabolic systemwhere y,z are the states and u ∈ L^∞(ω × (0,T)) is the control function which acts on the system through the subset ω. (y₀, z₀) is given function in (W^1,∞(Ω)∩System is said to be null-controllable, i.e., for each (y₀,z₀), there exists a control u ∈ L^∞(Q), such that the solution (y,z) ∈( L^∞(0,T; L²(Ω)))² satisfies(y,z)(·,T)=0 in Ω.System is said to be approximately controllable in L²(Ω) × L²(Ω), i.e., for each (y_d, z_d) ∈ L²(Ω) × L²(Ω) and any ε > 0, there exists a control u, such that the solution (y,z) ∈ L^∞(0,T; L²(Ω)))² satisfiesThe nonlinear terms f(x,t,y,z,(?)y),g(x,t,y,z,(?)y,(?)z) are locally Lip-schitz continuous. Moreover, the control forces act on a single equation of the systems, controlling a system with a minimum number of forces or by forces satisfying an algebraic or any other type relation is a common problem in the control theory. The key ingredients of our proof are global Carleman inequality and Observability estimate for the linearized system and then to use a fixed point theorem. The main difficulty is to prove the Observability estimate for the linearized system corresponding to the control by a single force which in avery different way from the case that two control force localized on the same subdomain.The above problems will be discussed as the following four parts:Part 1. Introduce the background of the controllability roblem and the relavant research progress, what's more, we state our main result.Part 2. Get global Carleman inequality and Observability estimate for the linearized system.Part 3. We prove the main results of null controllability.Part 4. Prove approximate controllability of the system.