Null Controllability Of Degenerate Parabolic Equations

Controllability theory of nondegenerate parabolic equations has been widely in-vestigated over the last 40 years and there have been a great number of results, such as exact controllability, exact null controllability and approximate controllability. There has grown in various fields of the controllability theory.It has been applied in the semi-linear parabolic problems, problems in undounded domains, and fluid models, such as the Euler, Stokes and Navier-Stokes equations.On the contrary, the study on the controllability of degenerate parabolic equa-tions just began several years ago and very few results have been known. However, many problems that are relevant for applications are described by degenerate parabolic equations where degeneracy occurs at the boundary of the space domain. For exam-ple, Prandtl equation can be transformed to the Crocco equation by the "Crocco" transformation [26]. The linearization of the Crocco equation is in the form of whereΩ= (0, L)×(0,1),fand u1 depend on the incident velocity of the flow, the coefficients a, b and c are positive and regular. In addition, a and b are degenerate at the boundary of the domain, that isBesides, such degenerate problems also appear in economics. For example, the Black-Scholes model can be transformed into the following equation [28] where S is the value of the underlying asset, V is the value of the option which has the S underlying asset,σis volatility, r is risk-free interest rate and t is the current time. And the Budyko-Sellers model in climatology is where k> 0 is a constant, u and v are the control functions [29]. The common point of these problems is that the degeneracy of the parabolic operator occurs at the boundary of the domain. Hence, it is important to investigate the controllability theory of the degenerate parabolic problems.For the nondegenerate parabolic equations, we take the simplest heat equation as an example. Consider the linear system whereΩis an open, bounded and smooth set of Rn,ωis an open subset ofΩ, and Xωis the characteristic function ofω, u0∈L2(Ω) is arbitrary but fixed. Then it is well known that the system (1) is null controllability, exactly controllability and approximate controllability. See in particular [1] and [13]. WhileΩis an unbounded domain, the system is approximate controllability, but whether it is null controllability or not is determined byω. More precisely, whenΩ\ωis a bounded domain, the system (1) is null controllability, whileωis a bounded set, the system (1) is not null controllability. See in particular [2]-[5]. Consider the nonlinear system whereΩ,ω,χω,u0 are defined as above,and f is locally Lipschitz continuous with f(0)=0.Of course,the existence of a solution to(2)is not assured for arbitrary u0 and h in the whole interval[0,T],unless some conditions are imposed to f. some of these conditions can ensure the null controllability,and some cannot.See in particular [15]-[19].For the degenerate parabolic equaltions,the following equation is considered ut-(a(x)ux)x=h(t,x)χω(x),(t,x)∈Q=(0,T)×(0,1), where a∈C([0,1])∩C1((0,1))and is positive in(0,1).A typical choice of a(x)is a(x)=xα.,0≤x≤1, withα>0,namely, ut-(xαux)x=h(t,x)χω(x),(t,x)∈Q=(0,T)×(0,1), (3) which is degenerate at x=0.As we know,the well-posed problems for the parabolic equations with boundary degeneracy are different from the common ones.The de-generacy of(3)is divided into weak one and strong one according to the value ofα, and different boundary conditions are proposed for the two cases.More precisely,the boundary value condition is u(0,t)=u(1,t)=0, t∈(0,T) in the weakly degenerate case with O<α<1,while is Xαux(0,t)=u(t,1)=0, t∈(0,T) in the strongly degenerate case withα≥1. The initial value condition of (3) for both cases is u(x,0)= uo(x), x∈(0,1).Then the system is null controllable if 0< a< 2, while not ifα> 2. But for the case a≥2, the system is approximately controllable [21] and also regional controllable [9]. Concerning the nonlinear degenerate equation, see [7],[10],[22].Finally, we introduce the proof of null controllability. The approach of proving the null controllability that is widely used is based on Carleman estimates. Since the observability is equivalent to null controllability, and Carleman estimates is able to derive the observability, the proof is complete. Additionally, for degenerate parabolic equations, Hardy inequality plays an important role in the process of proving Carleman estimates.