Exact controllability and stabilizability of the Korteweg-de Vries equation

Abstract

In this paper, we consider distributed control of the system described by the Korteweg-de Vries equation (i) ∂ t u + u ∂ x u + ∂ x 3 u = f \begin{equation*} \partial _t u + u \partial _x u + \partial _x^3 u = f \tag {i} \end{equation*} on the interval 0 ≤ x ≤ 2 π , t ≥ 0 0\leq x\leq 2\pi , \, t\geq 0 , with periodic boundary conditions (ii) ∂ x k u ( 2 π , t ) = ∂ x k u ( 0 , t ) , k = 0 , 1 , 2 , \begin{equation*} \partial ^k_x u(2\pi , t ) = \partial ^k_x u(0,t) , \quad k=0,1,2, \tag {ii} \end{equation*} where the distributed control f ≡ f ( x , t ) f\equiv f(x,t) is restricted so that the “volume” ∫ 0 2 π u ( x , t ) d x \int ^{2\pi }_0 u(x,t) dx of the solution is conserved. Both exact controllability and stabilizibility questions are studied for the system. In the case of open loop control, if the control f f is allowed to act on the whole spatial domain ( 0 , 2 π ) (0,2\pi ) , it is shown that the system is globally exactly controllable, i.e., for given T > 0 T> 0 and functions ϕ ( x ) \phi (x) , ψ ( x ) \psi (x) with the same “volume”, one can alway find a control f f so that the system (i)–(ii) has a solution u ( x , t ) u(x,t) satisfying \[ u ( x , 0 ) = ϕ ( x ) , u ( x , T ) = ψ ( x ) . u(x,0) = \phi (x) , \qquad \quad u(x,T) = \psi (x) . \] If the control f f is allowed to act on only a small subset of the domain ( 0 , 2 π ) (0,2\pi ) , then the same result still holds if the initial and terminal states, ψ \psi and ϕ \phi , have small “amplitude” in a certain sense. In the case of closed loop control, the distributed control f f is assumed to be generated by a linear feedback law conserving the “volume” while monotonically reducing ∫ 0 2 π u ( x , t ) 2 d x \int ^{2\pi }_0 u(x,t)^2 dx . The solutions of the resulting closed loop system are shown to have uniform exponential decay to a constant state. As in the open loop control case, a small amplitude assumption is needed if the control is allowed to act on only a small subdomain. The smoothing property of the periodic (linear) KdV equation discovered recently by Bourgain has played an important role in establishing the exact controllability and stabilizability results presented in this paper.