Stabilization methods for the Korteweg-de Nonlinear System Boundary - PowerPoint PPT Presentation

Control System Internal Control Linear System Stabilization methods for the Korteweg-de Nonlinear System Boundary Control Vries equation from the right Finite Dimensional Case Infinite Dimensional Case Application to our Problem Numerical Simulations Eduardo Cerpa Boundary Control from the left Control Design Universidad T´ ecnica Federico Santa Mar´ ıa Linear System Nonlinear System Valpara´ ıso, Chile Grenoble, September 2012

Table of contents Control System Control System Internal Control Linear System Internal Control Nonlinear System Boundary Control Linear System from the right Nonlinear System Finite Dimensional Case Infinite Dimensional Case Application to our Problem Numerical Simulations Boundary Control from the right Boundary Control Finite Dimensional Case from the left Control Design Infinite Dimensional Case Linear System Nonlinear System Application to our Problem Numerical Simulations Boundary Control from the left Control Design Linear System Nonlinear System

Control System Internal Control Linear System Nonlinear System Boundary Control from the right Finite Dimensional Case Infinite Dimensional Case Control System Application to our Problem Numerical Simulations Boundary Control from the left Control Design Linear System Nonlinear System

Korteweg-de Vries equation 1895 Control System Internal Control Linear System Nonlinear System Boundary Control from the right Finite Dimensional Case Infinite Dimensional Case Application to our Problem Numerical Simulations Boundary Control from the left Control Design Function u = u ( t , x ) models for a time t the amplitude of the Linear System Nonlinear System water wave at position x . The nonlinear dispersive partial differential equation, named Korteweg-de Vries equation and abbreviated as KdV, describes approximately long waves in water of relatively shallow depth u t + u xxx + uu x = 0 , x ∈ R , t ∈ R

Korteweg-de Vries equation on a bounded domain Control System On a bounded interval, the extra term u x should be incorporated Internal Control Linear System in the equation in order to obtain an appropriate model for Nonlinear System water waves in a uniform channel when coordinates x is taken Boundary Control from the right with respect to a fixed frame. Thus, for L > 0 the equation Finite Dimensional Case Infinite Dimensional Case considered here is Application to our Problem Numerical Simulations Boundary Control u t + u x + u xxx + uu x = 0 , x ∈ [ 0 , L ] , t ≥ 0 from the left Control Design Linear System + Boundary conditions, for instance posed on Nonlinear System t ≥ 0 u ( t , 0 ) = u ( t , L ) = u x ( t , L ) = 0 , + Initial data u ( 0 , x ) = u 0 ∈ L 2 ( 0 , L )

Asymptotic behaviour Control System We are interested in the long-time behavior of the energy Internal Control Linear System � L Nonlinear System | u ( t , x ) | 2 dx . E ( t ) = Boundary Control from the right 0 Finite Dimensional Case Infinite Dimensional Case More precisely we want to prove the exponential decay of E ( t ) Application to our Problem Numerical Simulations as t goes to infinity. Boundary Control from the left E ( t ) ≤ Ce − ω t E ( 0 ) , ∀ t ∈ [ 0 , ∞ ) Control Design Linear System Nonlinear System Let us start considering the linear equation u t + u x + u xxx = 0 , u ( t , 0 ) = u ( t , L ) = u x ( t , L ) = 0 , u ( 0 , · ) = u 0

Asymptotic behaviour Control System Internal Control Linear System By performing integration by parts in the equation Nonlinear System Boundary Control � L from the right Finite Dimensional Case ( u t + u x + u xxx ) u dx = 0 Infinite Dimensional Case 0 Application to our Problem Numerical Simulations we get Boundary Control from the left Control Design � L d Linear System | u ( t , x ) | 2 dx = −| u x ( t , 0 ) | 2 ≤ 0 . Nonlinear System dt 0 The energy is non-increasing, but is it strictly decreasing? Remember we are looking for an exponential decay.

Solutions with constant energy Control System The energy is not decreasing. In fact there are solutions with Internal Control constant energy! Linear System Nonlinear System Boundary Control from the right For instance, if L = 2 π and Finite Dimensional Case Infinite Dimensional Case Application to our Problem u 0 = ( 1 − cos ( x )) , Numerical Simulations Boundary Control from the left the solution of the linear KdV u t + u x + u xxx = 0 is stationary Control Design Linear System Nonlinear System u ( t , x ) = ( 1 − cos ( x )) which satisfies u x ( t , 0 ) = 0 for any t ≥ 0 and then � L E ( t ) = d ˙ | u ( t , x ) | 2 dx = 0 dt 0

Critical domains Control System For the linear KdV equation there exist constant energy Internal Control solutions if and only if Linear System Nonlinear System � � � k 2 + k ℓ + ℓ 2 Boundary Control from the right ; k , ℓ ∈ N ∗ L ∈ N := 2 π . Finite Dimensional Case 3 Infinite Dimensional Case Application to our Problem Numerical Simulations This phenomena is linked to the controllability of a linear KdV Boundary Control from the left from the boundary. Control Design Linear System (Controllability) Nonlinear System Take a look at the linear control system u t + u x + u xxx = 0 u ( t , 0 ) = u ( t , L ) = 0 , u x ( t , L ) = κ ( t ) , u ( 0 , · ) = 0

(Controllability) Control System ◮ Linear KdV is controllable ⇔ the following map is onto Internal Control Linear System B : κ ∈ L 2 ( 0 , T ) �→ u ( T , · ) ∈ L 2 ( 0 , L ) . Nonlinear System Boundary Control from the right ◮ The map B is onto ⇔ the following inequality holds Finite Dimensional Case Infinite Dimensional Case Application to our Problem � B ∗ ( φ T ) � L 2 ( 0 , T ) ≥ C � φ T � L 2 ( 0 , L ) (Obs) Numerical Simulations Boundary Control from the left ◮ The map B is onto ⇔ its adjoint system is observable, i.e. Control Design Linear System Nonlinear System � φ x ( t , L ) � L 2 ( 0 , T ) ≥ C � φ T � L 2 ( 0 , L ) (Obs) where φ = φ ( t , x ) satisfies,  φ t + φ x + φ xxx = 0 ,  φ ( t , 0 ) = φ ( t , L ) = φ x ( t , 0 ) = 0 , (Adj)  φ ( T , · ) = φ T .

(Controllability) Control System Internal Control Linear System Nonlinear System Boundary Control Theorem (Rosier 97) from the right Finite Dimensional Case ◮ The linear KdV system is controllable iff L / ∈ N . Infinite Dimensional Case Application to our Problem Numerical Simulations ◮ If L / ∈ N , the nonlinear system (KdV) is locally exactly Boundary Control controllable. from the left Control Design Linear System Nonlinear System Theorem (Coron-Cr´ epeau 04, EC 07, EC-Cr´ epeau 09) Let L ∈ N , there exists T L ≥ 0 such that (KdV) is locally exactly controllable in L 2 ( 0 , L ) if T ≥ T L .

Back to stabilization We will design some feedback control laws in order to get Control System E ( t ) ≤ Ce − ω t E ( 0 ) , Internal Control ∀ t ≥ 0 . Linear System Nonlinear System Internal control: Boundary Control from the right Finite Dimensional Case u t + u x + u xxx + uu x = F ( u ) , u ( 0 , · ) = u 0 , Infinite Dimensional Case Application to our Problem u ( t , 0 ) = 0 , u ( t , L ) = 0 , u x ( t , L ) = 0 , Numerical Simulations Boundary Control from the left Boundary control from the right: Control Design Linear System Nonlinear System u t + u x + u xxx + uu x = 0 , u ( 0 , · ) = u 0 , u ( t , 0 ) = 0 , u ( t , L ) = 0 , u x ( t , L ) = F ω ( u ) , Boundary control from the left: u t + u x + u xxx + uu x = 0 , u ( 0 , · ) = u 0 , u ( t , 0 ) = K ω ( u ) , u ( t , L ) = 0 , u x ( t , L ) = 0 ,

Control System Internal Control Linear System Nonlinear System Boundary Control from the right Finite Dimensional Case Infinite Dimensional Case Internal Control Application to our Problem Numerical Simulations Boundary Control from the left Control Design Linear System Nonlinear System

Internal Control Equation with internal control Control System Internal Control u t + u x + u xxx + uu x = F Linear System Nonlinear System Boundary Control We consider a feedback law in the form from the right Finite Dimensional Case Infinite Dimensional Case F ( u ) = − au Application to our Problem Numerical Simulations Boundary Control where a ∈ L ∞ ( 0 , L ; R + ) satisfies from the left Control Design � a ( x ) ≥ a 0 > 0 , Linear System ∀ x ∈ O , Nonlinear System where O is nonempty open subset of ( 0 , L ) . Closed-loop system u t + u x + u xxx + a ( x ) u + uu x = 0 , u ( t , 0 ) = u ( t , L ) = u x ( t , L ) = 0 , u ( 0 , · ) = u 0 ( · ) .

Internal Control - Linear Control System A natural strategy is to consider first the linearized equation Internal Control Linear System around the origin Nonlinear System Boundary Control from the right u t + u x + u xxx + au = 0 , Finite Dimensional Case Infinite Dimensional Case u ( t , 0 ) = u ( t , L ) = u x ( t , L ) = 0 , (1) Application to our Problem u ( 0 , · ) = u 0 ( · ) , Numerical Simulations Boundary Control from the left and prove the exponential decay of its solutions. Control Design Linear System Nonlinear System Theorem (Perla-Vasconcellos-Zuazua 02) Let L > 0 and a = a ( x ) as before. There exist C , ω > 0 : � u ( t , · ) � L 2 ( 0 , L ) ≤ Ce − ω t � u 0 � L 2 ( 0 , L ) , ∀ t ≥ 0 for any solution of (1) with u 0 ∈ L 2 ( 0 , L ) .