Global stabilization of a Korteweg-de Vries equation with saturating - PowerPoint PPT Presentation

Problem statement Main results Stability analysis Simulation Perspectives Global stabilization of a Korteweg-de Vries equation with saturating distributed control Swann MARX 1 A joint work with Eduardo CERPA 2 , Christophe PRIEUR 1 and Vincent ANDRIEU 3 . 1 GIPSA-lab, Grenoble, France. 2 Universidad Técnica Federico Santa María, Valparaíso, Chile. 3 LAGEP , Lyon, France and Bergische Universität Wuppertal, Wuppertal, Germany. Coron fest 1/22

Problem statement Main results Stability analysis Simulation Perspectives Table of contents Problem statement 1 Main results 2 Stability analysis 3 Simulation 4 5 Perspectives 2/22

Problem statement Main results Stability analysis Simulation Perspectives KdV equation with a distributed control The Korteweg-de Vries describes approximately long waves in water of relatively shallow depth. For all L > 0, it is described as follows  y t + y x + y xxx + yy x + u = 0   y ( t , 0 ) = y ( t , L ) = y x ( t , L ) = 0 (KdV-u)  y ( 0 , x ) = y 0  Stabilization References : [Perla Menzala et al., 2002], [Rosier and Zhang, 2006], [Pazoto, 2005] In the following, we will focus on (KdV-u). 3/22

Problem statement Main results Stability analysis Simulation Perspectives KdV equation with a distributed control The Korteweg-de Vries describes approximately long waves in water of relatively shallow depth. For all L > 0, it is described as follows  y t + y x + y xxx + yy x + u = 0   y ( t , 0 ) = y ( t , L ) = y x ( t , L ) = 0 (KdV-u)  y ( 0 , x ) = y 0  Stabilization References : [Perla Menzala et al., 2002], [Rosier and Zhang, 2006], [Pazoto, 2005] In the following, we will focus on (KdV-u). The paper [Cerpa, 2014] is a good introduction to the control of this equation. 3/22

Problem statement Main results Stability analysis Simulation Perspectives Case without control : critical length phenomenon  y t + y x + y xxx = 0   y ( t , 0 ) = y ( t , L ) = y x ( t , L ) = 0 (LKDV)  y ( 0 , x ) = y 0 ( x )  Critical length set for the linear KdV equation [Rosier, 1997] � � � � k 2 + kl + l 2 k , l ∈ N ∗ If L ∈ 2 π , there exist solutions of 3 (LKDV) for which the energy does not decay to zero. 4/22

Problem statement Main results Stability analysis Simulation Perspectives Case without control : critical length phenomenon Critical length set for the linear KdV equation [Rosier, 1997] � � � � k 2 + kl + l 2 k , l ∈ N ∗ If L ∈ 2 π , there exist solutions of 3 (LKDV) for which the energy does not decay to zero. With L = 2 π , y e = 1 − cos ( x ) is an equilibrium solution. Indeed y e t + y e x + y e xxx = 0 Thus, with y 0 ( x ) = 1 − cos ( x ) , the solution does not decay to zero. 4/22

Problem statement Main results Stability analysis Simulation Perspectives Case without control : critical length phenomenon Critical length set for the linear KdV equation [Rosier, 1997] � � � � k 2 + kl + l 2 k , l ∈ N ∗ If L ∈ 2 π , there exist solutions of 3 (LKDV) for which the energy does not decay to zero. Local asymptotic stability of 0 with L = 2 π [Chu, Coron and Shang, 2015] Let us assume that L = 2 π and u = 0. Then 0 ∈ L 2 ( 0 , L ) is (locally) asymptotically stable for (KdV-u). Thus the nonlinearity yy x improves the stability. Note that the stability is local . 4/22

Problem statement Main results Stability analysis Simulation Perspectives Global stabilization of y = 0 with a distributed control without constraint : general case In [Pazoto, 2005] and [Rosier and Zhang, 2006], the authors use a control u ( t , x ) = a ( x ) y ( t , x ) with a defined as follows � 0 < a 0 ≤ a ( x ) ≤ a 1 , ∀ x ∈ ω, a = where ω is a nonempty open subset of ( 0 , L ) . (loc-control) They prove that the origin of (KdV-u) is globally asymptotically stabilized with such a control. 5/22

Problem statement Main results Stability analysis Simulation Perspectives Saturation function : finite dimension Usual saturation For all s ∈ R , the function sat satisfies  − u 0 if s ≤ − u 0  sat ( s ) = s if − u 0 ≤ s ≤ u 0 , u 0 if s ≥ u 0 .  where u 0 denotes the saturation level. Saturating a controller can lead to catastrophic behavior for the stability of the system. 6/22

Problem statement Main results Stability analysis Simulation Perspectives Saturation in infinite dimension Saturation operator For any function s and all x ∈ [ 0 , L ] , the operator sat satisfies sat ( s )( x ) = sat ( s ( x )) (SAT- loc ) 7/22

Problem statement Main results Stability analysis Simulation Perspectives Illustration of the saturation 1 0.8 0.6 0.4 cos(x) and sat(cos(x)) 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 0 0.5 1 1.5 2 2.5 3 x F IGURE : x ∈ [ 0 , π ] . Red : sat ( cos ( x )) and u 0 = 0 . 5, Blue : cos ( x ) . 8/22

Problem statement Main results Stability analysis Simulation Perspectives Distributed control saturated System under consideration  y t + y x + y xxx + yy x + sat ( ay ) = 0 ,   y ( t , 0 ) = y ( t , L ) = 0 ,   (KdV-sat) y x ( t , L ) = 0 ,     y ( 0 , x ) = y 0 ( x ) . Remark : A similar work has been done on the wave equation [Prieur, Tarbouriech and Gomes da Silva Jr, 2016] and the linear KdV equation [SM, Cerpa, Prieur and Andrieu, 2015]. 9/22

Problem statement Main results Stability analysis Simulation Perspectives Well-posedness theorem Theorem (Well posedness (SM-Cerpa-Prieur-Andrieu)) For any initial conditions y 0 ∈ L 2 ( 0 , L ) , there exists a unique mild solution y ∈ B ( T ) := C ( 0 , T ; L 2 ( 0 , L )) ∩ L 2 ( 0 , T ; H 1 ( 0 , L )) to (KdV-sat) . B ( T ) is endowed with the following norm � 1 / 2 �� T � y ( t ) � 2 � y � B ( T ) := max t ∈ [ 0 , T ] � y ( t ) � L 2 ( 0 , L ) + H 1 ( 0 , L ) dt . 0 � � �� L Recall : L 2 ( 0 , L ) = 0 f ( x ) 2 dx < ∞ f , � f � L 2 ( 0 , L ) := � � H 1 ( 0 , L ) = f , � f � H 1 ( 0 , L ) := � f � L 2 ( 0 , L ) + � f ′ � L 2 ( 0 , L ) < ∞ 10/22

Problem statement Main results Stability analysis Simulation Perspectives Global asymptotic stability theorem Theorem (Global asymptotic stability(SM-Cerpa-Prieur-Andrieu)) Recall : α is said to be a There exist class K ∞ function if a positive value µ ⋆ , it is nonnegative, a class K ∞ function α , it is strictly such that, for any initial condition y 0 ∈ increasing, L 2 ( 0 , L ) , every solution y to (KdV-sat) lim r → + ∞ α ( r ) = + ∞ satisfies, for all t ≥ 0 α ( 0 ) = 0. � y ( t , . ) � L 2 ( 0 , L ) ≤ α ( � y 0 � L 2 ( 0 , L ) ) e − µ ⋆ t , 11/22

Problem statement Main results Stability analysis Simulation Perspectives Global asymptotic stability theorem Theorem (Global asymptotic stability(SM-Cerpa-Prieur-Andrieu)) There exist a positive value µ ⋆ , Example : The function a class K ∞ function α , r �→ α ( r ) = r such that, for any initial condition y 0 ∈ L 2 ( 0 , L ) , every solution y to (KdV-sat) is a class K ∞ function. satisfies, for all t ≥ 0 � y ( t , . ) � L 2 ( 0 , L ) ≤ α ( � y 0 � L 2 ( 0 , L ) ) e − µ ⋆ t , 11/22

Problem statement Main results Stability analysis Simulation Perspectives Global asymptotic stability theorem Theorem (Global asymptotic stability(SM-Cerpa-Prieur-Andrieu)) There exist a positive value µ ⋆ , We have in fact a class K ∞ function α , Globally asymptotically stable + Semi-globally such that, for any initial condition y 0 ∈ L 2 ( 0 , L ) , every solution y to (KdV-sat) exponentially stable satisfies, for all t ≥ 0 � y ( t , . ) � L 2 ( 0 , L ) ≤ α ( � y 0 � L 2 ( 0 , L ) ) e − µ ⋆ t , 11/22

Problem statement Main results Stability analysis Simulation Perspectives Semi-global exponential stability Semi-global exponential stability The origin for the system (KdV-sat) is said to be semi-globally exponentially stable in L 2 ( 0 , L ) if for any r > 0 there exist two constants C = C ( r ) > 0 and µ = µ ( r ) > 0 such that for any y 0 ∈ L 2 ( 0 , L ) such that � y 0 � L 2 ( 0 , L ) ≤ r the weak solution y = y ( t , x ) to (KdV-sat) satisfies � y ( t , . ) � L 2 ( 0 , L ) ≤ C � y 0 � L 2 ( 0 , L ) e − µ t ∀ t ≥ 0 . 12/22

Problem statement Main results Stability analysis Simulation Perspectives Sector condition Sector condition Let r be a positive value. Let a be defined by � 0 < a 0 ≤ a ( x ) ≤ a 1 , ∀ x ∈ ω, a = where ω is a nonempty open subset of ( 0 , L ) . Given s ∈ L ∞ ( 0 , L ) satisfying, for all x ∈ [ 0 , L ] , | s ( x ) | ≤ r , we have � � sat ( a ( x ) s ( x )) − k ( r ) a ( x ) s ( x ) s ( x ) ≥ 0 , ∀ x ∈ [ 0 , L ] , (sec-cond- loc ) with � u 0 � k ( r ) = min a 1 r , 1 . 13/22

Problem statement Main results Stability analysis Simulation Perspectives Some estimates Bounded initial conditions Let r be a positive value such that � y 0 � L 2 ( 0 , L ) ≤ r . Some estimates � T � y ( T , . ) � 2 L 2 ( 0 , L ) = � y 0 � 2 | y x ( σ, 0 ) | 2 dt L 2 ( 0 , L ) − 0 (Stability) � T � L − 2 sat ( ay ) ydxdt 0 0 L 2 ( 0 , T ; H 1 ( 0 , L )) ≤ 8 T + 2 L L 2 ( 0 , L ) + TK � y � 2 � y 0 � 2 27 � y 0 � 4 L 2 ( 0 , L ) . 3 (Regularity) 14/22