Global stabilization of a Korteweg-de Vries equation with saturating distributed control

Transcription

1 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

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

3 KdV equation with a distributed control The Korteweg-de Vries describes approximately long waves in water of relatively shallow depth. For all L >, it is described as follows y t + y x + y xxx + yy x + u = y(t, ) = y(t, L) = y x (t, L) = (KdV-u) y(, x) = y Stabilization References : [Perla Menzala et al., 22], [Rosier and Zhang, 26], [Pazoto, 25] In the following, we will focus on (KdV-u).

4 KdV equation with a distributed control The Korteweg-de Vries describes approximately long waves in water of relatively shallow depth. For all L >, it is described as follows y t + y x + y xxx + yy x + u = y(t, ) = y(t, L) = y x (t, L) = (KdV-u) y(, x) = y Stabilization References : [Perla Menzala et al., 22], [Rosier and Zhang, 26], [Pazoto, 25] In the following, we will focus on (KdV-u). The paper [Cerpa, 214] is a good introduction to the control of this equation.

5 Case without control : critical length phenomenon y t + y x + y xxx = y(t, ) = y(t, L) = y x (t, L) = y(, x) = y (x) (LKDV) Critical length set for the linear KdV equation [Rosier, 1997] { / } If L 2π k, l N, there exist solutions of k 2 +kl+l 2 3 (LKDV) for which the energy does not decay to zero.

6 Case without control : critical length phenomenon Critical length set for the linear KdV equation [Rosier, 1997] { / } If L 2π k, l N, there exist solutions of k 2 +kl+l 2 3 (LKDV) for which the energy does not decay to zero. With L = 2π, y e = 1 cos(x) is an equilibrium solution. Indeed y e t + y e x + y e xxx = Thus, with y (x) = 1 cos(x), the solution does not decay to zero.

7 Case without control : critical length phenomenon Critical length set for the linear KdV equation [Rosier, 1997] { / } If L 2π k, l N, there exist solutions of k 2 +kl+l 2 3 (LKDV) for which the energy does not decay to zero. Local asymptotic stability of with L = 2π [Chu, Coron and Shang, 215] Let us assume that L = 2π and u =. Then L 2 (, L) is (locally) asymptotically stable for (KdV-u). Thus the nonlinearity yy x improves the stability. Note that the stability is local.

8 Global stabilization of y = with a distributed control without constraint : general case In [Pazoto, 25] and [Rosier and Zhang, 26], the authors use a control u(t, x) = a(x)y(t, x) with a defined as follows { < a a(x) a 1, x ω, a = where ω is a nonempty open subset of (, L). (loc-control) They prove that the origin of (KdV-u) is globally asymptotically stabilized with such a control.

9 Saturation function : finite dimension Usual saturation For all s R, the function sat satisfies u if s u sat(s) = s if u s u, u if s u. where u denotes the saturation level. Saturating a controller can lead to catastrophic behavior for the stability of the system.

10 Saturation in infinite dimension Saturation operator For any function s and all x [, L], the operator sat satisfies sat(s)(x) = sat(s(x)) (SAT-loc)

11 Illustration of the saturation cos(x) and sat(cos(x)) x FIGURE: x [, π]. Red : sat(cos(x)) and u =.5, Blue : cos(x).

12 Distributed control saturated System under consideration y t + y x + y xxx + yy x + sat(ay) =, y(t, ) = y(t, L) =, y x (t, L) =, y(, x) = y (x). (KdV-sat) Remark : A similar work has been done on the wave equation [Prieur, Tarbouriech and Gomes da Silva Jr, 216] and the linear KdV equation [SM, Cerpa, Prieur and Andrieu, 215].

13 Well-posedness theorem Theorem (Well posedness (SM-Cerpa-Prieur-Andrieu)) For any initial conditions y L 2 (, L), there exists a unique mild solution y B(T ) := C(, T ; L 2 (, L)) L 2 (, T ; H 1 (, L)) to (KdV-sat). B(T ) is endowed with the following norm y B(T ) := max t [,T ] y(t) L 2 (,L) + ( ) 1/2 T y(t) 2 H 1 (,L) dt. { } Recall : L 2 (, L) = f, f L 2 (,L) := L f (x)2 dx < } H 1 (, L) = {f, f H 1 (,L) := f L 2 (,L) + f L 2 (,L) <

14 Global asymptotic stability theorem Theorem (Global asymptotic stability(sm-cerpa-prieur-andrieu)) There exist a positive value µ, a class K function α, such that, for any initial condition y L 2 (, L), every solution y to (KdV-sat) satisfies, for all t y(t,.) L 2 (,L) α( y L 2 (,L) )e µ t, Recall : α is said to be a class K function if it is nonnegative, it is strictly increasing, lim r + α(r) = + α() =.

15 Global asymptotic stability theorem Theorem (Global asymptotic stability(sm-cerpa-prieur-andrieu)) There exist a positive value µ, a class K function α, such that, for any initial condition y L 2 (, L), every solution y to (KdV-sat) satisfies, for all t Example : The function r α(r) = r is a class K function. y(t,.) L 2 (,L) α( y L 2 (,L) )e µ t,

16 Global asymptotic stability theorem Theorem (Global asymptotic stability(sm-cerpa-prieur-andrieu)) There exist a positive value µ, a class K function α, such that, for any initial condition y L 2 (, L), every solution y to (KdV-sat) satisfies, for all t We have in fact Globally asymptotically stable + Semi-globally exponentially stable y(t,.) L 2 (,L) α( y L 2 (,L) )e µ t,

17 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 (, L) if for any r > there exist two constants C = C(r) > and µ = µ(r) > such that for any y L 2 (, L) such that y L 2 (,L) r the weak solution y = y(t, x) to (KdV-sat) satisfies y(t,.) L 2 (,L) C y L 2 (,L) e µt t.

18 Sector condition Sector condition Let r be a positive value. Let a be defined by { < a a(x) a 1, x ω, a = where ω is a nonempty open subset of (, L). Given s L (, L) satisfying, for all x [, L], s(x) r, we have ( ) sat(a(x)s(x)) k(r)a(x)s(x) s(x), x [, L], with { } u k(r) = min a 1 r, 1. (sec-cond-loc)

19 Some estimates Bounded initial conditions Let r be a positive value such that Some estimates y L 2 (,L) r. T y(t,.) 2 L 2 (,L) = y 2 L 2 (,L) y x (σ, ) 2 dt y 2 L 2 (,T ;H 1 (,L)) T 2 L sat(ay)ydxdt (Stability) 8T + 2L 3 y 2 L 2 (,L) + TK 27 y 4 L 2 (,L). (Regularity)

20 Claim Claim For any T > and any r > there exists a positive constant C = C(r, T ) > 1 such that for any solution y to (KdV-sat) with an initial condition y L 2 (, L) such that y L 2 (,L) r, it holds that ( T y 2 L 2 (,L) C y x (t, ) 2 dt T +2 L sat(ay(t, x))y(t, x)dtdx ). (Claim) Proving this Claim allows to prove the semi-global exponential stability of the origin of the equation when using u = sat(ay).

21 Assuming the Claim Indeed, if this claim holds, we obtain with (Stability) y(kt,.) 2 L 2 (,L) γk y 2 L 2 (,L) k where 1 1 C := γ (, 1). Once again with (Stability), we have y(t,.) L 2 (,L) y(kt,.) L 2 (,L) for kt t (k + 1)T. Therefore, if we assume the Claim, we have, for all t, y(t,.) 2 L 2 (,L) 1 γ y L 2 (,L) e log γ T t

22 Proof of the Claim Another useful estimate T y 2 L 2 (,L) T +2 L T y(t, x) 2 dxdt + (T t) L T sat(ay)ydxdt. We prove also the Claim with ( T y 2 L 2 (,T ;L 2 (,L)) C 1 y x (t, ) 2 dt T +2 L (T t) y x (t, ) 2 dt sat(ay(t, x))y(t, x)dtdx (Stability-2) ),

23 Proof of the Claim We proceed by contradiction. Suppose the claim fails to be true. Then there exists a sequence of solution y n B(T ) of (KdV-sat) such that Bounded initial conditions y n (,.) L 2 (,L) r (IC-bounded) Contradiction argument lim n + T y n 2 L 2 (,T ;L 2 (,L)) y x n (t, ) 2 dt + 2 T L sat(ay n )y n dxdt = +. (Contra-argu)

24 Using the sector condition Note that in a first hand we have, from the fact that the solution is bounded in L 2 (, L) and from the hidden regularity (Regularity) Therefore, y n 2 L 2 (,T ;H 1 (,L)) L (, L)-regularity x [, L], T 8T + 2L β := r 2 + TK 3 27 r 4. y n (t, x) 2 dt L y n 2 L 2 (,T ;H 1 (,L)) Lβ. (L -reg)

25 Using the sector condition Now let us consider Ω i [, T ] defined as follows { } Ω i = t [, T ], sup y(t, x) > i x [,L] In the following, we will denote by Ω c i its complement. It is defined by { } Ω c i = t [, T ], sup y(t, x) i x [,L].. (Space-sec-cond)

26 Using the sector condition Since the function t sup x [,L] y n (t, x) 2 is a nonnegative function, we have T sup y n (t, x) 2 dt sup y n (t, x) 2 dt i 2 ν(ω i ), x [,L] Ω i x [,L] where ν(ω i ) denotes the Lebesgue measure of Ω i. Therefore, with T y n (t, x) 2 dt Lβ, we obtain ν(ω i ) Lβ i 2. We deduce from the previous equation that ( max T Lβ ) i 2, ν(ω c i ) T. (Lebesgue-measure)

27 Using the sector condition Moreover, since y(t, x) i in Ω c i, then we can use the sector condition ( ) sat(a(x)s(x)) k(r)a(x)s(x) s(x), x [, L], and we obtain, for all i N T L sat(ay n )y n dtdx Ω c i L ak(i)(y n ) 2 dtdx. (Stability-sec-cond)

28 Using the sector condition Let λ n := y n L 2 (,T ;L 2 (,L)) and v n (t, x) = y n (t,x) λ. Notice that λ n n is bounded. Hence, up to extracting a subsequence, we may assume that λ n λ. Then v n fullfills vt n + vx n + vxxx n + λ n v n vx n + sat(aλn v n ) λ n =, v n (t, ) = v n (t, L) = vx n (t, L) =, v n L 2 (,T ;L 2 (,L)) = 1.

29 Using the sector condition Due to the contradiction argument, that is lim n + T then we have T y n 2 L 2 (,T ;L 2 (,L)) y x n (t, ) 2 dt + 2 T L sat(ay n )y n dxdt = +, v n x (t, ) 2 dt + 2 T L sat(aλ n v n ) λ n v n dtdx.

30 Using the sector condition And since we have T L sat(aλ n v n ) λ n v n dtdx Ω c i L ak(i)(v n ) 2 dtdx. then we obtain that vx n (t, ) 2 dt + 2 T Ω c i L k(i)a(v n ) 2 dtdx

31 Conclusion Using a result of Aubin Lions, we prove that v n converges strongly to v L 2 (, T ; L 2 (, L)). Moreover, we have, for all i N v(t, x) =, x ω, t i N Ω c i, and v x(t, ) =, t (, T ). With we know that ( max T Lβ ) i 2, ν(ω c i ) T, ν ( i N Thus, for almost every t [, T ] Ω c i ) = T. v(t, x) =, x ω, and v x (t, ) =.

32 Conclusion Moreover, v solves v t + v x + v xxx + λvv x =, v(t, ) = v(t, L) = v x (t, L) =, v L 2 (,T ;L 2 (,L)) = 1. Thus v B(T ) and in particular v is continuous. Thus v(t, x) =, x ω, t [, T ], and v x (t, ) =, t (, T ).

33 Conclusion With a unique continuation argument given by [Saut and Scheurer, 1987], we prove that v(t, x) =, x [, L], t [, T ] It is in contradiction with v L 2 (,T ;L 2 (,L)) = 1. Thus, the claim is true and we have Semi-global exponential stability Let y be a solution to the KdV equation with a saturated control. Thus, for every initial condition y satisfying y L 2 (,L) r, there exist positive values µ := µ(r) and K := K (r) such that y L 2 (,L) Ke µt y L 2 (,L), t

34 y (x) = 1 cos(x), L = 2π and ω = [ 1 3 L, 2 3 L] y w (t,x) y(t, x) t 6 2 x t 6 2 x 4 FIGURE: Solution y(t,x) with a localized feedback law without saturation FIGURE: Solution y(t,x) with a localized feedback law saturated ; u =.5

35 y (x) = 1 cos(x), L = 2π and ω = [ 1 3 L, 2 3 L] sat loc (ay)(t,x) y 2 L 2 (,L) y w 2 L 2 (,L) E t 6 2 x t FIGURE: Control f = sat(ay) where ω = [ 1 3 L, 2 3 L], u =.5 FIGURE: Energy functions of the solutions

36 Perspectives Very few papers deal with stabilization of PDEs with bounded boundary controls (see [Lasiecka and Seidman, 23]). For instance, is the origin for the following linear hyperbolic equation { zt + Λz x =, where λ(λ) >, stable? z(t, ) = Hz(t, 1) + Bsat(Kz(t, 1)),

37 THANK YOU FOR YOUR ATTENTION! HAPPY BIRTHDAY JEAN-MICHEL!

38 Cerpa, E. (214). Control of a Korteweg-de Vries equation : a tutorial. Mathematical Control and Related Fields, 4(1) : Lasiecka, I. and Seidman, T. I. (23). Strong stability of elastic control systems with dissipative saturating feedback. Systems & Control Letters, 48 : Marx, S., Cerpa, E., Prieur, C., and Andrieu, V. (July 215). Stabilization of a linear Korteweg-de Vries with a saturated internal control. In Proceedings of the European Control Conference, page To appear, Linz, AU. Pazoto, A. (25). Unique continuation and decay for the Korteweg-de Vries equation with localized damping.

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40 ESAIM : Control, Optimisation and Calculus of Variations, 2 : Rosier, L. and Zhang, B.-Y. (26). Global stabilization of the generalized Korteweg de Vries equation posed on a finite domain. SIAM Journal on Control and Optimization, 45(3) : Saut, J.-C. and Scheurer, B. (1987). Unique continuation for some evolution equations. J. Differential Equations, 66 :