Numerical control and inverse problems and Carleman estimates

Transcription

1 Numerical control and inverse problems and Carleman estimates Enrique FERNÁNDEZ-CARA Dpto. E.D.A.N. - Univ. of Sevilla from some joint works with A. MÜNCH Lab. Mathématiques, Univ. Blaise Pascal, C-F 2, France D.A. SOUZA Dpto. E.D.A.N. - Univ. of Sevilla and others: J. LIMACO, L.A. MEDEIROS, S. MENEZES, M.C. SANTOS,...

2 Outline 1 Background, controllability, examples and basic results Controllability problems Examples and basic results 2 Numerics: methods and results The Fursikov-Imanuvilov strategy for the heat equation Extensions to some nonlinear problems Exact controllability to the trajectories for the Navier-Stokes equations Final comments

3 Background, controllability, examples and basic results Controllability problems Controllability problems: yt A(y) =B(v) y(0) =y 0 + (maybe) other conditions y :[0, T ] 7! H is the state, v 2 U ad is the control The goal: For any y 0, find v with... Null controllability (NC):... y(t )=0 Exact controllability to the trajectories (ECT):...y(T )=ŷ(t ), where...

4 Controllability problems, examples and basic results Examples and basic results N-dimensional heat, R N : 8 < y t y = v1!, (x, t) 2 (0, T ) (H N ) y(x, t) =0, (x, t) (0, T ) : y(x, 0) =y 0 (x), x 2 We assume:! (small) NC? ECT? OK, equivalent [Lebeau-Robbiano 1995], [Fursikov-Imanuvilov 1996] Stokes and Oseen: 8 < y t +(z r)y y + rp = v1!, r y = 0, (x, t) 2 (0, T ) (Os) y(x, t) =0, (x, t) (0, T ) : y(x, 0) =y 0 (x), x 2 NC? ECT OK, it suffices z 2 L 1, z t 2 L 2 t (L x ), r z = 0 [Fursikov-Imanuvilov 1999], [Imanuvilov 2001] [EFC, Guerrero, Imanuvilov & Puel 2004] Applications: Heating and cooling, controlling a (creeping) fluid

5 Controllability problems, examples and basic results Examples and basic results Nonlinear-nonlocal parabolic system: 8 y t a( R >< y, R z) y = f (y, z) + v1!, (x, t) 2 (0, T ) z t b( R (NN) y, R z) y = g(y, z), (x, t) 2 (0, T ) y(x, t) =z(x, t) =0, (x, t) (0, T ) >: y(x, 0) =y 0 (x), z(x, 0) =z 0 (x), x 2 Several difficulties, mainly: Nonlinear a, b, f, g Nonlocal nonlinearities Only one control Applications: Controlling reacting media, interacting populations, among others

6 Controllability problems, examples and basic results Examples and basic results Nonlinear-nonlocal parabolic system: 8 y t a( R >< y, R z) y = f (y, z)+v1!, (x, t) 2 (0, T ) z t b( R (NN) y, R z) y = g(y, z), (x, t) 2 (0, T ) y(x, t) =z(x, t) =0, (x, t) (0, T ) >: y(x, 0) =y 0 (x), z(x, 0) =z 0 (x), x 2 We assume: a, b, f, g 2 Cb, 1 a a 0 > 0, b b 0 > yg(0, 0) 6= 0 NC? ECT? OPEN What we know: local NC (local ECT is unknown) Theorem [Clark-EFC-Limaco-Medeiros] 9" > 0 such that k(y 0, z 0 )k L 2 apple " )9null controls For the proof (ideas from Imanuvilov & Yamamoto... ) 1 (NC) = F (y, z, v) =0 in an appropriate space, with F 1 (y, z, v) :=y t a( R y, R z) y f (y, z) v1! F 2 (y, z, v) :=z t b( R y, R z) y g(y, z) F 3 (y, z, v) :=y(0) y 0 F 4 (y, z, v) :=z(0) z 0 2 Apply Liusternik s local inversion Theorem

7 Controllability problems, examples and basic results Examples and basic results Navier-Stokes: (NS) 8 < : y t +(y r)y y + rp = v1!, r y = 0 y(x, t) =0, (x, t) (0, T ) y(x, 0) =y 0 (x) A variant - Boussinesq: 8 y t +(y r)y y + rp = k + v1!, r y = 0 >< t + y r = w1! (B) y(x, t) =0, (x, t) =0, (x, t) (0, T ) >: y(x, 0) =y 0 (x), (x, 0) = 0 (x) Applications: Controlling a real fluid

8 Controllability problems, examples and basic results Examples and basic results Navier-Stokes: (NS) 8 < : NC? ECT? OPEN What we know: local ECT y t +(y r)y y + rp = v1!, r y = 0 y(x, t) =0, (x, t) (0, T ) y(x, 0) =y 0 (x) Theorem [EFC-Guerrero-Imnuvilov-Puel] Fix a solution (y, p), with y 2 L 1 9" > 0 such that ky 0 y(0)k H 1 0 apple " )9controls such that y(t )=y(t ) For the proof: 1 Reduce ECT to NC 2 Again: (NC) = F(y, v) =0 in an appropriate space 3 Again: apply Liusternik s Theorem

9 Controllability problems, examples and basic results Examples and basic results The Ladyzhenskaya-Smagorinsky model: 8 < y t +(y r)y a( R ry 2 ) y + rp = v1!, r y = 0 (LS) y(x, t) =0, (x, t) (0, T ) : y(x, 0) =y 0 (x) Applications: Controlling a turbulent fluid We assume: a 2 C 1 b, a a 0 > 0 NC? ECT? OPEN What we know: local NC Theorem [EFC-Limaco-Menezes] 9" > 0 such that ky 0 k L 2 apple " )9null controls Arguments of the same kind Attention: local ECT is again open

10 Some controllability results and open questions Nonlinear problems THINGS WORK REASONABLY FOR LINEAR PROBLEMS CONTRARILY: VERY FEW IS KNOWN FOR NONLINEAR PROBLEMS In general, only local results: Near the target at t = 0 ) controllability at t = T Two general techniques: fixed-point and inverse function Both relying on linearization! Many recent contributions: Zuazua, Fursikov, Imanuvilov, Puel, Coron, Yamamoto, Cannarsa,... and many open questions! In the sequel: numerics

11 Numerics: methods and results The Fursikov-Imanuvilov strategy for the heat equation The problem: (H 1 ) y t y xx = v1! +... Goal: Good v such that y(t )=0 numerically, i.e. ky(t )k L Glowinski, JL Lions, Boyer-Hubert-Le Rousseau, Münch... The classical way: Minimize RR! (0,T ) v 2 dx dt Subject to (H 1 ), y(t )=0 Eventually: penalize the functional and/or the PDEs But: This leads to oscillations!

12 Numerics: methods and results The Fursikov-Imanuvilov strategy for the heat equation ALTERNATIVE METHOD: introduce, 0 e C(x)/(T t) and solve The weighted (FI) formulation of the NC problem: Minimize RR ( 2 y 2 + 1! 2 0 v 2 ) Subject to (H 1 ), y(t )=0 Notation: Ly = y t y xx, L p = p t p xx Euler-Lagrange characterization of the optimal (v, y): with 8 >< >: y = 2 L p, v = 2 0 p! (0,T ) L( 2 L p) !p = 0, (x, t) 2 (0, 1) (0, T ) p(0, t) =p(1, t) =0, t 2 (0, T ) ( 2 L p) x=0 =( 2 L p) x=1 = 0, t 2 (0, T ) ( 2 L p) t=0 = y 0 (x), ( 2 L p) t=t = 0, x 2 (0, 1) Attention: 2nd order in time, 4th order in space

13 Numerics: methods and results The Fursikov-Imanuvilov strategy for the heat equation Weak (Lax-Milgram) formulation: RR ( 2 L p L ' !p ') dx dt = R y 0 (x) '(x, 0) dx 8' 2 P; p 2 P m(p, ') =h`, 'i 8' 2 P; p 2 P P = {' : RR ( 2 L ' ! ' 2 ) < +1, ' x=0 ' x=1 0} 9! solution in view of Carleman: ) I(') := RR 2 0 ' 2 + apple Cm(', ') 8' 2 P The ellipticity of m(, ) in a subspace of L 2 loc and... The continuity of ` in P Then: y = 2 L p, v = 2 0 p1!

14 Numerics: methods and results The Fursikov-Imanuvilov strategy for the heat equation 8 < : RR ( 2 L p L ' !p ') dx dt = R y 0 (x) '(x, 0) dx 8' 2 P; p 2 P y = 2 L p, v = 2 0 p1! Standard finite element approximation: RR ( 2 L p h L ' h !p h ' h ) dx dt = R y 0 (x) ' h (x, 0) dx 8' h 2 P h ; p h 2 P h i.e. Attention: m(p h, ' h )=h`, ' h i 8' h 2 P h ; p h 2 P h We need P h P = {' : RR ( 2 L ' ! ' 2 ) < +1, ' x=0 ' x=1 0} i.e. necessarily C 1 in x and C 0 in t finite elements (relatively bad news) Standard FEM framework (good news): convergence in P for usual polynomial P h with [ h P h = P [EFC-Münch 2013], [Cindea-EFC-Münch 2013], [EFC-Santos 2013] (resp. heat, waves, Schrödinger 1D)

15 Numerics: methods and results The Fursikov-Imanuvilov strategy for the heat equation A first numerical experiment =(0, 1),! =(0.2, 0.8), T = 0.5, y 0 (x) sin( x), y t ay xx = v1!, a = 10 1 Approximation: P 3,x P 1,t, C 1 in x, C 0 in t x t Figure:! =(0.2, 0.8). The state

16 Numerics: methods and results The Fursikov-Imanuvilov strategy for the heat equation x t Figure:! =(0.2, 0.8). The control

17 Numerics: methods and results The Fursikov-Imanuvilov strategy for the heat equation An alternative to the standard finite element approximation Weak and equivalent weak mixed formulations: RR ( 2 L p L p !p p0 ) dx dt = R y 0 (x) p 0 (x, 0) dx 8p 0 2 P; p 2 P 8 < : RR (z z !p p0 )+ RR ( 1 L p 0 z 0 ) = R RR y 0 (x) p 0 (x, 0) ( 1 L p z) 0 = 0 8(z 0, p 0, 0 ) 2 Z P Z ; (z, p, ) 2 Z P Z 8 < : a((z, p), (z 0, p 0 )) + b((z 0, p 0 ), )=h`,(z 0, p 0 )i b((z, p), 0 )=0 8(z 0, p 0, 0 ) 2 Z P Z ; (z, p, ) 2 Z P Z Here: b((y, p), ):= RR [( 1 ( p t z) + p x( 1 ) x] 9! solution, again from Carleman and inf-sup, [Münch-Souza 2013] Nonconformal approximation (now p h is C 0 in x and t): 8 < a((z h, p h ), (zh, 0 ph)) 0 + b((z h, 0 ph), 0 h) =h`, (zh, 0 ph)i 0 00 b((z : h, p h ), h )=0 8(zh, 0 ph, 0 0 h) 2 Z h P h Z h ; (z h, p h, h) 2 Z h P h Z h

18 Numerics: methods and results The Fursikov-Imanuvilov strategy for the heat equation A numerical experiment for the 2D classical heat equation (H 2 ) 8 < : y t y = v1!, (x, t) 2 (0, T ) y(x, t) =0, (x, t) (0, T ) y(x, 0) =y 0 (x), x 2 We assume:! (small)

19 Numerical null controllability for the 2D heat equation Some numerical results A numerical experiment for the 2D classical heat equation The mesh Initial state y Final time T = 1

20 Numerical null controllability for the 2D heat equation Some numerical results The control

21 Numerical null controllability for the 2D heat equation Some numerical results The control

22 Numerical null controllability for the 2D heat equation Some numerical results The state

23 Numerical null controllability for the 2D heat equation Some numerical results The state

24 Numerics: methods and results Extensions to some nonlinear problems A third experiment, nonlinear-nonlocal system: 8 y t a( R >< y, R z) y = f (y, z)+v1!, (x, t) 2 (0, T ) z t b( R (NN) y, R z) y = g(y, z), (x, t) 2 (0, T ) y(x, t) =z(x, t) =0, (x, t) (0, T ) >: y(x, 0) =y 0 (x), z(x, 0) =z 0 (x), x 2 a, b, f, g 2 C 1 b, a a 0 > 0, b b 0 > yg(0, 0) 6= 0 =(0, 1),! =(0.2, 0.8), T = 0.5, y 0 (x) sin( x), z 0 (x) sin(2 x), f A 1 (1 + sin y) y + B 1 (1 + sin z) z, g A 2 (1 + sin y) y + B 2 (1 + sin z) z a a 0 (1 +(1 + r 2 + s 2 ) 1 ), b b 0 (1 +(1 + r 2 + s 2 ) 1 ). Formulation F (y, z, v) =0 + Quasi-Newton method Only F 0 (0, 0, 0)! Convergence is ensured At every step: NC for a linear parabolic system (1 control) Approximation: P 1 in (x, t) + multipliers (mixed formulation), C 0 in (x, t) freefem++ & mesh adaptation

25 Numerics: methods and results Extensions to some nonlinear problems MESH ADAPTATION Figure: The initial mesh. Number of vertices: 232. Number of triangles: 402. Total number of unknowns: = 1392.

26 Numerics: methods and results Extensions to some nonlinear problems MESH ADAPTATION Figure: The final adapted mesh. Number of vertices: Number of triangles: Total number of unknowns: =

27 Numerics: methods and results Extensions to some nonlinear problems Figure: The computed null control.

28 Numerics: methods and results Extensions to some nonlinear problems Figure: The computed state y.

29 Numerics: methods and results Extensions to some nonlinear problems Figure: The computed state z.

30 Exact controllability to the trajectories for the Navier-Stokes equations The strategy and the method Again Navier-Stokes, local NC: 8 < y t +(y r)y y + rp = v1!, r y = 0 (NS) y(x, t) =0, (x, t) (0, T ) : y(x, 0) =y 0 (x) Goal: Find v such that y(t )=0 Strategy: Reformulation: NC Fixed point: The task is reduced to control Stokes-like systems y t +(z r)y y + rp = v1!, r y = 0, etc.

31 Some ideas for 2D heat equations and Stokes-like systems The case of a 2D Stokes-like problem Set Ly := y t +(z ry) y, L p := p t (z rp) p Weak and equivalent mixed weak formulations: RR ( 2 (L p + rq)(l p 0 + rq 0 ) !p p0 ) dx dt = R y 0 (x) p 0 (x, 0) dx 8(p 0, q 0 ) 2 P 0 Q 0 ; (p, q) 2 P 0 Q 0 Then, y = 2 (L p + rq), v = 2 0 p! (0,T ) Attention: P 0 is a space of functions p with r p 0 8 >< >: RR (zz !p p0 ) + RR (( 1 (L p 0 + rq 0 ) z 0 ) + p 0 r ) = R y 0 (x) p 0 (x, 0) RR (( 1 (L p + rq) z) 0 + p r ) =0 8(z 0, p 0, q 0, 0, 0 ) 2...; (z, p, q,, ) 2... Two multipliers: related to z = 1 (L p + rq), r p = 0 8 < a((z, p, q), (z 0, p 0, q 0 )) + b((z 0, p 0, q 0 ), (, )) =h`, q 0 i b((z, p, q), ( : 0, 0 )) = 0 8(z 0, p 0, q 0, 0, 0 ) 2...;(z, p, q,, ) 2... Less regular p, more regular Again, Carleman, inf-sup... ) 9!

32 Some ideas for 2D heat equations and Stokes systems The case of the 2D Stokes problem Nonconformal approximation (all variables are C 0 in x and t): 8 < a((z h, p h, q h ), (zh, 0 ph, 0 qh)) 0 + b((z h, 0 ph, 0 qh), 0 ( h, h ))=h`, qhi 0 b((z : h, p h, q h ), ( 0 h, h)) 0 = 0 8(zh, 0 ph, 0 qh, 0 0 h, h) ;(z h, p h, q h, h, h ) 2... Now: b((z, p, q), (, )) = RR (( 1 ( p t + rq) z) +rp r( 1 )+p r ) Similar tasks for the ECT problem (y(t )=y(t )...)

33 Exact controllability to the trajectories for the Navier-Stokes equations Some numerical results Test 1: Poiseuille flow y =(4x 2 (1 x 2 ), 0), p = 4x 1 (stationary) Figure: Poiseuille flow

34 Exact controllability to the trajectories for the Navier-Stokes equations Some numerical results Test 1: Poiseuille flow =(0, 5) (0, 1),! =(1, 2) (0, 1), T = 2, Re = 10, y 0 = y + mz, z = r, =(1 y) 2 y 2 (5 x) 2 x 2 (m =.1 ) Approximation: P 2 in (x 1, x 2, t) + multipliers... freefem++ Figure: The Mesh Nodes: 1830, Elements: 7830, Variables:

35 Exact controllability to the trajectories for the Navier-Stokes equations Some numerical results Test 1: Poiseuille flow Figure: The initial State

36 Exact controllability to the trajectories for the Navier-Stokes equations Some numerical results Test 1: Poiseuille flow Figure: The State at t = 1.1

37 Exact controllability to the trajectories for the Navier-Stokes equations Some numerical results Test 1: Poiseuille flow Figure: The State at t = 1.7 ZPoisseuille.edp

38 Exact controllability to the trajectories for the Navier-Stokes equations Some numerical results Test 2: Taylor-Green (vortex) flow y =(sin(2x 1 ) cos(2x 2 )e 8t, cos(2x 1 ) sin(2x 2 )e 8t ) Figure: Taylor-Green flow

39 Exact controllability to the trajectories for the Navier-Stokes equations Some numerical results Test 2: Taylor-Green (vortex) flow Figure: The Taylor-Green velocity field

40 Exact controllability to the trajectories for the Navier-Stokes equations Some numerical results Test 2: Taylor-Green (vortex) flow =(0, ) (0, ),! =( /3, 2 /3) (0, 1), T = 1, Re = y 0 = y + mz, z = r, =( y) 2 y 2 ( x) 2 x 2 (m << 1) Approximation: P 2 in (x 1, x 2 ) and t + multipliers... freefem++ Figure: The mesh Nodes: 3146, Elements: 15900, Variables:

41 Exact controllability to the trajectories for the Navier-Stokes equations Some numerical results Test 2: Taylor-Green (vortex) flow Figure: The initial state

42 Exact controllability to the trajectories for the Navier-Stokes equations Some numerical results Test 2: Taylor-Green (vortex) flow Figure: The state at t = 0.6

43 Exact controllability to the trajectories for the Navier-Stokes equations Some numerical results Test 2: Taylor-Green (vortex) flow Figure: The state at t = 0.9 Taylor-Green Vortex.edp

44 Numerics: methods and results Final comments ADDITIONAL COMMENTS: Many open questions remain Carleman allows for theoretical and numerical results (control oscillations disappear... ) Numerical analysis and convergence results for these and other problems: in progress... Variants and extensions: non-scalar parabolic systems, 3D heat, 3D Stokes, Navier-Stokes, boundary control, nonlinear waves, etc. In particular: Navier-Stokes systems, higher Re 2D heat equation with C 1 in space approximation The same for 2D Stokes and Navier-Stokes systems

45 Additional results and comments Final comments REFERENCES: CLARK, H.,FERNÁNDEZ-CARA, E.,LÍMACO, J.AND MEDEIROS, L.A. Theoretical and numerical local null controllability for a parabolic system with local and nonlocal nonlinearities. Applied Mathematics and Computation 223 (2013) FERNÁNDEZ-CARA, E.,LÍMACO, J.AND MENEZES, S.B. On the theoretical and numerical control of a Ladyzhenskaya-Smagorinsky model of turbulence. J. Math. Fluid Mech. 17 (2015), FERNÁNDEZ-CARA, E.AND MÜNCH, A. Numerical null controllability of semi-linear 1D heat equations : fixed point, least squares and Newton methods. MCRF Volume 2, Number 3, September FERNÁNDEZ-CARA, E.AND MÜNCH, A. Strong convergent approximations of null controls for the 1D heat equation. SeMA J (2013) 61:49 78.

46 Additional results and comments Final comments REFERENCES (Cont.): FERNÁNDEZ-CARA, E.AND MÜNCH, A. Numerical exact controllability of the 1D heat equation: duality and Carleman weights. J Optim Theory Appl (2014) 163: FERNÁNDEZ-CARA, E.;MÜNCH, A.AND SOUZA, D.A. On the numerical controllability of the two-dimensional heat, Stokes and Navier-Stokes equations. Submitted.

47 THANK YOU VERY MUCH...