A mixed formulation for the approximation of the HUM control for the wave equation

Transcription

1 Nicolae Cîndea Approimation of the HUM control for the wave equation / 6 A mied formulation for the approimation of the HUM control for the wave equation Nicolae Cîndea joint work with Arnaud Münch Université Blaise Pascal Clermont-Ferrand Benasque, 3/8/3

2 The wave equation with boundary control We consider the following wave equation: y tt (c()y ) + d(, t)y =, (, t) (, ) (, T ) y(, t) =, y(, t) = v(t), t (, T ) y(, ) = y (), y t (, ) = y (), (, ). c C 3 ([, ]) with c() c > in [, ] d L ((, ) (, T )) y L (, ) and y H (, ) We search a control v = v(t) such that y(t ) =, y t (T ) =. () Aim For a controllability time T > large enough and for every y, y, give a numerical approimation of the control v of minimal L -norm. Nicolae Cîndea Approimation of the HUM control for the wave equation / 6

3 Nicolae Cîndea Approimation of the HUM control for the wave equation 3/ 6 Hilbert Uniqueness Method - a brief recall T Minimize J(y, v) = v(t) dt Subject to (y, v) C(y, y ; T ) where C(y, y ; T ) denotes the linear manifold { (y, v) : v L C(y, y ; T ) = (, T ), y solves the wave equation and satisfies y(t ) = y t (T ) = () }. min (ϕ,ϕ ) H (,) L (,) J (ϕ, ϕ ) J (ϕ, ϕ ) = T ϕ (, t) dt + ϕ t(, )d, ϕ(, ) H,H

4 Nicolae Cîndea Approimation of the HUM control for the wave equation 3/ 6 Hilbert Uniqueness Method - a brief recall T Minimize J(y, v) = v(t) dt Subject to (y, v) C(y, y ; T ) where C(y, y ; T ) denotes the linear manifold { (y, v) : v L C(y, y ; T ) = (, T ), y solves the wave equation and satisfies y(t ) = y t (T ) = By duality arguments this minimization problem is equivalent to the following one min (ϕ,ϕ ) H (,) L (,) J (ϕ, ϕ ) J (ϕ, ϕ ) = T ϕ (, t) dt + y ()ϕ t (, )d y, ϕ(, ) H,H () }.

5 Nicolae Cîndea Approimation of the HUM control for the wave equation 3/ 6 Hilbert Uniqueness Method - a brief recall Dual minimization problem reads as : J (ϕ, ϕ ) = min (ϕ,ϕ ) H (,) L (,) J (ϕ, ϕ ) T ϕ (, t) dt + y ()ϕ t (, )d y, ϕ(, ) H,H Lϕ = in Q T, ϕ = on Σ T (ϕ(, T ), ϕ t (, T )) = (ϕ, ϕ ), in Ω.

6 Nicolae Cîndea Approimation of the HUM control for the wave equation 3/ 6 Hilbert Uniqueness Method - a brief recall J (ϕ, ϕ ) = min (ϕ,ϕ ) H (,) L (,) J (ϕ, ϕ ) T ϕ (, t) dt + y ()ϕ t (, )d y, ϕ(, ) H,H Lϕ = in Q T, ϕ = on Σ T (ϕ(, T ), ϕ t (, T )) = (ϕ, ϕ ), in Ω. The coercivity of J is the consequence of the following observability estimate : there eists a constant k T > such that ϕ(, ), ϕ t (, ) V k T ϕ (, ) L (,T ), (ϕ, ϕ ) V, () where V = H (, ) L (, ).

7 Nicolae Cîndea Approimation of the HUM control for the wave equation 4/ 6 Hilbert Uniqueness Method - a reformulation Since ϕ is completely and uniquely determined by (ϕ, ϕ ), we consider the following etremal problem: min Ĵ (ϕ), subject to Lϕ =, ϕ Φ where { Φ = ϕ L (Q T ), ϕ = on Σ T such that Lϕ L (Q T ) ϕ (, ) L (, T ) }. Remark Φ is an Hilbert space endowed with the inner product (ϕ, ϕ) Φ = T for any fied η >. c()ϕ (, t)ϕ (, t) dt + η LϕLϕ d dt. Q T

8 Nicolae Cîndea Approimation of the HUM control for the wave equation 5/ 6 Hilbert Uniqueness Method - a mied reformulation We consider the following mied formulation : find (ϕ, λ) Φ L (Q T ) solution of { a(ϕ, ϕ) + b(ϕ, λ) = l(ϕ), ϕ Φ b(ϕ, λ) =, λ L (Q T ), (3) where T a : Φ Φ R, a(ϕ, ϕ) = c()ϕ (, t)ϕ (, t)dt b : Φ L (Q T ) R, b(ϕ, λ) = Lϕ(, t)λ(, t)ddt Q T l : Φ R, l(ϕ) = y () ϕ t (, )d + y, ϕ(, ),.

9 Nicolae Cîndea Approimation of the HUM control for the wave equation 6/ 6 { a(ϕ, ϕ) + b(ϕ, λ) = l(ϕ), ϕ Φ b(ϕ, λ) =, λ L (Q T ), (3) Theorem. The mied formulation (3) is well-posed.. The unique solution (ϕ, λ) Φ L (Q T ) is the unique saddle-point of the Lagrangian L : Φ L (Q T ) R defined by L(ϕ, λ) = a(ϕ, ϕ) + b(ϕ, λ) l(ϕ). (4) 3. The optimal function ϕ is the minimizer of Ĵ over Φ while the optimal function λ L (Q T ) is the state of the controlled wave equation in the weak sense.

10 Nicolae Cîndea Approimation of the HUM control for the wave equation 6/ 6 { a(ϕ, ϕ) + b(ϕ, λ) = l(ϕ), ϕ Φ b(ϕ, λ) =, λ L (Q T ), (3) Theorem. The mied formulation (3) is well-posed.. The unique solution (ϕ, λ) Φ L (Q T ) is the unique saddle-point of the Lagrangian L r : Φ L (Q T ) R defined by L r (ϕ, λ) = a r(ϕ, ϕ) + b(ϕ, λ) l(ϕ). (4) 3. The optimal function ϕ is the minimizer of Ĵ over Φ while the optimal function λ L (Q T ) is the state of the controlled wave equation in the weak sense.

11 Nicolae Cîndea Approimation of the HUM control for the wave equation 7/ 6 Ingredients of the proof We easily check that : a is continuous over Φ Φ, symmetric and positive b is continuous over Φ L (Q T ) assuming c A and T > T (c), l is continuous over Φ (as a direct consequence of an apropriate Carleman estimate see it ) The well-posedness of the mied formulation is a consequence of the following two properties (see Brezzi and Fortin 99 book): a is coercive on N (b), where: N (b) = {ϕ Φ such that b(ϕ, λ) = for every λ L (Q T )}. b satisfies the usual inf-sup condition over Φ L (Q T ): inf sup λ L (Q T ) ϕ Φ b(ϕ, λ) ϕ Φ λ L (Q T ) δ.

12 Nicolae Cîndea Approimation of the HUM control for the wave equation 8/ 6 Numerical approimation Let Φ h and M h be two finite dimensional spaces parametrized by the variable h such that Φ h Φ, M h L (Q T ), h >. We introduce the following approimated problems : find (ϕ h, λ h ) Φ h M h solution of { ar (ϕ h, ϕ h ) + b(ϕ h, λ h ) = l(ϕ h ), ϕ h Φ h b(ϕ h, λ h ) =, λ h, M h. where for any given r >. a r (ϕ, ϕ) = a(ϕ, ϕ) + r Lϕ d dt Q T

13 We introduce a triangulation T h such that Q T = K Th K and we assume that {T h } h> is a regular family. We define t t Nicolae Cîndea Approimation of the HUM control for the wave equation 9/ 6 t Φ h = {ϕ h C (Q T ) : ϕ h K P(K) K T h, ϕ h = on Σ T } where P(K) may be chosen as The Bogner-Fo-Schmit (BFS for short) C element defined for rectangles. The reduced Hsieh-Clough-Tocher (HCT for short) C element defined for triangles. M h = {λ h C (Q T ) : λ h K Q(K) K T h },

14 Nicolae Cîndea Approimation of the HUM control for the wave equation / 6 Mied formulation as a linear system Let n h = dim Φ h, m h = dim M h and let the real matrices A r,h R n h,n h, B h R m h,n h and L h R n h be defined by a r (ϕ h, ϕ h ) =< A r,h {ϕ h }, {ϕ h } > R n h,r n h, ϕ h, ϕ h Φ h, b(ϕ h, λ h ) =< B h {ϕ h }, {λ h } > R m h,r m h, ϕ h Φ h, λ h M h, l(ϕ h ) =< L h, {ϕ h } >, ϕ h Φ h With these notations, the finite dimensional mied problem reads as follows : find {ϕ h } R n h and {λ h } R m h such that ( Ar,h B T h B h ) R n h +m h,n h +m h ( {ϕh } {λ h } ) R n h +m h = ( Lh ) R n h +m h. v h (t) = c()π t (φ h, (, )).

15 Nicolae Cîndea Approimation of the HUM control for the wave equation / 6 A numerical eample y () = 4 (,/) (), y () =, T =.4. The control of minimal L -norm is known eactly : v(t) = ( t) /, 3/ (t).

16 Nicolae Cîndea Approimation of the HUM control for the wave equation / 6 A numerical eample v h (t) v(t) t Figure: Control of minimal L -norm v and its approimation v h on (, T ).

17 Nicolae Cîndea Approimation of the HUM control for the wave equation / 6 A numerical eample λ(, t).5 t.5.5 Figure: The primal variable λ h in Q T ; h =.46 ; r =.

18 Nicolae Cîndea Approimation of the HUM control for the wave equation / 6 A numerical eample BFS HCT uniform HCT non uniform.5 h Figure: Evolution of v v h L (,T ) w.r.t. h for BFS finite element ( ), HCT-uniform mesh ( ) and HCT- non uniform mesh ( ); r =.

19 t t t Nicolae Cîndea Approimation of the HUM control for the wave equation / 6 t t A numerical eample - mesh adaptation Figure: Iterative refinement of the triangular mesh over Q T with respect to the variable λ h : 4, 4, 54, 556, 4 75 triangles; r = 3.

20 Nicolae Cîndea Approimation of the HUM control for the wave equation / 6 A numerical eample - mesh adaptation 3 λ(, t).5.5 t.5.5 Figure: Primal variable λ h in Q T.

21 Nicolae Cîndea Approimation of the HUM control for the wave equation 3/ 6 Another numerical eample - distributed control case y () = e 5(.), y () = T =., ω = (.,.4) and a non-constant function c = c() C ([, ]) with. [,.45] c() = [., 5.] (c () > ), (.45,.55) 5. [.55, ]. (5)

22 Nicolae Cîndea Approimation of the HUM control for the wave equation 3/ 6 Another numerical eample - distributed control case t.5.5 λ(, t) t.5.5 Figure: Left : Triangular mesh of q T and of Q T. Right : The primal variable λ h in Q T.

23 t t Some perspectives Optimization in spacetime of the support of the control inf-sup condition is uniform with respect to h? Nicolae Cîndea Approimation of the HUM control for the wave equation 4/ 6

24 Nicolae Cîndea Approimation of the HUM control for the wave equation 4/ 6 Some perspectives inf λ h M h b(ϕ h, λ h ) sup δ h. ϕ h Φ h ϕ h Φh λ h Mh Optimization in spacetime of the support of the control inf-sup condition is uniform with respect to h? Figure: BFS finite element - Evolution of the inf-sup constante δ h with respect to h for r = 4 (<), r = 3 ( ), r = ( ) and r = ( ).

25 Nicolae Cîndea Approimation of the HUM control for the wave equation 5/ 6 N. Cîndea, A. Munch, A mied formulation for the direct approimation of the control of minimal L -norm for linear type wave equations, submitted August 3. Thank you for the attention!

26 Nicolae BackCîndea Approimation of the HUM control for the wave equation 6/ 6 Proposition (C, Fernandez-Cara, Münch, ESAIM COCV 3) Let <, c > and c A(, c ). Let β > and let us consider the function φ(, t) := βt + M and g(, t) := e λφ(,t). Finally, let us assume that T > β ma [,] c() / ( ). Then there eist positive constants s and M, such that, for all s > s, one has s T T M e sg ( w t + w ) d dt + s 3 T T T e sg Lw d dt + Ms { A(, c ) = c C 3 ([, ]) : c() c >, ( ) min c() + ( )c () [,] T T T < min [,] e sg w d dt e sg w (, t) dt. ( c() + ( )c ()) },