Workshop ’Quantum Control’ Paris, IHP, December 2010 Controllability of Cubic Schroedinger Equation via Low-Dimensional Source Term Andrey Sarychev Dipartimento di Matematica per le Decisioni, Universit` a di Firenze, Italia
Introduction Main theme: Lie algebraic approach to controllability of dis- tributed parameter systems . Example of such approach - approximate controllability and con- trollability in finite-dimensional projections criteria ∗ for 2D and 3D Navier-Stokes/Euler equation of fluid motion controlled by low-dimensional forcing . Goal: develop similar technique for cubic defocusing Schroedinger equation i∂ t u ( t, x ) + ∆ u ( t, x ) = | u ( t, x ) | 2 u ( t, x ) + F ( t, x ) (NLS) ∗ A.Agrachev, A.Sarychev, S.Rodrigues,A.Shirikyan, H.Nersisyan 1
controlled via low-dimensional source term F . We consider di- mension 2 and periodic boundary conditions; x ∈ T 2 . Problem setting is distinguished by the type of control ; it is ap- plied via source term, which is ’linear combination of few func- tions’: v k ( t ) F k ( x ) , K 1 is finite . � F ( t, x ) = k ∈K 1 In the periodic case we take F k ( x ) = e ik · x , k ∈ Z 2 , F being trigonometric polynomial in x . The control functions v k ( t ) , t ∈ [0 , T ] , k ∈ K 1 are chosen freely from L ∞ [0 , T ].
Preliminaries on existence and uniqueness of trajectories For our goals it suffices to deal with NLS equation evolving in Sobolev space H = H 2 ( T 2 ). Our source term F is trigonometric k v k ( t ) e ik · x is measurable polynomial in x and t �→ F ( t, x ) = � essentially bounded map in t . Local existence of solutions in this setting is standard and is proved by fixed point argument for contracting map in C ([0 , T ]; H 2 ( T 2 )). The same argument remains valid for equa- tion with more general nonlinearity. 2
Preliminaries ctd. Proposition 1. Given equation i∂ t u ( t, x ) + ∆ u ( t, x ) = | u ( t, x ) | 2 u ( t, x ) + P 2 ( u, ¯ u, t, x ) , where P 2 ( u, ¯ u, t ) is second degree polynomial in u, ¯ u with coeffi- cients f ( t, x ) from L ∞ ([0 , T ] , H 2 ( T 2 )). Then for each B > 0 and each ˜ u with � u � H 2 ≤ B there exists T B > 0 such that there ∃ unique strong solution u ( · ) ∈ C ([0 , T B ] , H 2 ( T 2 )) of the Cauchy problem for (NLS) with the initial condition u (0) = ˜ u. � 3
Preliminaries-3 Global existence/uniqueness result for cubic NLS with source term: i∂ t u ( t, x ) + ∆ u ( t, x ) = | u ( t, x ) | 2 u ( t, x ) + F ( t, x ) , Proposition 2. For the source term F ( t, x ) from L ∞ ([0 , T ] , H 2 ( T 2 )). u ∈ H 2 the Cauchy problem with the initial condition for each ˜ u possesses unique strong solution u ( · ) ∈ C ([0 , T ] , H 2 ( T 2 )) . � u (0) = ˜ (A stronger version of) results on continuous dependence of tra- jectories on the r.-h. side will appear later. 4
Controlled NLS equation: controllability problem settings We will study controllability in finite-dimensional projections mean- ing that proper control v k ( t ) , k ∈ K 1 may steer i∂ t u ( t, x ) + ∆ u ( t, x ) = | u ( t, x ) | 2 u ( t, x ) + v k ( t ) e ik · x � k ∈K 1 in time T > 0 from u 0 ∈ H 2 ( T 2 ) to a point with preassigned orthogonal projection on a given finite-dimensional subspace L ⊂ H 2 ; and approximate controllability, meaning that set of ’points’ attain- able from each u 0 ∈ H 2 ( T 2 ) is dense in L 2 . 5
Controllability of NLS equation: main result Theorem. There exists set K = { m 1 , m 2 , m 3 , m 4 } , consisting of 4 modes such that cubic defocusing Schroedinger equation 4 v α ( t ) e im α · x i∂ t u ( t, x ) + ∆ u ( t, x ) = | u ( t, x ) | 2 u ( t, x ) + � α =1 is controllable in each finite-dimensional projection and approximately controllable. � 6
Outline of approach from geometric control viewpoint Our study of controllability of NLS equation is based (as well as previous work on Navier-Stokes/Euler equation) on method of iterated Lie extensions . Lie extension of a control system ˙ x = f ( x, u ) , u ∈ U allows us to join (’almost maintaining’ controllability properties) to the r.-h. side additional vector fields, which are expressed via Lie brackets of f ( · , u ) for various u ∈ U . If after a series of extensions one arrives to a system, which is then the original system also would be. Controlled NLS equation is a particular type of infinite-dimensional control-affine system u = f 0 ( u ) + f k ( u ) v k ( t ) . � ˙ k ∈K 7
We proceed with Lie extensions, at each step of which following Lie brackets appear: [ f m , [ f m , f 0 ]] , [ f n , [ f m , [ f m , f 0 ]]] , m, n ∈ K . The 3rd-order Lie brackets [ f m , [ f m , f 0 ]] are obstructions to con- trollability; they have to be ’compensated’. The 4th-order Lie bracket [ f n , [ f m , [ f m , f 0 ]]] are directions along which the extended control acts.
Geometric control in infinite-dimension Obstacles : • r.-h. sides of equations (’vector fields’) include unbounded operators • instead of flows one often has to deal with semigroups of nonlinear operators; • lack of adequate infinite-dimensional differential geometry: manifolds, distributions, integrability etc. 8
’In practice’ we use fast-oscillating controls, which underly Lie extensions method. Specially designed resonances between such controls result in a motion which provides ( approximates ) motion in ex- tending direction, along a Lie bracket. Choosing special coordinates (Fourier Ansatz) on torus we will feed fast-oscillating controls into the r.-h. sides of equations for the components q m , q n , m, n ∈ K 1 ⊂ Z 2 in such a way that it will produce effect of control for the dynamics of certain component q ℓ with ℓ �∈ K 1 and (asymptotically) will not affect the dynamics of other components. This is called extension of control. Final result is obtained by (finite) iteration of such steps. 9
Cubic Schroedinger equation on T 2 as infinite-dimensional system of ODE Invoking Fourier Ansatz we seek for solution of the NLS equation i∂ t u ( t, x ) + ∆ u ( t, x ) = | u ( t, x ) | 2 u ( t, x ) + F ( t, x ) (NLS) in the form of a series expansion q k ( t ) e i ( kx + | k | 2 t ) . � u ( t, x ) = k ∈ Z 2 with respect to modes e k = e i ( kx + | k | 2 t ) . The source term can be represented as e i ( kx + | k | 2 t ) v k ( t ) , � F ( t, x ) = k ∈K 1 ⊂ Z 2 notation v k ( t ) is kept for controls. The set of controlled modes K 1 is finite. 10
Substituting the expansions of u and F into NLS equation we get infinite system of ODE’s for the coefficients q ( t ): i∂ t q k ( t ) = S k ( q, t ) = − q k | q k | 2 + 2 q k | q j | 2 + � j ∈ Z 2 q k 2 q k 3 e iω ( K ) t , k ∈ Z 2 . ( NLSODE ) � + q k 1 ¯ k 1 − k 2 + k 3 = k ; k � = k 1 ,k 3 ω ( K ) = | k 1 | 2 − | k 2 | 2 + | k 3 | 2 − | k | 2 . k ∈K 1 v k ( t ) e ik 2 t e ikx , then con- If we add controlling source term � trols v k ( t ) appear in the equations, indexed by k ∈ K 1 . We proceed with extension . 11
Sketch of the extension step Assume that the set of controlled modes is { m, n } ⊂ Z 2 . We will show how choosing in clever way controls in these modes, one gets an extended control for the mode (2 m − n ) ∈ Z 2 . Feed into the r.-h. side of the ODEs for q m , q n control func- tions ˙ v m ( t ) + ˜ v n , ˙ v m ( t ) + ˜ v n respectively, where v m ( t ) , v n ( t ) are Lipschitzian functions. We get i∂ t q m ( t ) = S m ( q, t ) + ˙ v m ( t ) + ˜ v m , i∂ t q n ( t ) = S n ( q, t ) + ˙ v n ( t ) + ˜ v n . Introduce new variables q ∗ ℓ by relations q m = q ∗ m − iv r ( t ) , q n = q ∗ n − iv n ( t ) , q ∗ k = q k , for k � = m, n, 12
or q = q ∗ + V ( t ) = q ∗ + v m ( t ) e m + v n ( t ) e n . (SUB1) The equations for components of q ∗ are: � S j ( q + V ( t ) , t ) + ˜ v j , j ∈ { m, n } ; i∂ t q ∗ j ( t ) = S j ( q + V ( t ) , t ) , j �∈ { m, n } . Impose isoperimetric constraints v m (0) = v n (0) = 0 , v m ( T ) = v n ( T ) = 0 , in order to preserve the end-points of the trajectory: q (0) = q ∗ (0) , q ( T ) = q ∗ ( T ) .
Controllability of equations for q ∗ ⇒ controllability of the original system. Calculating S j ( q + V ( t ) , t ) at the r-h. side we get i∂ t q ∗ k ( t ) = − ( q ∗ k + δ k,mn v k ) | q ∗ k + δ k,mn v k | 2 + � V � 2 + +2( q ⋆ | q ∗ s | 2 + � k + δ k,mn v k ) s ∈ Z 2 ( q ∗ q ∗ v k 2 )( q ∗ k 3 + δ k 3 ,mn v k 3 ) e iω ( K ) t , � + k 1 + δ k 1 ,mn v k 1 )(¯ k 2 + δ k 2 ,mn ¯ δ k,mn = 1, whenever k ∈ { m, n } , otherwise δ k,mn = 0. The result is cubic polynomial with respect to v m , v n , ¯ v m , ¯ v n .
Fast oscillations Now we introduce fast-oscillations, choosing the controls v m ( t ) , v n ( t ) of the form v m ( t ) = e i (1+ ερ ) t/ε ˆ v m ( t ) , v n ( t ) = e i (2+ εσ ) t/ε ˆ v n ( t ) , (SUB2) where ˆ v m ( t ) , ˆ v n ( t ) are functions of bounded variation, ρ, σ will be specified later and ε > 0 is small parameter. Taking all the monomials of degree ≤ 3 in v m , v n , ¯ v m , ¯ v n we clas- sify them into resonant and non-resonant . We call a monomial non-resonant if, after substitution of (SUB2) into it, we get a fast-oscillating factor e iβt/ε , β > 0. All other monomials are res- onant ; they are classified as bad resonances (obstructions) and good resonances - extending controls. 13
Recommend
More recommend
