New phenomena for the null controllability of parabolic systems F.Ammar Khodja, M. González-Burgos & L. de Teresa Aix-Marseille Université, CNRS, Centrale Marseille, l2m, UMR 7373, Marseille, France assia.benabdallah@univ-amu.fr Optimal Control for Evolutionary PDEs and Related Topics Cortona-June 2-24, 216
Controllability of systems: The finite dimensional case (1) { t y = Ly + Bv y() = y L M n (R), B M n,m (R), m n. Definition System (1) is controllable at time T > if y, y 1 R n, v L 2 (, T) m such that y(t; y, v) = y 1
Controllability of systems: The finite dimensional case (2) { t y = Ly + Bv y() = y L M n (R), B M n,m (R). Proposition ( The Fattorini-Hautus test) System (2) (or (L, B)) is controllable if and only if Proposition ker(s L ) ker(b ) = {}, s C. System (2) is controllable at time T > if and only if it is controllable at any time.
Controllability of systems: The infinite dimensional case (3) { t y = Ly + Bv y() = y (L, D(L)) generator of a semigroup of class C on H, Hilbert space, B : V H is a bounded operator, V is also a Hilbert space Definition 1 System (3) is exactly controllable at time T > if y, y 1 H, v L 2 ((, T); V) such that y(t; y, v) = y 1 2 System (3) is approximately controllable at time T > if y, y 1 H, ε >, v L 2 ((, T); V) such that y(t; y, v) y 1 < ε 3 System (3) is null controllable at time T > if y H, v L 2 ((, T); V) such that y(t; y, v) =
Wave equation : Exact controllability Ω R N be a bounded domain, N 1, γ Ω (4) tty 2 = y, in Ω (, T) y = v1 γ, on Ω (, T), y(, ) = y 1, t y(, )) = y 2, in Ω Theorem Let x R N and m(x) = x x. Let γ = {x Ω ; m(x).ν(x) > }, R = max x Ω m(x), T = 2R. For all T > T, for all (y 1, y 2 ) L 2 (Ω) H 1 (Ω), (y 11, y 12 ) L 2 (Ω) H 1 (Ω), there exists v L 2 ( Ω (, T)) such that y solution of (4) satisfies: y(, T) = y 11 ( ), t y(, T) = y 12 ( ), on Ω.
Hyperbolic behavior in control of Hyperbolic equations Control of hyperbolic equations requires 1 Minimal time of control. 2 Geometrical condition on the location of the control.
Heat equation: Approximate and null controllability Ω R N a bounded domain, N 1, γ Ω a relative open subset, T > (5) { t y y = in Ω (, T), y = v1 γ on Ω (, T), y(, ) = y in Ω. γ Ω Theorem (Boundary controllability : results) 1 For any y, y 1 H 1 (Ω), any ε >, there exists v L 2 ( Ω (, T)) s.t. the solution y to (5) satisfies y(, T) y 1 H 1 (Ω) ε. 2 For any y H 1 (Ω) there exists v L 2 ( Ω (, T)) s.t. the solution y to (5) satisfies y(, T) = in Ω.
Heat equation: Approximate and null controllability 1 No exact controllability (due to the regularizing effect). 2 Approximate controllability null controllability. 3 No minimal time, no geometric condition on the location of the control.
Heat equation: Approximate and null controllability Four IMPORTANT REFERENCES 1 H.O. FATTORINI, D.L. RUSSELL, Exact controllability theorems for linear parabolic equations in one space dimension, Arch. Rational Mech. Anal. 43 (1971), 272 292. 2 S. DOLECKI, Observability for the one-dimensional heat equation, Studia Mathematica. 48 (1973), 292-35. 3 G. LEBEAU, L. ROBBIANO, Contrôle exact de l équation de la chaleur, Comm. P.D.E. 2 (1995), no. 1-2, 335 356. 4 A. FURSIKOV, O. YU. IMANUVILOV, Controllability of Evolution Equations, Lecture Notes Series 34, Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1996.
Hyperbolic phenomena in control: 1 Minimal time >. 2 Condition on the location of the domain of control. 3 Approximate controllability Null controllability. Issue: Do " Hyperbolic phenomena" occur in control of parabolic equations?
Contents 1 A first example : pointwise control of the heat equation 2 A second example later : Boundary control of parabolic systems 3 Hyperbolic behavior in distributed control of parabolic equations Existence of a control : the Fattorini-Russell method A non controllability result Is it possible to have a positive minimal time of control?
Pointwise control of the heat equation Let x (, π) t y xxy 2 = δ (x x ) v(t) in (, π) (, T) y(, ) = y(π, ) = on (, T) y(, ) = y in (, π) S. DOLECKI, Observability for the one-dimensional heat equation, Studia Math. 48 (1973), 291 35.
Pointwise control of the heat equation Theorem (Pointwise control for parabolic systems-s. Dolecki, 73 ) Let Then: T(x ) := lim sup log sin(kx ) k 2 1 System (6) is null controllable for T > T(x ) 2 System (6) is not null controllable for T < T(x ) (6) t y xxy 2 = δ (x x ) v(t) in (, π) (, T) y(, ) = y(π, ) = on (, T) y(, ) = y in (, π)
Boundary control for parabolic systems F. AMMAR KHODJA, A. BENABDALLAH, M. GONZÁLEZ-BURGOS, L. DE TERESA, Minimal time for the null controllability of parabolic systems: the effect of the condensation index of complex sequences, J. Funct. Anal. 267 (214), no. 7, 277 2151. (7) t y ( D 2 xx + A ) y = y(, ) = Bv, y(π, ) = on (, T), y(, ) = y in Q = (, π) (, T), in (, π), where T > is a given time, ( ) 1 D = (with d > ), A = d ( 1 ) (, B = 1 ) M 2,1 (R)
Results Theorem (F. Ammar Khodja, A. Benabdallah, M. González-Burgos, L. de Teresa, 214) Let d 1 1 T > : Approximate controllability if and only if d Q 2 T = c(λ) [, + ] such that 1 The system is null controllable at time T if d Q and T > T 2 Even if d Q, if T < T, the system is not null controllable at time T c(λ) is the index of condensations of the sequence Λ = (k 2, dk 2 ) k 1.
Index of condensation : some background The index of condensation of a sequence Λ = (λ k ) C is a real number c (Λ) [, + ] associated with this sequence and which "measures" the condensation at infinity. This notion has been : introduced by V.l. Bernstein in 1933: Leçons sur les progrès récents de la théorie des séries de Dirichlet for real sequences, extended by J. R. Shackell in 1967 for complex sequences.
Index of condensation : some background Let Λ = (λ k ) C be a sequence with pairwise distinct elements and: δ > : R (λ k ) δ λ k >, k 1, such that the sequence Λ is measurable: there exists D [, [ (the density of Λ) such that n lim = D, n λ n
Index of condensation: some background Definition The index of condensation of Λ is: c (Λ) = lim sup k ln E (λ k ) Rλ k [, + ]. E (λ k ) = 2 λ k j k ( 1 λ2 k λ 2 j )
Is it possible to have a minimal time of control >? T >? For Λ = (k 2, dk 2 ) k 1 with d Q, is it possible that c (Λ) >? Theorem (F. Ammar Khodja, A. Benabdallah, M. González-Burgos, L. de Teresa, 214) Let Λ = ( k 2, d k 2). For any δ [, ], there exists d R\Q such that c (Λ) = δ
Distributed controllability problem y t y xx + q(x)ay = Bv1 ω in Q, y(, ) =, y(π, ) = on (, T), y(, ) = y, in (, π), ( ) ( ) 1 1 D =, A =, B = 1 ( 1 ) F. AMMAR KHODJA, A. BENABDALLAH, M. GONZÁLEZ-BURGOS, L. DE TERESA, New phenomena for the null controllability of parabolic systems: Minimal time and geometrical dependence to appear in Jour. Math. Anal. App (216).
Distributed controllability problem Approximate controllability F. BOYER AND G. OLIVE, Approximate controllability conditions for some linear 1D parabolic systems with space-dependent coefficients, Math. Control Relat. Fields 4 (214), no 3, 263 287. Theorem (F. Boyer and G. Olive) Let ω = (a, b) and q L (, π), a function satisfying (8) Supp q ω =. Then, system is approximately controllable at time T > if and only if (9) I k (q) + I 1,k (q) k 1. I k (q) := 2 π π q(x) sin(kx) 2 dx, I 1,k (q) := 2 π a q(x) sin(kx) 2 dx
Fattorini-Russell method Let ϕ be a solution of the adjoint problem: t ϕ = ( D xx 2 + A ) ϕ, in Q = (, π) (, T), ϕ(, ) = ϕ(π, ) = on (, T), ϕ(, T) = ϕ L 2 (, π; R 2 ). If y is a solution of the direct problem, then y(t), ϕ y, ϕ() T = v(t)b D x ϕ(, t) dt Thus y(t) = if, and only if, there exists v such that T ω v B ϕ dx dt = y, ϕ(), ϕ L 2 (, π; R 2 ) B = ( 1 )
Fattorini-Russell method Material at our disposal L := d2 dx 2 Id + q(x)a, σ(l ) = {k 2 : k 1}. Given k 1, if Φ 1,k := ( ϕk ψ k ) (, Φ 2,k := ϕ k where ψ k is the unique solution of the non-homogeneous Sturm-Liouville problem: ψ xx k 2 ψ = [I k (q) q(x)] ϕ k in (, π), ψ() =, ψ(π) =, (1) π ψ(x)ϕ k (x) dx =, ),
Fattorini-Russell method Material at our disposal ( L k 2 I d ) Φ 1,k = I k (q)φ 2,k and ( L k 2 I d ) Φ 2,k = ). { Λ1 := {k 1 : I k (q) }, Λ 2 := {k 1 : I k (q) = }, In particular, if k Λ 1 then k 2 is a simple eigenvalue and Φ 2,k and Φ 1,k are, respectively, an eigenfunction and a generalized eigenfunction of the operator L associated to k 2, while if k Λ 2 then Φ 1,k and Φ 2,k are both eigenfunctions of L associated to k 2. {Φ i,k, i = 1, 2, k 1} is a Riesz basis of L2 (, π; R 2 ).
Fattorini-Russell method y(t) = y(t), Φ i,k =, k 1, i = 1, 2 Choosing ϕ = Φ 2,k, we have ϕ 2,k (x, t) = e k2 (T t) Φ 2,k (x) and ϕ 2,k (x, ) = e k2t Φ 2,k(x) Choosing ϕ = Φ 1,k, we have ϕ 1,k (x, t) = e k2 (T t) (Φ 1,k (x) (T t)i k(q)φ 2,k ) and ϕ 1,k (x, ) = e k2t (Φ 1,k(x) (T t)i k (q)φ 2,k) Thus y(t) = if, and only if, there exists v L 2 (Ω (, T)) such that Q T v(x, t)1 ω B ϕ i,k (x, t) dx dt = y, ϕ i,k (, ), i = 1, 2, k 1
The first main idea Search controls under the particular form v(x, t) = f 1 (x)v 1 (T t) + f 2 (x)v 2 (T t), Supp f 1, Supp f 2 ω = (a, b) The moment problem leads to T T f 1,k v 1 (t)e k2t dt + f 2,k T T f 1,k v 1 (t)e k2t dt + f 2,k v 2 (t)e k2t dt f i,k := π v 2 (t)e k2t dt = e k2 T y, Φ 2,k T T I k (q)f 1,k v 1 (t) te k2t dt I k (q)f 2,k v 2 (t) te k2t dt = e k2 T ( y, Φ 1,k TIk (q) y, Φ 2,k ), f i (x)ϕ k (x) dx, f i,k := π f i (x)ψ k (x) dx, i = 1, 2, k 1.
A vectorial moment method The moment problem can be written as A k V k + Ã k Ṽ k = F k k 1, ( ) ( ) f1,k f 2,k A k =, Ã k = f 1,k f 2,k I k (q)f 1,k I k (q)f 2,k T T v 1 (t)e k2t dt V k := T, Ṽ v 1 (t)te k2t dt k := T v 2 (t)e k2t dt v 2 (t)te k2 k 2t dt F k = e k2 T ( y, Φ 1,k y, Φ 2,k TI k (q) ) y, Φ 2,k.,
A vectorial moment method Can we find v 1, v 2 solving the previous system of moments such that v 1, v 2 L 2 (, T)?
Construction of the functions f 1, f 2 There exist functions f 1, f 2 L 2 (, π) satisfying Supp f 1, Supp f 2 ω and such that there exists positive constants C 1 and C 2 (only depending on f 1 and f 2 ) such that I 1,k (q) := I 1,k (q) I k (q) det A k C 1 k 6 C 2, k 1. k a q(x) ϕ k (x) 2 dx, I k (q) := π q(x) ϕ k (x) 2 dx.
A minimal time Let T (q) := lim sup k min{ log I 1,k (q), log I k (q) } k 2. Theorem The moment problem leads to T T v i (t)e k2t dt = e k2t M (k) 1,i (y ), v i (t) te k2t dt = e k2t M (k) 2,i (y ), i = 1, 2 k 1 M (k) i,j (y ) C ε e k2 (T (q)+2ε) y L 2 (,π;r 2 ), k 1, 1 i, j 2.
Biorthogonal family E. FERNÁNDEZ-CARA, M. GONZÁLEZ-BURGOS, L. DE TERESA, Boundary controllability of parabolic coupled equations, J. Funct. Anal. 259 (21), no. 7, 172 1758. The sequence { e 1,k := e k2t, e 2,k := te k2 t } k 1 admits a biorthogonal family {q 1,k, q 2,k } k 1 in L 2 (, T) (11) T e r,k q s,j (t) dt = δ kj δ rs, k, j 1, 1 r, s 2, which satisfies that for every ε > there exists a constant C ε,t > such that (12) q i,k L 2 (,T) C ε,t e εk2, k 1, i = 1, 2.
Conclusion T > T (q) M (k) i,j (y ) T T v i (t)e k2t dt = e k2t M (k) 1,i (y ), v i (t) te k2t dt = e k2t M (k) 2,i (y ), C ε e k2 (T (q)+2ε) y L 2 (,π;r 2 ), k 1, 1 i, j 2. v i (t) = ( ) e k2 T M (k) 1,i (y )q 1,k (t) + M (k) 2,i (y )q 2,k (t), i = 1, 2. k 1 e k2t M (k) l,i (y )q l,k (t) C L 2 ε,t e k2 (T T (q) 3ε) y L (,T) 2 (,π;r 2 ), for any k 1 and l, i : 1 l, i 2. ( ε, T T ) (q) v i L 2 (, T), i = 1, 2. 3
The controllability result We have proved There exists T (q) [, + ] such that the system is null controllable at time T > T (q) T > T (q)) a necessary condition? What happens if T < T (q)?
Is it possible to have T (q) >? Theorem For any τ [, ], there exists a function q L (, π) satisfying I k (q) + I 1,k (q) k 1, such that T (q) = τ.
A non controllability result The null controllability property is equivalent to the following observability inequality: ϕ () L 2 (,π;r 2 ) C T T ω B ϕ (x, t) 2 dt. for all ϕ solution to { t ϕ = ( xx 2 + A ) ϕ, in Q = (, π) (, T), ϕ(, ) = ϕ(π, ) = on (, T), B = ( 1 )
The minimal time of control depends on ω The minimal time of control depends on ω M. GONZÁLEZ-BURGOS, L. DE TERESA, Controllability results for cascade systems of m coupled parabolic PDEs by one controll force, Port. Math. 67 (21), no. 1, 91 113. Assume that there exist an open subset ω ω and σ > such that q σ > in ω, then the null controllability occurs at any time T >. The moment method gives a sufficient condition of null controllability This is due to the fact that, in this method, we have restricted the control to a particular form v(x, t) = f 1 (x)v 1 (T t) + f 2 (x)v 2 (T t), Supp f 1, Supp f 2 ω = (a, b)
Supp q ω = : The negative controllability result Assume that T (q) >, otherwise there is nothing to prove. Argue by contradiction and suppose the observability estimate ϕ () L 2 (,π;r 2 ) C T T ω B ϕ (x, t) 2 dt holds true for all ϕ solution to { t ϕ = ( xx 2 + A ) ϕ, in Q = (, π) (, T), ϕ(, ) = ϕ(π, ) = on (, T), B = ( 1 )
Supp q ω = : The negative controllability result For θ = a k Φ 1,k + b kφ 2,k, with (a k, b k ) R 2 and Φ i,k the eigenvectors of the operator L := xx + qa, the previous inequality reads as A 1,k CA 2,k, { A 1,k := e 2k2 T a k 2 + [ a k 2 ψ k 2 L 2 (,π) + (b k Ta k I k (q)) 2]} A 2,k := T ω e 2k2t a k ψ k (x) + b k ϕ k (x) ta k I k (q)ϕ k (x) 2 dx. ψ xx k 2 ψ = [I k (q) q(x)] ϕ k in (, π), ψ() =, ψ(π) =, π ψ(x)ϕ k (x) dx =,
Use the assumption on ω Theorem Let ω = (a, b) (, π) and q L (, π) be a function satisfying Supp q ω =. Then, for any k 1: with τ k a positive constant and g k (x) = I k(q) k x ψ k (x) = τ k ϕ k (x) + g k (x), x ω, sin(k(x ξ))ϕ k (ξ) dξ π I 1,k (q) cos(kx), x ω, 2 k
Use the assumption on ω By choosing a k = 1 and b k = τ k, we get: g k (x) = I k(q) k x The observability estimate becomes a k ψ k (x) + b k ϕ k (x) = g k (x) π I 1,k (q) sin(k(x ξ))ϕ k (ξ) dξ cos(kx) 2 k 1 Ce 2k2 T ( I 1,k (q) 2 + I k (q) 2) Ce 2k2 h 1 k 2 min( log I 1,k(q), log I k (q) ) T i.
The negative null controllability result From the definition of T (q) there exists a subsequence of indices {k n } n 1 N satisfying: T (q) = lim n min ( log I 1,kn (q), log I kn (q) ) kn 2. If T (q) <, as a consequence, we deduce that for any ε > there is n ε 1 such that min ( log I 1,kn (q), log I kn (q) ) k 2 n T (q) ε, n n ε. Coming back to the previous inequality, we obtain 1 Ce 2k2 n[t (q) ε T], n n ε, which gives a contradiction if we take ε (, T (q) T). In the case in which T (q) =, the reasoning is easier and we also get a contradiction.
Is it possible to have a positive minimal time of control? T (q) >? For ω = (a, b) (, π) and q L (, π), is it possible that T (q) >?
Is it possible to have a positive minimal time of control? T (q) >? For ω = (a, b) (, π) and q L (, π), is it possible that T (q) >? Theorem For any τ [, ], there exists a function q L (, π) satisfying Supp q ω = and I k (q) + I 1,k (q) k 1, such that T (q) = τ.
Comments F. ALABAU-BOUSSOUIRA, M. LÉAUTAUD, Indirect controllability of locally coupled wave-type systems and applications, J. Math. Pures Appl. (9) 99 (213), no. 5, 544 576. F. ALABAU-BOUSSOUIRA, Insensitizing exact controls for the scalar wave equation and exact controllability of 2-coupled cascade systems of PDE s by a single control, Math. Control Signals Systems 26 (214), no. 1, 1 46. L. ROSIER, L. DE TERESA, Exact controllability of a cascade system of conservative equations, C. R. Math. Acad. Sci. Paris 349 (211), no. 5-6, 291 296. q and q or q in (, π). These results have been obtained as a consequence of the corresponding hyperbolic results by using the transmutation strategy.
Comments This minimal time also arises in other parabolic problems K. BEAUCHARD, P. CANNARSA, R.GUGLIELMI, Null-controllability of Grushin-type operators in dimension two, J. Eur. Math. Soc. 16 (214), no. 1, pp. 67 11.
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