Observability and measurable sets

Transcription

1 Observability and measurable sets Luis Escauriaza UPV/EHU Luis Escauriaza (UPV/EHU) Observability and measurable sets 1 / 41

2 Overview Interior: Given T > 0 and D Ω (0, T ), to find N = N(Ω, D, T ) > 0 such that for all u 0 in L 2 (Ω) there is f in L 2 (D) verifying f L 2 (D) N u 0 L 2 (Ω), and such that the solution u to u t u = f χ D, in Ω (0, T ), u = 0, in Ω [0, T ], u(0) = u 0, in Ω, satisfies u(t ) = 0. Luis Escauriaza (UPV/EHU) Observability and measurable sets 2 / 41

3 Boundary: For given T > 0 and J Ω (0, T ) to find N = N(Ω, J, T ) such that for all u 0 in L 2 (Ω) there is h in L 2 (J ) verifying h L 2 (J ) N u 0 L 2 (Ω) and such that the solution to u t u = 0, in Ω (0, T ), u = h χ J, in Ω [0, T ], u(0) = u 0, in Ω, satisfies u(t ) 0. Luis Escauriaza (UPV/EHU) Observability and measurable sets 3 / 41

4 Previous results D.L. Russell (1973) proved that provided the wave equation is exactly controllable at some time T > 0, with controls acting over D = ω (0, T ), ω Ω an open set, then the heat equation can be null-controlled with controls acting over the same region. G. Lebeau and L. Robbiano (1995) proved the null controllability of the heat equation, both at the interior and at the boundary, and by means of an explicit construction of the control functions. Luis Escauriaza (UPV/EHU) Observability and measurable sets 4 / 41

5 Let e 1, e 2,..., e n,... and ω 2 1 ω ω2 n,... denote the eigenfunctions and eigenvalues for with zero Dirichlet boundary conditions in Ω { e j + ω 2 j e j = 0, en Ω, e j = 0, en Ω u(x, y) = ω j µ ( aj e ω j y + b j e ω j y ) e j (x), when µ ω 1 and a j, b j R. Spectral inequalities: If ω Ω is an open set, there is N = N(ω, Ω) such that ω j µ a 2 j + b 2 j e 2Nµ 3 4 for all µ ω 1, a j, b j R. 1 4 ω ω j µ(a j e yω j + b j e yω j )e j 2 dxdy, Luis Escauriaza (UPV/EHU) Observability and measurable sets 5 / 41

6 If γ Ω is an open set, there is N = (γ, Ω) such that ω j µ a 2 j + b 2 j e 2Nµ 3 4 for all µ ω 1, a j, b j R. 1 4 γ (a j e yω j + b j e yω j ) e j ν 2 dσdy, ω j µ Luis Escauriaza (UPV/EHU) Observability and measurable sets 6 / 41

7 G. Lebeau and L. Robbiano used quantitative results of unique continuation for elliptic equations: quantification of the unique continuation from open sets for u(x, y) = ω j µ ( aj e ω j y + b j e ω j y ) e j (x), They proved the following quantitative estimates of propagation of smallness. Luis Escauriaza (UPV/EHU) Observability and measurable sets 7 / 41

8 There are N > 0 and θ (0, 1) such that u L 2 (Ω ( 1 8, 5 8 )) N u θ L 2 (Ω (0,1)) u 1 θ L 2 (ω ( 1 4, 3 4 )), u L 2 (Ω ( 1 8, 5 8 )) N u θ u L 2 (Ω (0,1)) ν 1 θ L 2 (γ ( 1 4, 3 )), 4 hold, when { u + y 2 u = 0, in Ω R, u = 0, in Ω R. G. Lebeau and E. Zuazua also proved the spectral inequality aj 2 e Nµ a j e j 2 dx, ω j µ ω j µ B R (x 0) when B 4R (x 0 ) Ω, µ ω 1, a j, b j R. Luis Escauriaza (UPV/EHU) Observability and measurable sets 8 / 41

9 The Hilbert uniqueness method (HUM) by J.L. Lions shows that the null controllability is equivalent to the corresponding observability inequality for solutions to the heat equation. Observability inequalities: Let D Ω (0, T ). Then, there is N = N(Ω, D, T ) such that e T f L2 (Ω) N e t f L2 (D), for all f L 2 (Ω), where u(t) = e t f is the solution to { u t u = 0, in Ω (0, T ), u(0) = f, in Ω (0, T ). Luis Escauriaza (UPV/EHU) Observability and measurable sets 9 / 41

10 Let J Ω (0, T ). Then, there is N = N(Ω, J, T ) such that for all f L 2 (Ω). e T f L 2 (Ω) N ν et f L 2 (J ), The observability inequalities are global quantitative results of parabolic unique continuation. Luis Escauriaza (UPV/EHU) Observability and measurable sets 10 / 41

11 O. Fursikov, O. Yu. Imanulivov, Controllability of Evolution Equations (1996). O. Yu. Imanulivov, M. Yamamoto, On Carleman inequalities for parabolic solutions in Sobolev spaces of negative order and exact controllability for semilinear parabolic equations (2000). E. Fernandez Cara y S. Guerrero, Global Carleman inequalities for parabolic systems and applications to null controllability (2006) M. Gonzalez Burgos, Controlabilidad de E.D.P. Parabólicas (2001). Luis Escauriaza (UPV/EHU) Observability and measurable sets 11 / 41

12 In all of these works the control regions are always of the forms D = ω (0, T ) and J = γ (0, T ), with ω Ω and γ Ω open sets. The cost of the controllability depends on the radius of the largest ball contained in ω, the geometry of Ω and T. Something similar happens with the cost of the controllability at the boundary. Luis Escauriaza (UPV/EHU) Observability and measurable sets 12 / 41

13 The techniques for the proofs of the observably inequalities make use of energy methods and of global parabolic Carleman inequalities. They require the construction of suitable Carleman weights. τ 3 2 e τα(x)/t(t t) t 3 2 (T t) 3 2 ϕ L 2 (Ω [0,T ]) N e τα(x)/t(t t) ( t ϕ ϕ) L 2 (Ω [0,T ]) + Nτ 3 2 e τα(x)/t(t t) t 3 2 (T t) 3 2 ϕ L 2 (ω [0,T ]). holds for τ N and ϕ C (Ω [0, T ]), with ϕ = 0 in Ω [0, T ]. Luis Escauriaza (UPV/EHU) Observability and measurable sets 13 / 41

14 The proof of these inequalities requires smoothness of the Carleman weight function and to make sure that it has its extreme values located in suitable regions associated to D = ω (0, T ), Ω (0, T ) and T > 0. The construction of such functions seems not possible when ω does not contain an open ball or it is measurable set. Luis Escauriaza (UPV/EHU) Observability and measurable sets 14 / 41

15 Null controllability and Measurable Sets We prove the interior and boundary null controllability of the heat equation with controls acting over space-time measurable sets with positive measure. When D = ω (0, T ) and J = γ (0, T ) with ω and γ measurable, we can use the methods developed by G. Lebeau and L. Robbiano. When D Ω (0, T ) and J Ω (0, T ) are arbitrary measurable sets, its possible time dependency does not allow to use the G. Lebeau and L. Robbiano methods and must establish the existence of control functions by proving observability inequalities. The same methods apply to prove the same results for more general parabolic evolutions associated to self-adjoint elliptic operators with space-analytic coefficients. Luis Escauriaza (UPV/EHU) Observability and measurable sets 15 / 41

16 Teorema Let n 2. Then, t can be null controlled at all times T > 0, with interior controls acting over D Ω (0, T ), with D a measurable set with positive measure, when = (A(x) ) + V (x), is a self-adjoint elliptic operator, A is smooth in Ω, V is bounded in Ω and both are real-analytic in a neighborhood of B 2R (x 0 ) Ω, with D B R (x 0 ) (0, T ). The same holds when n = 1 for = 1 ρ(x) [ x (a(x) x ) + b(x) x + c(x)], D = ω (0, T ), provided that a, b, c, ρ and ω are measurable in Ω = (0, 1). Luis Escauriaza (UPV/EHU) Observability and measurable sets 16 / 41

17 Teorema Let n 2. Then, t is null controllable at all times T > 0 with boundary controls acting over J Ω (0, T ), with J measurable with positive surface measure in Ω (0, T ), when = (A(x) ) + V (x), is a self-adjoint elliptic operator, the matrix A is regular in Ω, V is bounded in Ω and A, V and Ω are real-analytic in a neighborhood of J in Ω. The minimal regularity of A away from the neighborhoods of real-analyticity around D or J is Lipschitz. Luis Escauriaza (UPV/EHU) Observability and measurable sets 17 / 41

18 Methods The previous results are consequences of the following facts: 1. The quantification of the real-analyticity at the interior and at the boundary of solutions to elliptic equations with analytic coefficients. F. John (1955), C.B. Morrey (1966), C. B. Morrey, L. Nirenberg (1957). 2. The quantification of reasonings by Landis and Oleinik which imply the space-analyicity of solutions to parabolic equations with zero Dirichlet lateral data. 3. The following estimate of propagation of smallness from measurable sets due S. Vessella (1999). Luis Escauriaza (UPV/EHU) Observability and measurable sets 18 / 41

19 S. Vessella (1999) Let f : Ω R n R be a real-analytic function with Ω bounded open set in R n, α f (x) M α!ρ α, when x Ω, α N n, for some M > 0, 0 < ρ 1 and E Ω be a measurable set with positive measure. Then, there are N = N(ρ, E ) and θ = θ(ρ, E ), 0 < θ < 1, such that ( θ f L (Ω) N f dx) M 1 θ. E Luis Escauriaza (UPV/EHU) Observability and measurable sets 19 / 41

20 Classic Inequality of Propagation of Smallness: { u + 2 y u = 0, in Ω R, u = 0, in Ω R. Let (x 0, τ) Ω R and 0 < r 1 < r 2 < r 3 < + be such that B r3 (x 0, τ) (Ω R) is star-shaped. Then, there are θ = θ(r 1, r 2, r 3 ), 0 < θ < 1 and N = N(Ω, r 1, r 2, r 3 ) such that u L 2 (B r2 (x 0,τ) (Ω R)) N u θ L 2 (B r1 (x 0,τ) (Ω R)) u 1 θ L 2 (B r3 (x 0,τ) (Ω R)). For us the more interesting case is when B r1 (x 0, τ) Ω R is well located inside Ω R at positive distance from Ω R and with r 2 > d(x 0, Ω). Luis Escauriaza (UPV/EHU) Observability and measurable sets 20 / 41

21 x 0 Τ B r1 B r2 B r3 Luis Escauriaza (UPV/EHU) Observability and measurable sets 21 / 41

22 The simplest way to derive the latter estimate follows from the logarithmic convexity of H(r) = 1 r n u 2 (x, y)dxdy, B r (x 0,τ) Ω R with respect to the variable t = log r, when B r3 (x 0, τ) Ω R is star-shaped and { u + 2 y u = 0, en Ω R, u = 0, en Ω R. The frequency function of A. S. Almgren, Fang-Hua Lin, N. Garofalo and I. Kukavica. Luis Escauriaza (UPV/EHU) Observability and measurable sets 22 / 41

23 The local quantification of the real-analyticity of solutions to elliptic equations gives the following: Let (x 0, τ) Ω R, 0 < R 1, 0 < ρ 1 and u verify { u + 2 y u = 0, in Ω R, then u = 0, in Ω R, x α y β u L (B R (x 0,τ) (Ω R)) N α!β!(ρr) ( α β u 2 dxdy B 2R (x 0,τ) (Ω R) for some N and ρ. ) 1 2 Luis Escauriaza (UPV/EHU) Observability and measurable sets 23 / 41

24 The latter and Vessella s theorem are seen to imply the spectral inequalities ω j µ a 2 j + b 2 j e 2Nµ ω ω j µ(a j e yω j + b j e yω j )e j 2 dxdy, for µ ω 1, a j, b j R, and when ω Ω is a measurable set with positive measure. Luis Escauriaza (UPV/EHU) Observability and measurable sets 24 / 41

25 Thoerem (Observability inequality at the interior) Ω is a bounded Lipschitz domain in R n such the the spectral inequalities aj 2 e Nµ a j e j 2 dx, (1) ω j µ ω j µ B R (x 0 ) hold when µ ω 1, a j R and B 4R (x 0 ) Ω, 0 < R 1, for some N = N(Ω, B R (x 0 )). Then, given T > 0 and a measurable set D B R (x 0 ) (0, T ) with D > 0, there is B = B(Ω, T, B R (x 0 ), D, D ) such that e T f L 2 (Ω) e B e t f (x) dxdt, holds for all f in L 2 (Ω). D Luis Escauriaza (UPV/EHU) Observability and measurable sets 25 / 41

26 The spectral inequality (1) and the special decay of the L 2 (Ω)-norm of e t f, when the first µ frequencies of f are zero, imply the following global two-spheres and one-cylinder inequality of propagation of smallness: e t f L 2 (Ω) when 0 s < t T. Define then ( ) θ N e N/(t s) e t f L 2 (B R (x 0 )) e s f 1 θ L 2 (Ω), D t = {x Ω : (x, t) D} and E = {t (0, T ) : D t D (2T ) }. From Fubini s theorem, E is measurable in (0, T ) and E D /2 Luis Escauriaza (UPV/EHU) Observability and measurable sets 26 / 41

27 α x e t f L (B 2R (x 0 )) e NR2 /(t s) α!(rρ) α e s f L 2 (Ω). From Vessella s theorem, one may replace B R (x 0 ) by D t and to derive the following: Theorem There are N = N(Ω, B R (x 0 ), D /T ) and θ = θ(ω, B R (x 0 ), D /T ), 0 < θ < 1 such that ( ) θ e t f L 2 (Ω) e N/(t s) e t f L 1 (D t) e s f 1 θ L 2 (Ω), when 0 s < t, t E and f L 2 (Ω). Luis Escauriaza (UPV/EHU) Observability and measurable sets 27 / 41

28 Lema Let l be a density point of E. Then, for each z > 1, there is l 1 = l 1 (z, E) in (l, T ) such that the sequence T > l 1 > l 2 >... l m > l m+1 > > l defined as l m+1 = l + z m (l 1 l), m = 1, 2,..., verifies that E (l m+1, l m ) 1 3 (l m l m+1 ), si m 1. Luis Escauriaza (UPV/EHU) Observability and measurable sets 28 / 41

29 γ(z m ) e lm f L 2 (Ω) γ(z m 1 ) e l m+1 f L 2 (Ω) N lm l m+1 χ E (s) e s f L 1 (D s) ds, when m 1. γ(t) = e A/t and A = A(Ω, R, E, D / (T B R )) = Finally, because 2 (N + 1 θ)2 θ (l 1 l). e T f L 2 (Ω) e l1 f L 2 (Ω), sup e t f L 2 (Ω) < +, t 0 we get e T f L 2 (Ω) e B for all f L 2 (Ω) and for some B 0. D e t f (x) dxdt, ĺım γ(t) = 0, t 0+ Luis Escauriaza (UPV/EHU) Observability and measurable sets 29 / 41

30 A mixture of ideas from L. Miller, A direct Lebeau-Robbiano strategy for the observability of heat-like semigroups (2010) and K. D. Phung y G. Wang, An observability estimate for parabolic equations from a measurable set in time and its applications (2011). The telescoping series method. Luis Escauriaza (UPV/EHU) Observability and measurable sets 30 / 41

31 Locally star-shaped domain A Lipschitz domain Ω in R n is locally star-shaped when for each p Ω, there are x p in Ω and r p > 0 such that In particular, p x p < r p and B rp (x p ) Ω is star-shaped with center x p. (q x p ) ν q 0, for a.e. q B rp (x p ) Ω. Luis Escauriaza (UPV/EHU) Observability and measurable sets 31 / 41

32 Teorema Let Ω a bounded, Lipschitz and locally star-shaped domain in R n. Then, Ω verifies the spectral inequalities (1). Lema Ω We prove it by combining Algrem s frequency function with a boundary Carleman inequality. Let Ω be a Lipschitz domain in R n with Ω B R, for some R > 0, 0 Ω and τ > 0. Then, x 2τ u 2 dx R2 4τ 2 x 2τ+2 ( u) 2 dx R2 q ν q 2τ ( ) u 2 2τ ν dσ, for all u in C 2 (Ω) with u = 0 in Ω. Ω Ω Luis Escauriaza (UPV/EHU) Observability and measurable sets 32 / 41

33 Lemma Let Ω be a Lipschitz domain in R n, q Ω and r > 0. Then, there is N = N(m) and θ = θ(m), 0 < θ < 1, such that u L 2 (B r (q) Ω) Nr 3θ 2 u ν θ L 2 ( 6r (q)) u 1 θ L 2 (B 8r (q) Ω), when { u = 0, in B 8r (q) Ω, 6r (q) = B 6r (q) Ω. u = 0, in B 8r (q) Ω. Luis Escauriaza (UPV/EHU) Observability and measurable sets 33 / 41

34 The spectral inequalities (1) have been shown when Ω is of class C 2. We can replace B R (x 0 ) Ω by any measurable set ω B R (x 0 ) with positive measure. These include all the C 1 -domains, Lipschitz polyhedron in R n, n 3, all the convex domains in R n and all the Lipschitz polygons in the plane. Also Lipschitz domains with Lipschitz constant less than 1 2. Luis Escauriaza (UPV/EHU) Observability and measurable sets 34 / 41

35 Theorem (Observability inequality at the boundary) Ω a bounded Lipschitz domain in R n where the spectral inequalities (1) hold, T > 0, q Ω and R (0, 1], are such that 4R (q) is analytic. Then, for each J R (q) (0, T ) with J > 0, there is 2 B = B(Ω, T, R, J, J ), such that e T f L 2 (Ω) e B ν et f (x) dσdt, when f L 2 (Ω). J 4R (q) = B 4R (q) Ω. Luis Escauriaza (UPV/EHU) Observability and measurable sets 35 / 41

36 Lema Here we use the Carleman inequality: Let Ω be a bounded Lipschitz domain in R n, 0 / Ω and σ(t) = te t. Then, τ σ(t) τ e x 2 /8t h L 2 (Ω (0,+ )) t 1/2 σ(t) τ e x 2 /8t ( + t )h L 2 (Ω (0,+ )) + q 1/2 σ(t) τ e q 2 /8t h ν L 2 ( Ω (0,+ )), when τ 1 and h C 0 (Ω [0, + )) with h = 0 in Ω [0, + ). Luis Escauriaza (UPV/EHU) Observability and measurable sets 36 / 41

37 Lemma There are N = N(Ω, R, n) and θ = θ(ω, n), 0 < θ < 1, such that e t f L 2 (Ω) when f L 2 (Ω) and 0 s < t T. ( ) θ N e N/(t s) ν eτ f L 2 ( R (q) [s,t]) e s f 1 θ L 2 (Ω), Because 4R (q) is analytic, there are N and ρ, 0 < ρ 1, such that α x β t e t f (x) enr2 /(t s) α! β! (Rρ) α ((t s) /4) β es f L 2 (Ω), when x 2R (q), 0 s < t, α N n and β 0. By Vessella s theorem, we can replace R (q) [s, t] above by any measurable set inside R (q) [s + ɛ (t s), t ɛ (t s)], when 2 0 < ɛ < 1/2; and then use the telescoping series method as before. Luis Escauriaza (UPV/EHU) Observability and measurable sets 37 / 41

38 Applications The observability inequalities over measurable sets imply the following results. 1 Existence of interior controls in L (D) and of boundary controls in L (J ), when D and J are measurable sets with positive measure in Ω (0, T ) and Ω (0, T ) respectively. 2 The bang-bang properties and uniqueness of the L -optimal control functions. 3 The bang-bang property and uniqueness of the L -controls associated to the time optimal control problems of the first and second kind. Luis Escauriaza (UPV/EHU) Observability and measurable sets 38 / 41

39 The same ideas show that one can prove the null controllability of other time independent parabolic evolutions whose corresponding spectral inequalities (or observability inequalities) are otherwise unknown over the control regions D = ω (0, T ) or J = γ (0, T ), and with ω and γ open or measurable sets in Ω and Ω respectively. { t u Lu = 0, en Ω (0, T ), u = 0, en Ω, with u(t) = e tl u 0 and L = (L α u) m α=1, with u = (u1,..., u m ), m 2 and L α u = i,j,α,β m n β=1 i,j=1 i (a αβ ij (x) j u β ), α = 1,..., m, a αβ ij C ω (Ω). a αβ ij (x)ξi α ξ β j δ ξi α 2, when ξ R nm and x R n. i,α Luis Escauriaza (UPV/EHU) Observability and measurable sets 39 / 41

40 Parabolic evolutions of higher order: t u + ( 1) m m u, m = 2,..., with zero Dirichlet lateral boundary conditions; i.e. u = u = = m 1 u = 0, in Ω. Luis Escauriaza (UPV/EHU) Observability and measurable sets 40 / 41

41 It is not known how to derive with classical methods (Carleman inequalities or frequency functions) inequalities of propagation of smallness of Hölder type, which propagate up the boundary, the possible smallness from interior open sets closed to the boundary of solutions to general elliptic systems of second order or to higher order elliptic equations with local zero Dirichlet data. These are not even known with C boundaries. Luis Escauriaza (UPV/EHU) Observability and measurable sets 41 / 41