Observability and null-controllability for parabolic equations in Lp

arXiv:2005.14503v2 [math.FA] 21 Jan 2021

Observability and null-controllability for parabolic equations in L _p -spaces

Clemens Bombach¹, Dennis Gallaun², Christian Seifert^2,3, and Martin Tautenhahn¹

1Technische Universit¨at Chemnitz, Fakult¨at f¨ur Mathematik, 09107 Chemnitz, Germany

2Technische Universit¨at Hamburg, Institut f¨ur Mathematik, 21073 Hamburg, Germany

3Technische Universit¨at Clausthal, Institut f¨ur Mathematik, 38678 Clausthal-Zellerfeld, Germany

Abstract

We study (approximate) null-controllability of parabolic equations inLp(R^d) and provide explicit bounds on the control cost. In particular we consider systems of the form ˙x(t) = −Apx(t) +1Eu(t), x(0) = x⁰ ∈ Lp(R^d), with interior control on a so-called thick set E ⊂ R^d, where p ∈ [1,∞), and where A is an elliptic operator of orderm∈NinLp(R^d). We prove null-controllability of this system via duality and a sufficient condition for observability. This condition is given by an uncertainty principle and a dissipation estimate. Our result unifies and generalizes earlier results obtained in the context of Hilbert and Banach spaces. In particular, our result applies to the casep= 1.

Mathematics Subject Classification (2020). 47D06, 35Q93, 47N70, 93B05, 93B07.

Keywords. Null-controllability, Banach space, non-reflexive,C0-semigroups, elliptic operators, observability estimate,Lp-spaces.

1. Introduction

We consider parabolic control systems on Lp(R^d),p∈[1,∞), of the form

˙

x(t) =−A_px(t) +1_Eu(t), t∈(0, T], x(0) =x₀ ∈L_p(R^d), (1) where −A_p is a strongly elliptic differential operator of order m ∈ N with constant coefficients, 1_E: L_p(E) → L_p(R^d) is the embedding from a measurable set E ⊂ R^d to R^d,T > 0, and where u∈L_r((0, T);L_p(E)) with somer ∈[1,∞]. Hence, the influence of the control functionuis restricted to the subsetE. Note that we allow for lower order terms in the strongly elliptic differential operator. The focus of this paper is laid on null- controllability, that is, for any initial condition x₀ ∈L_p(R^d) there is a control function u∈L_r((0, T);L_p(E)) such that the mild solution of (1) at timeT equals zero. We will also be concerned with the notion of approximate null-controllability, which means that for any ε >0 and any x₀ ∈L_p(R^d) we can find a control functionu ∈L_r((0, T);L_p(E)) such that the mild solution of (1) at timeT has norm smaller thanε; in reflexive spaces,

these two notions agree (see [Car88]). By linearity, (approximate) null-controllability implies that any target state in the range of the semigroup generated by −A_p can be reached (up to an errorε) within timeT.

We will show that ifE is a so-called thick set, then the system is approximately null- controllable for p= 1 and null-controllable if p∈(1,∞). Note that the casep ∈(1,∞) is already covered by [GST20]. However, our new proof unifies the cases p = 1 and p ∈ (1,∞). Moreover, we provide explicit upper bounds on the control cost, i.e. on the norm of the control function u which steers the system (approximately) to zero at time T, which are explicit in terms of geometric properties of the thick set E and of the final time T. Controllability for systems on L_p(Ω), where Ω is a bounded domain and p∈[1,∞), has been studied earlier in the literature, for instance in [FPZ95] in the context of semilinear heat equations, which include as a special case approximate null- controllability for the linear heat equation. For further results in this direction, we refer to [FCZ00] and the survey article [Zua06]. In comparison, we obtain (approximate) null- controllability of linear differential operators of higher orders withu∈L_r((0, T);L_p(R^d)) where r∈[1,∞]. Note that from the physical point of view the case p= 1 is probably most interesting, since we can then interpret the states as heat densities (and its norms will be the total heat content).

An equivalent formulation of approximate null-controllability is final-state observabil- ity of the adjoint system to (1). This means that there is a constant C_obs≥0 such that for all ϕ∈L_p(R^d)^′ we have

S_T^′ ϕ

Lp(Rd)^′ ≤







C_obsR_T

0 k(S_t^′ϕ)|_Ek^r_L^′_p(E)′dt1/r^′

if r^′∈[1,∞), C_obsess sup

t∈[0,T]

k(S_t^′ϕ)|_Ek_Lp(E)′ if r^′=∞,

where (S_t)_t≥0 is theC₀-semigroup generated by−A_p and r^′∈[1,∞] is such that 1/r+ 1/r^′ = 1. This equivalence follows from Douglas’ lemma, see [Dou66] for Hilbert spaces, and [Emb73,DR77,Har78,CP78,Car85,Car88,For14] for Banach spaces.

In Section 2 we formulate our results on final-state observability in Theorem 2.2 and (approximate) null-controllability in Theorem2.3. The proof of Theorem2.2rests on an abstract observability estimate stated in the appendix (see TheoremA.1) and is provided in Section3.

The main strategy we follow to prove observability has first been described in [LR95, LZ98, JL99] for the Hilbert space case (i.e. p = r = 2), and further studied, e.g., in [Mil10,TT11,WZ17,BPS18,NTTV20]. However, far less is known on its generalization to Banach spaces; to the best of our knowledge, we are only aware of [GST20]. Note that strong continuity of the semigroup (S_t)_t≥0 is assumed there. However, being interested in approximate null-controllability in L1 requires observability in L∞, and there strong continuity of semigroups is rather rare [Lot85]. Theorem A.1 provides a generalization of [GST20] to not necessarily strongly continuous semigroups.

2. Observability and Null-controllability in Lp-Spaces

In order to formulate our main theorems we review some basic facts from Fourier analysis.

For details we refer, e.g., to the textbook [Gra14]. We denote by S(R^d) the Schwartz space of rapidly decreasing functions, which is dense in L_p(R^d) for all p ∈[1,∞). The space of tempered distributions, i.e. the topological dual space of S(R^d), is denoted by S^′(R^d). Forf ∈ S(R^d) letFf:R^d→Cbe the Fourier transform of f defined by

Ff(ξ) :=

Z

Rd

f(x)e^−iξ·xdx.

ThenF:S(R^d)→ S(R^d) is bijective, continuous and has a continuous inverse, given by F⁻¹f(x) = 1

(2π)^d Z

R^d

f(ξ)e^ix·ξdξ

for all f ∈ S(R^d). For u ∈ S^′(R^d) the Fourier transform is again denoted by F and is given by (Fu)(φ) =u(Fφ) for φ∈ S(R^d). By duality, the Fourier transform is bijective on S^′(R^d) as well.

Let m∈Nand

a(ξ) = X

|α|₁≤m

a_αξ^α, ξ∈R^d,

be a polynomial of degree m with coefficients a_α ∈ C. We say that the polynomial a is strongly elliptic if there exist constants c > 0 and ω ∈ R such thata satisfies for all ξ∈R^dthe lower bound

Rea(ξ)≥c|ξ|^m−ω. (2)

Note that strong ellipticity implies that m is even.

Given a strongly elliptic polynomial a and p ∈[1,∞], we define the associated heat semigroupS : [0,∞)→ L(L_p(R^d)) by

S_tf =F⁻¹e^−taFf =F⁻¹e^−ta∗f. (3) Note that the second equality holds since e^−ta∈ S(R^d). It is well known that the operator semigroup (S_t)_t≥0 is strongly continuous if p∈[1,∞). For p=∞ the semigroup is the dual semigroup of a strongly continuous semigroup onL1(R^d) and hence it is only weak*- continuous in general. For details we refer, e.g., to [Are04]. By [TR96], the integral kernel k_t = F⁻¹e^−ta satisfies the following heat kernel estimate: There exist c₁, c₂ > 0 such that for allx∈R^d and t >0 we have

|k_t(x)| ≤c₁e^ωtt^−d/me^−c2

_|x|m t

_m−1¹

. (4)

This implies that there is M ≥1 and ω∈R such that for allt≥0 we have

kS_tk ≤Me^ωt, t≥0. (5) In order to formulate our main result we introduce the notion of a thick subsetE ofR^d.

Definition 2.1. Letρ∈(0,1] andL∈(0,∞)^d. A set E⊂R^dis called (ρ, L)-thick if E is measurable and for all x∈R^d we have

E∩

×

(0, Li) +x

!

≥ρ

d

Y

i=1

Li.

Here,|·|denotes Lebesgue measure in R^d.

The following theorem yields a final-state observability estimate for (S_t)_t≥0 on thick sets.

Theorem 2.2. Let m∈N, a:R^d→C a strongly elliptic polynomial of order m, c >0 and ω ∈ R as in (2), and (S_t)_t≥0 as in (3). Let ρ ∈ (0,1], L ∈ (0,∞)^d, E ⊂ R^d a (ρ, L)-thick set, p, r∈[1,∞], and T >0. Then we have for all f ∈Lp(R^d)

kS_Tfk_Lp(Rd)≤









 C_obs

Z T 0

k(S_tf)|_Ek^r_Lp(E)dt ^1/r

if r∈[1,∞), C_obsess sup

t∈[0,T]

k(S_tf)|_Ek_Lp(E) if r=∞, where

C_obs = Ka

T^1/r K_d

ρ

Kd(1+|L|1λ^∗)

exp Km(|L|1ln(K_d/ρ))^m/(m−1)

(cT)^1/(m−1) +Kmax{ω,0}T

! .

Here, λ^∗ = (2^m+4max{ω,0}/c)^1/m, K >0is an absolute constant, and Ka, K_d, Km >0 are constants depending only on the polynomial a, on d, or onm, respectively.

By duality, we thus obtain (approximate) null-controllability for (1).

Theorem 2.3. Let m∈N, a:R^d→C a strongly elliptic polynomial of order m, c >0 and ω ∈ R as in (2), and (S_t)_t≥0 as in (3). Let ρ ∈ (0,1], L ∈ (0,∞)^d, E ⊂ R^d a (ρ, L)-thick set, r∈[1,∞], and T >0.

(a) For any f ∈L₁(R^d) and any ε >0 there exits u∈L_r((0, T);L₁(E))with kuk_{Lr((0,T);L1(E))}≤C_obskfk_L1(R^d)

such that

S_Tf + Z T

0

S_T−t1_Eu(t)dt L1(R^d)

< ε.

(b) Letp∈(1,∞). Then for any f ∈Lp(R^d) there exits u∈Lr((0, T);Lp(E)) with kuk_{Lr((0,T);Lp(E))}≤C_obskfk_Lp(R^d)

such that

S_Tf + Z _T

0

S_T−t1_Eu(t)dt= 0.

Here, C_obs is as in Theorem 2.2 with r replaced byr^′ where r^′ ∈[1,∞] such that 1/r+ 1/r^′ = 1.

Remark 2.4 (Discussion on observability and null-controllability). Forp∈[1,∞) let−A_p be the generator of theC0-semigroup (St)t≥0 onLp(R^d). Note that for allf ∈ S(R^d) we have

A_pf = X

|α|₁≤m

a_α(−i)^|α|∂^αf.

Then, the statement of Theorem 2.2corresponds to a final-state observability estimate for the system

˙

x(t) =−A_px(t), t∈(0, T], x(0) =x₀ ∈L_p(R^d), y(t) =x(t)|_E, t∈[0, T].

Let us now turn to the discussion on null-controllability. For a measurable setE ⊂R^d and T >0 we consider the linear control problem

˙

x(t) =−A_px(t) +1_Eu(t), t∈(0, T], x(0) =x₀ ∈L_p(R^d)

where u ∈ L_r((0, T);L_p(E)) with r ∈ [1,∞]. The unique mild solution is given by Duhamel’s formula

x(t) =S_tx₀+B^tu, where B^tu= Z t

0

S_t−τ1_Eu(τ)dτ.

Then, the statement (a) of Theorem2.3corresponds toapproximately null-controllability in timeT, that is, for allε >0 and andx₀ ∈L₁(R^d), there exists anu∈L_r((0, T);L_p(E)) such that kx(T)k_Lp(R^d) =kS_Tx₀+B^Tuk_Lp(R^d) < ε. The statement (b) of Theorem 2.3 corresponds to null-controllability in time T, that is, for all x₀ ∈ L_p(R^d), p ∈ (1,∞), there exists an u∈ L_r((0, T);L_p(E)) such thatx(T) = 0. Note that in case p∈(1,∞) null-controllability and approximate null-controllability are equivalent, see, e.g., [Car88].

It is a standard duality argument that Theorem 2.3 follows from Theorem 2.2 by means of Douglas’ lemma. For the sake of completeness we give a short proof.

Proof of Theorem 2.3. Let p ∈ [1,∞), r ∈ [1,∞] and B^T: L_r((0, T);L_p(E)) → L_p(R^d) be given by

B^Tu= Z _T

0

S_T−t1_Eu(t)dτ.

Then, by [Vie05, Theorem 2.1] we have for all g∈L_p^′(R^d) k(B^T)^′gk_{Lr((0,T);Lp(E))}^′ = sup

τ∈[0,T]

k(S_T^′ _−τg)|Ek_Lp′(E)= sup

t∈[0,T]

k(S_t^′g)|Ek_Lp′(E)

if r= 1, and

k(B^T)^′gk_{Lr((0,T);Lp(E))}′ = Z T

0

k(S_t−τ^′ g)|_Ek^r_L^′

p′(E)dτ 1/r^′

= Z T

0

k(S_t^′g)|_Ek^r_L^′

p′(E)dt 1/r^′

if r ∈ (1,∞], where r^′ ∈ [1,∞] is such that 1/r + 1/r^′ = 1 and p^′ ∈ (1,∞] is such that 1/p+ 1/p^′ = 1. Since FS_t^′ = e^−ta(−·)F, we have that (S_t^′)_t≥0 is associated to the symbol a(−·) which is strongly elliptic with the same constant c > 0. Moreover, since the associated heat kernel is given by (F⁻¹e^−ta)(−·), we have kS^′_tk ≤ Me^ωt with the same M and ω as in (5). Thus, Theorem 2.2 and the above equalities imply for all g∈L_p^′(R^d)

kS_T^′ gk_Lp′ ≤C_obsk(B^T)^′gk_{Lr′((0,T);Lp′(E))}=C_obsk(B^T)^′gk_{Lr((0,T);Lp(E))}^′,

where C_obs is as in Theorem 2.2 with r replaced by r^′. By Douglas’ lemma, see e.g.

[Har78,Car85,Car88], we conclude

{STf: kfk_Lp(Rd)≤1} ⊂ {B^Tu:kuk_{Lr((0,T);Lp(E))}≤C_obs} if p= 1 and

{S_Tf:kfk_Lp(R^d) ≤1} ⊂ {B^Tu: kuk_{Lr((0,T);Lp(E))}≤C_obs} if p∈(1,∞).

By scaling, this implies the statement of the theorem.

3. Proof of Theorem 2.2

For the proof of Theorem 2.2 we apply the abstract observability estimate in Theo- remA.1. For this purpose, we define a familiy of operatorsP_λ, and verify the uncertainty principle (13) and the dissipation estimate (14).

We start with defining the operators P_λ. Let η ∈ C_c^∞([0,∞)) with 0 ≤ η ≤ 1 such that η(r) = 1 for r ∈[0,1/2] and η(r) = 0 for r ≥1. For λ > 0 we defineχ_λ:R^d →R by χ_λ(ξ) =η(|ξ|/λ). Since χ_λ ∈ S(R^d), we have F⁻¹χ_λ ∈ S(R^d) ⊂L₁(R^d) and for all p∈[1,∞] we defineP_λ:Lp(R^d)→Lp(R^d) byP_λf = (F⁻¹χ_λ)∗f. By Young’s inequality we have for all f ∈L_p(R^d)

kP_λfk_Lp(Rd)=k(F⁻¹χ_λ)∗fk_Lp(Rd)≤ kF⁻¹χ_λk_L1(Rd)kfk_Lp(Rd).

Moreover, the norm kF⁻¹χ_λk_L1(R^d) is independent of λ > 0. Indeed, by the scaling property of the Fourier transform and by change of variables we have for allλ >0

kF⁻¹χ_λk_L1(R^d)=|λ|^dk(F⁻¹χ₁)(λ·)k_L1(R^d)=kF⁻¹χ₁k_L1(R^d). (6) Hence, for allλ >0 the operatorP_λ is a bounded linear operator and the family (P_λ)_λ>0 is uniformly bounded bykF⁻¹χ₁k_L1(R^d).

Next, we observe that the uncertainty principle (13) is a consequence of the following Logvinenko–Sereda theorem from [Kov01], see also [LS74,Kov00] for predecessors.

Theorem 3.1 (Logvinenko–Sereda theorem). There exists K ≥ 1 such that for all p ∈ [1,∞], λ > 0, ρ ∈ (0,1], L ∈ (0,∞)^d, (ρ, L)-thick sets E ⊂ R^d, and f ∈ L_p(R^d) satisfying suppFf ⊂[−λ, λ]^d we have

kfk_Lp(Rd)≤d0e^d1λkfk_Lp(E), where

d0 = e^Kdln(Kd/ρ) and d1 = 2|L|1ln(K^d/ρ). (7)

Concerning the dissipation estimate (14), we first consider the semigroups associated to powers of the Laplacian.

Proposition 3.2. Let m∈N be even, andGt:Lp(R^d)→Lp(R^d) be given by G_tf =F⁻¹e^−t|·|mFf .

Then for all p∈[1,∞], f ∈L_p(R^d), λ >0, and t≥0 we have k(Id−P_λ)G_tfk_Lp(Rd)≤K_m,de^{−2−m−2tλm}kfk_Lp(Rd), where K_m,d >0 is a constant depending only on m and d.

Proof. Let us set a=|·|^m. Hence we have for all f ∈L_p(R^d) that (Id−P_λ)G_tf =F⁻¹((1−χ_λ)e^−ta)∗f,

and by Young’s inequality we obtain for allλ, t >0 and allf ∈Lp(R^d) k(Id−P_λ)G_tfk_Lp(R^d) ≤ kF⁻¹((1−χ_λ)e^−ta)k_L1(R^d)kfk_Lp(R^d).

For µ > 0 we define k_µ: R^d → R by k_µ = F⁻¹((1−χ_µ)e^−a). By substitution first in Fourier space, and then in direct space we obtain, using |t^1/mξ|^m = |t| |ξ|^m, for all λ, t >0

kF⁻¹((1−χ_λ)e^−ta)k_L1(R^d)

= Z

R^d

1 (2π)^d

1 t^d/m

Z

R^d

e^{ix·(t−1/mξ)}(1−χ_t¹/mλ(ξ))e^−|ξ|mdξ

dx

= Z

R^d

1 (2π)^d

Z

R^d

e^iy·ξ(1−χ_t¹/mλ(ξ))e^−|ξ|mdξ

dy=kk_t¹/mλk_L1(R^d). We denote by K_m,d >0 constants which depend only on m and the dimension d. We allow these constants to change with each occurrence. By Young’s inequality and (6), we have

kF⁻¹(χ_µe^−a)k_L1(Rd)=kF⁻¹χ_µ∗ F⁻¹e^−ak_L1(Rd) ≤K_m,d. Hence we find for allµ >0 the uniform bound

kk_µk_L1(R^d)≤ kF⁻¹e^−ak_L1(R^d)+kF⁻¹(χ_µe^−a)k_L1(R^d)≤K_m,d. (8) Next we show that the L1-norm of kµ decays even exponentially as µtends to infinity.

For this purpose, let now µ ≥ 1, α ∈ N^d

0 with |α|₁ ≤ d+ 1, and denote by M_α the multiplication withx^α. By differentiation properties of the Fourier transform we have

M_αk_µ=M_αF⁻¹[(1−χ_µ)e^−a] =F⁻¹D^α_ξ[(1−χ_µ)e^−a]

and hence for allx∈R^d

|x^αk_µ(x)|=

1 (2π)^d

Z

R^d

e^ix·ξD_ξ^α[(1−χ_µ(ξ))e^−|ξ|m]dξ

≤ 1 (2π)^d

Z

Rd

D_ξ^α[(1−χ_µ(ξ))e^−|ξ|m]

dξ. (9) On the integrand of the right-hand side we apply the product rule and the triangle inequality to obtain

D^α_ξ[(1−χ_µ(ξ))e^−|ξ|m] ≤ X

β∈Nd 0 β≤α

α β

D_ξ^α−β(1−χ_µ(ξ))

D^β_ξe^−|ξ|m

. (10)

For all β ∈N^d

0 and β≤α we have

|D_ξ^βe^−|ξ|m| ≤K_m,d(1 +|ξ|)^|β|1(m−1)e^−|ξ|m≤K_m,de^−|ξ|m/2,

where for the last inequality we used that ξ 7→ (1 +|ξ|)^|β|1(m−1)e^−|ξ|m/2 is bounded on R^d. Since µ≥1, for all β ∈N^d

0,β ≤α we have

D_ξ^α−β(1−χµ(ξ))

≤µ^{|β|1−|α|1}(D_ξ^α−βχ1)(ξ/µ)≤sup

γ≤α

sup

ξ∈R^d

|(D^γ_ξχ1)(ξ)|1Rd\BRd(µ/2)(ξ) and hence

D_ξ^α−β(1−χ_µ(ξ))

D_ξ^βe^−|ξ|m

≤K_m,de^−|ξ|m/21R^d\B_Rd(µ/2)(ξ)≤K_m,de^−|ξ|m/4e^−µm/2m+2. Thus, (10) and|α|1 ≤d+ 1 imply for all ξ∈R^dthat

D_ξ^α[(1−χ_µ(ξ))e^−|ξ|m]

≤K_m,de^−|ξ|m/4e^−µm/2m+2 X

β∈Nd 0 β≤α

α β

≤K_m,de^−|ξ|m/4e^−µm/2m+2.

Hence, from (9), for all x∈R^dwe obtain

|x^αk_µ(x)| ≤K_m,de^−µm/2m+2 Z

R^d

e^−|ξ|m/4dξ =K_m,de^−µm/2m+2. (11) In particular, forj∈ {1,2, . . . , d}andα_j = (d+1)e_j, wheree_j denotes thej-th canonical unit vector in R^d, we obtain |x_j|^d+1|k_µ(x)| ≤ K_m,de^−µm/2m+2, hence kxk^d+1_∞ |k_µ(x)| ≤ K_m,de^−µm/2m+2, and consequently for all x∈R^dand all µ≥1 we find

|x|^d+1|k_µ(x)| ≤K_m,de^−µm/2m+2. (12) From (11) withα= 0 and (12) we obtain for all µ≥1 that

kk_µk_L1(R^d)≤K_m,de^−µm/2m+2 Z

B_Rd(1)

dx+K_m,de^−µm/2m+2 Z

R^d\B_Rd(1)

|x|^−d−1dx

≤K_m,de^−µm/2m+2.

From this inequality and (8) we obtain for all µ >0 that kk_µk_L1(R^d)≤K_m,de^−µm/2m+2.

We are now in the position to show the dissipation estimate for the general case.

Proposition 3.3. Let m ∈ N, a:R^d → C a strongly elliptic polynomial of order m, c > 0 and ω ∈ R as in (2), (S_t)_t≥0 as in (3), and (P_λ)_λ>0 as above. Then for all p∈[1,∞], f ∈Lp(R^d), λ >(2^m+4max{ω,0}/c)^1/m, and t≥0 we have

k(Id−P_λ)S_tfk_Lp(R^d)≤K_ae^{−2−m−4ctλm}kfk_Lp(R^d),

where K_a ≥ 1 is a constant depending only on the polynomial a (and therefore also on m and d).

Proof of Proposition 3.3. We introduce ˜a:R^d→C, ˜a(ξ) = (c/2)|ξ|^m. Then ˜aand (a−˜a) are strongly elliptic polynomials of orderm∈N. Note that the semigroup associated to

˜

ais G_(c/2)t for allt≥0, where (G_t)_t≥0 is as in Proposition 3.2. Moreover, let T_t be the semigroup associated toa−˜a. Sincea=a−˜a+ ˜a, it follows thatS_t=T_tG_(c/2)t. One can obtain a corresponding heat kernel bound for the kernel of the semigroup (T_t)_t≥0 with the same growth rate ω as for (S_t)_t≥0 as follows: Since m has to be even, by Young’s inequality for products there exists σ,σ, C˜ ≥0 such that

Re(a−a)(ξ˜ + iη) = Rea(ξ+ iη)−(c/2)|ξ+ iη|^m

≥(3/4)c|ξ|^m−σ|η|^m−ω−(c/2)|ξ+ iη|^m

≥(3/4)c|ξ|^m−σ|η|^m−ω−(c/2)

m/2

X

k=0

m/2 k

(|ξ|²)^k(|η|²)^m/2−k

≥(3/4)c|ξ|^m−σ|η|^m−ω−(c/2)(1 + 1/4)|ξ|^m−C|η|^m

≥(1/8)c|ξ|^m−σ|η|˜ ^m−ω,

with yields (2) with a replaced by a−a˜ and ξ replaced by ξ+ iη. Then, arguing as in [TR96, Proposition 2.1], one can prove via a heat kernel bound as in (4) that there exists ˜M ≥1 such thatkT_tk ≤M˜e^ωt for all t≥0. By Proposition 3.2and since Fourier multipliers commute, we obtain for allf ∈L_p(R^d)

k(Id−P_λ)S_tfk_Lp(R^d)=kS_t(Id−P_λ)fk_Lp(R^d)

≤ kT_tkkG_(c/2)t(Id−P_λ)fk_Lp(R^d)

≤M K˜ _m,de^{−t(2−m−2(c/2)λm−ω)}kfk_Lp(R^d),

where K_m,d > 0 is a constant depending only on m and d. Since λ > (2^m+4max{ω, 0}/c)^1/m, we have 2^−m−2(c/2)λ^m−ω >2^−m−2cλ^m/4 = 2^−m−4cλ^m.

We can finally prove Theorem2.2.

Proof of Theorem 2.2. Let (P_λ)_λ>0 be the family of operators defined at the beginning of this section. Then we have suppF(P_λf)⊂[−λ, λ]^d for all λ >0 and allf ∈L_p(R^d).

Thus, Theorem3.1implies that for all f ∈L_p(R^d) and allλ >0 we have kP_λfk_Lp(Rd)≤d₀e^d1λkP_λfk_Lp(E),

where d₀ and d₁ are as in (7). Moreover, according to Proposition 3.3, for all λ > λ^∗ and allf ∈L_p(R^d) we have

k(I−P_λ)S_tfk_Lp(Rd)≤d₂e^−d3λmtkfk_Lp(E),

where λ^∗ = (2^m+4max{ω,0}/c)^1/m, d₂ ≥ 1 depends only on the polynomial a, and whered₃= 2^−m−4c. Moreover, the functiont7→ k(S_tf)|_Ek_Lp(E) is Borel-measurable for all f ∈L_p(R^d). Indeed, if p ∈[1,∞) the semigroup (S_t)_t≥0 is strongly continuous and the measurability follows. If p=∞, measurability is a consequence of duality and the representation of the norm in L_∞(E) by means of the Hahn–Banach theorem.

Hence we can apply Theorem A.1 with X =Lp(R^d), Y =Lp(E), C:X → Y given by the restriction map on E, and obtain that the statement of the theorem holds with C_obs replaced by

C˜_obs = C₁ T^1/r exp

C₂

T^m−11 +C₃T

,

whereT^1/r = 1 if r=∞, and C1 = (4M d0) maxn

4d2M²(d0+ 1)8/(e ln 2)

,e^4d12λ∗o , C₂ = 4 2·8^m−m1d^m₁ /d_3m−1¹

, C₃ = max{ω,0} 1 + 10/(e ln 2)

,

with M as in (5). We denote by K_d, Km, and Ka positive constants which depend only on the dimensiond, on m, or on the polynomiala, respectively. A straightforward calculation shows that

C₁ ≤K_a K_d

ρ

K_d(1+|L|1λ^∗)

and C₂ ≤ K_m(|L|₁ln(K_d/ρ))^m/(m−1)

c^1/(m−1) .

Thus we obtain C˜_obs ≤ K_a

T^1/r K_d

ρ

K_d(1+|L|1λ^∗)

exp K_m(|L|₁ln(K_d/ρ))^m/(m−1)

(cT)^1/(m−1) +C₃T

!

=:C_obs.

A. Sufficient criteria for observability in Banach spaces

We provide an abstract sufficient criteria for final-state observability in Banach space, which is a slight generalization of Theorem 2.1 in [GST20]. In particular, it does not assume strong continuity of the semigroup. For the proof we comment only on the necessary modifications compared to [GST20].

Theorem A.1. Let X and Y be Banach spaces, C:X →Y a bounded linear operator, (S_t)_t≥0 a semigroup onX, M ≥1 and ω ∈R such that kS_tk ≤Me^ωt for all t≥0, and assume that for allx∈X the mapping t7→ kCS_txk_Y is measurable. Further, letλ^∗≥0, (P_λ)_λ>λ^∗ a family of bounded linear operators inX,r ∈[1,∞],d0, d1, d3, γ1, γ2, γ3, T >0 with γ₁ < γ₂, and d₂≥1, and assume that

∀x∈X ∀λ > λ^∗: kP_λxkX ≤d0e^d1λγ1kCP_λxkY, (13) and

∀x∈X ∀λ > λ^∗ ∀t∈(0, T /2] : k(Id−P_λ)S_txk_X ≤d₂e^{−d3λγ2tγ3}kxk_X. (14) Then we have for all x∈X

kS_Txk_X ≤





 C_obs

RT

0 kCS_txk^r_Y dt1/r

if r∈[1,∞), C_obsess sup_t∈[0,T]kCS_txk_Y if r=∞, with

C_obs= C₁ T^1/r exp

C₂ T

γ1γ3 γ2−γ1

+C₃T

,

where T^1/r = 1 if r =∞, and C₁ = (4M d₀) maxn

4d₂M²(d₀kCk+ 1)8/(e ln 2)

,e^{4d1(2λ∗)γ1}o , C₂ = 4 2^γ1(2·4^γ3)

γ1γ2

γ2−γ1d^γ₁²/d^γ₃¹_γ2−γ1¹ , C₃ = max{ω,0} 1 + 10/(e ln 2)

.

The assumption in (13) is an abstract uncertainty principle (sometimes also called spectral inequality), while (14) is a dissipation estimate. Thus, Theorem A.1 can be rephrased that an uncertainty principle together with a dissipation estimate implies a final state observability estimate.

Remark A.2. In the situation of Theorem A.1, if we assume that t7→CS_tx is Bochner measurable, we can rewrite the statement of the theorem as

kS_Txk_X ≤C_obskCS_(·)xk_Lr((0,T);Y).

Proof of Theorem A.1. Since we do not assume the semigroup (S_t)_t≥0 to be strongly continuous, we cannot apply [GST20, Theorem 2.1] directly. The strong continuity of (St)t≥0 was assumed in [GST20] in order to ensure that for all x ∈X and λ > λ^∗ the functions

F(t) = S_tx

X, F_λ(t) =

P_λS_tx

X, F_λ^⊥(t) =

(Id−P_λ)S_tx X, G(t) =

CS_tx

Y, G_λ(t) =

CP_λS_tx

Y, G^⊥_λ(t) =

C(Id−P_λ)S_tx Y,

are measurable. The measurability of these six functions was used to obtain the estimate F(t)≤ 2Me^ω+Td₀e^d1λγ1

t

Z t t/2

G(τ)dτ +d₂M²e^{5ω+T /4}e^d1λγ1

e^{d3λγ2(t/4)γ3} (d₀kCk+ 1)F(t/4), where ω₊ = max{0, ω}. Such an inequality implies the statement of the theorem by iteration, see [GST20]. Thus it suffices to show the last displayed inequality by assuming merely measurability of the mapping t7→G(t). Let t >0,τ ∈[t/2, t] andx∈X. Since F(τ)≤F_λ(τ) +F_λ^⊥(τ), by our assumptions and by the semigroup property we obtain

F(τ)≤d₀e^d1λγ1G_λ(τ) +d₂e^{−d3λγ2(τ /2)γ3}F(τ /2).

Using G_λ(τ) ≤ G(τ) +G^⊥_λ(τ) ≤ G(τ) +kCkF_λ^⊥(τ), our assumption, e^d1λγ1 ≥ 1, and F(τ /2)≤Me^ω+t/4F(t/4) we obtain

F(τ)≤d₀e^d1λγ1G(τ) + (d₀kCk+ 1)d₂e^{−d3λγ2(τ /2)γ3}e^d1λγ1Me^ω+t/4F(t/4).

Since F(t)≤Me^ω+tF(τ), we obtain

F(t)≤Me^ω+td₀e^d1λγ1G(τ) + (d₀kCk+ 1)d₂e^{−d3λγ2(τ /2)γ3}e^d1λγ1M²e^ω+5t/4F(t/4).

Since the mapping τ 7→ G(τ) is measurable by assumption, we can integrate this in- equality with respect to τ, and obtain the desired estimate.

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