This article serves as a review of observability inequalities from measurable sets for solutions to the heat equation. The purpose of trying to obtain the two observability inequalities that we will see and prove in this article, was that in control theory there is a very well known result, the Hilbert Uniqueness Method, that assures that the null controllability of an equation is equivalent to obtain an observability inequality for the adjoint equation. This result is attributed to J.L. Lion. In our previous research we were studying the null controllability of parabolic equations over measurable sets, so, for the Hilbert Uniqueness Method reason, we focused on proving the observability inequalities (Theorems 1 and 2) that we will see in this article.
In the next lines of the Introduction we will establish the type of problem we will work on, remember some apriori estimates for the parabolic equations and recall some previous results about this kind of work.
Then, in Section 2, we will establish and prove Theorem 1 and 2 which will give us two observability inequalities. We will continue, in Section 3, showing some applications of the obserbavility inequalities, the bang-bang property for the minimal time, optimal time and minimal norm control problems. In Section 4, we will establish some open problems related to observability inequalities and their applications to control theory. Finally, with Section 5, we will finish the article giving some details of a definition and a proof requiered in Section 3.
Let Ω be a bounded Lipschitz domain in ℝ
with
for all
for all
In the case that 𝒟 = ω× (0,
In [2], we stablished the inequalities (2) and (3) when 𝒟 and 𝒥 were arbitrary subsets of positive measure and of positive surface measure in Ω × (0,
We will see how we proved the two above-mentioned inequalities. We start assuming that the Lebeau-Robbiano spectral inequality stands on Ω. To introduce it, we write
for the eigenvalues of −Δ with the zero Dirichlet boundary condition over ∂Ω, and {
For
where
Throughout this paper the following notations are used:
Our main results related to the observability inequalities are stated as follows, but, first, we will define the real-analyticity of the set Δ4
Here,
In the next two theorems, we establish two observability inequalities for the heat equation over Ω × (0,
Next, we will see some results that will be necessary in the proof of the previous Theorem 1.
From Fubini’s theorem,
□
The reader can find the proof of the following Lemma 2 in either [10, pp. 256-257] or [11, Proposition 2.1].
[Theorem 1]
Setting
recalling that
we have
Choose now
The choice of
with
Finally, because of
and (10), the addition of the telescoping series in (16) gives
which proves (7) with
Next, we will see some results that will be necessary in the proof of the previous Theorem 2.
Then, 𝒥
From Fubini’s theorem,
□
[Theorem 2] Let
Let
Then, we can use the same arguments as those in the proof of Theorem 1 to verify Theorem 2. □
We will now show some applications of the Theorems 1 and 2 in the control theory of the heat equation. Specifically, we will focus on the uniqueness and bang-bang properties of the minimal time, time optimal and minimal
In this section we assume that
First of all, we will show that Theorems 1 and 2 imply the null controllability with controls restricted over measurable subsets in Ω × (0,
and
where
From now on, we always denote by
We only prove the boundary controllability. Let
Let
We set
It is clear that |𝓜| > 0. The proof of Theorem 2, the change of variables
holds, when
for some
Since 𝓜 ⊂ ∂Ω ×[
verifies
From the Hahn-Banach theorem, there is a linear extension
Thus,
We extend
To prove (24), we first use the unique solvability for the problem
with lateral Dirichlet data
In this section, we apply Theorems 1 and 2 to get the bang-bang property for the minimal time control problems usually called the first type of time optimal control problems; they are stated as follows. Let
Let
and
where
Any solution of
Let {
It suffices to show that
For this purpose, let
and
Also, by standard interior parabolic regularity there is
when
This, along with (26), (28), (29) and (30) indicates that (27) holds for all
Now, we can use the same methods as those in [14], as well as in Lemma 9, to get the following consequences of Theorems 1 and 2 respectively.
Next, we make use of Theorems 1 and 2 to study the bang-bang property for the time optimal control problems where the interest is on retarding the initial time of the action of a control with bounded
and
where
Consider the time optimal control problems:
and
Any solution of
Now, we can use the same arguments as those in the proof of Theorem 3.4 in [11] to get the following consequences of Theorem 1 and Theorem 2 respectively:
In this section, we apply Theorems 1 and 2 to get the bang-bang property for the minimal norm control problems; they are stated as follows. Let 𝒟 and 𝒥 be the subsets given at the beginning of this section. Let
and
Consider the minimal norm control problems:
and
Any solution of (
We can use the same methods as those in [11] to get the following consequences of Theorem 1 and Theorem 2 respectively:
In tis section we will establish the heat equation with similar conditions to what we studied before, but in this case we will require it to verify other type of boundary conditions instead of Dirichlet boundary conditions.
Let Ω be a bounded Lipschitz domain in ℝ
with Neumann boundary condition and
with Robin boundary condition, where
We proved two obserbavility inequalities (Theorems 1 and 2) for these kind of equations over measurable sets with Dirichlet boundary conditions, but if we change that condition to now use Neumann or Robin conditions, would we be able to prove some similar observability inequalities? And, if that’s the case, could we apply them to prove some bang-bang properties?
The idea of facing these questions is to spread our mathematical knowledge about this kind of problems and also to discover new interesting ways or limitations in the techniques we are used to working with. It could also be physically interesting because of the physical meaning of these new boundary conditions, as we will see now.
The Dirichlet boundary condition states that we have a constant temperature at the boundary. This can be considered as a model of an ideal cooler in a good contact having infinitely large thermal conductivity.
With the Neumann boundary condition case for the heat flow, we can say that we have a constant heat flux at the boundary or that it corresponds to a perfectly insulated boundary. If the flux is equal to zero, the boundary condition describes the ideal heat insulator with the heat diffusion. For the Laplace equation and drum modes, we could think this corresponds to allowing the boundary to flap up and down but not move otherwise.
Finally, the Robin boundary condition is the mathematical formulation of Newton’s law of cooling where the heat transfer coefficient
Here, we will give the definition of a Lipschitz domain and complete the proof of the equation (24) that appeared in the proof of Corollary 8.
Let Ω be a bounded domain in ℝ
where
(Proof of (24)] For each (
The later are called respectively elliptic and parabolic non-tangential approach regions from the interior of Ω × (0,
When
Let
(See the beginning of Section 3 for the definition of the solution to this equation.)
Let
and let
From [6, Theorem 3.2] [3, Theorem 6.1] or [4, Theorem 2.9]
and the limits
exist and are finite for a.e. (
when
Let
and from [6, Theorems 1.3 and 1.4] or the proof of (40) and (41) in this appendix, there are
when 0 <
exists and is finite for a.e.
Integrating the above identity over [
Recall that
Because
when 0 <
uniformly for
Also, from (37) and (38),
because
Recalling that γ =
Hence, (24) is proved.
To verify that
whose parabolic non-tangential maximal function is in
For fixed
with
for all functions
with Ω
when
By the maximum principle and the above estimate
which shows that