Observability Inequalities for Parabolic Equations over Measurable Sets and Some Applications Related to the Bang-Bang Property for Control Problems

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 ℝⁿ and T be a fixed positive time. Consider the heat equation:

${\begin{matrix} \partial_{t} u - Δ u = 0, in Ω \times (0, 1), \\ u = 0, on \partial Ω \times (0, T), \\ u (0) = u_{0}, in Ω, \end{matrix}$ $$ \begin{equation}\label{1.1} \left\{\begin{array}{l} \partial_tu-\Delta u=0,\quad\mbox{in $\Omega\times(0,1)$,} \\[.8em]\quad u=0, \quad\mbox{on $\partial\Omega\times(0,T)$,} \\[1em] \displaystyle u(0)=u_0, \quad\mbox{in $\Omega$,} \end{array}\right. \end{equation} $$(1)

with u₀ in L²(Ω). The solution of (1) will be treated as either a function from [0, T] to L²(Ω) or a function of two variables x and t. Two important apriori estimates for the above equation are as follows:

$‖ u (T) ‖_{L^{2} (Ω)} \leq N (Ω, T, D) \int_{D} | u (x, t) | d x d t,$ $$ \begin{equation} \|u(T)\|_{L^2(\Omega)}\leq N(\Omega,T,\mathscr{D})\int_{\mathscr{D}}|u(x,t)|\,dxdt, \end{equation} $$(2)

for all u₀ ∈ L²(Ω), where 𝒟 is a subset of Ω × (0, T), and

$‖ u (T) ‖_{L^{2} (Ω)} \leq N (Ω, T, J) \int_{J} | \frac{\partial}{\partial ν} u (x, t) | d σ d t,$ $$ \begin{equation} \|u(T)\|_{L^2(\Omega)}\leq N(\Omega,T,\mathscr{J})\int_{\mathscr{J}}|\frac{\partial}{\partial\nu}u(x,t)|\,d\sigma dt, \end{equation} $$(3)

for all u₀ ∈ L²(Ω), where 𝒥 is a subset of ∂Ω × (0, T). Such apriori estimates are called observability inequalities.

In the case that 𝒟 = ω× (0,T) and 𝒥 = Γ × (0,T) with ω and Γ accordingly open and nonempty subsets of Ω and ∂Ω, both inequalities (2) and (3) (where ∂Ω is smooth) were essentially first established, via the Lebeau-Robbiano spectral inequalities in [8]. These two estimates were set up to the linear parabolic equations (where ∂Ω is of class C²), based on the Carleman inequality provided in [7]. In the case when 𝒟 = ω × (0,T) and 𝒥 = Γ × (0,T) with ω and Γ accordingly subsets of positive measure and positive surface measure in Ω and ∂Ω, both inequalities (2) and (3) were built up in [1] with the help of a propagation of smallness estimate from measurable sets for real-analytic functions first established in [13]. For 𝒟 = ω × E, with ω and E accordingly an open subset of Ω and a subset of positive measure in (0, T), the inequality (2) (when ∂Ω is smooth) was proved in [14] with the aid of the Lebeau-Robbiano spectral inequality, and it was then verified for heat equations (when Ω is convex) with lower terms depending on the time variable, through a frequency function method in [11]. When 𝒟 = ω × E, with ω and E accordingly subsets of positive measure in Ω and (0, T), the estimate (2) (when ∂Ω is real-analytic) was obtained in [15].

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, T) and ∂Ω × (0, T) respectively. Such inequalities not only are mathematically interesting but also have important applications in the control theory of the heat equation, such as the bang-bang control, the time optimal control, the null controllability over a measurable set and so on.

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

$0 < λ_{1} \leq λ_{2} \leq \dots \leq λ_{j} \leq \dots$ $$ 0\lt \lambda_1\le\lambda_2\le\dots\leq\lambda_j\le\cdots $$

for the eigenvalues of −Δ with the zero Dirichlet boundary condition over ∂Ω, and {e_j: j≥1 } for the set of L²(Ω)-normalized eigenfunctions, i.e.,

${\begin{matrix} Δ e_{j} + λ_{j} e_{j} = 0, in Ω, \\ e_{j} = 0, on \partial Ω . \end{matrix}$ $$ \begin{equation} \left\{\begin{array}{l} \Delta e_j+\lambda_je_j=0,\quad\mbox{in $\Omega$,} \\[1em] \displaystyle e_j=0, \quad\mbox{on $\partial\Omega$.} \end{array}\right. \end{equation} $$(4)

For λ > 0 we define

$ℰ_{λ} f = \sum_{λ_{j} \leq λ} (f, e_{j}) e_{j} and ℰ_{λ}^{⊥} f = \sum_{λ_{j} > λ} (f, e_{j}) e_{j},$ $$ \mathscr E_\lambda f=\sum_{\lambda_j\le\lambda}( f,e_j)\,e_j\quad \mbox{and}\quad \mathscr E_\lambda^\perp f=\sum_{\lambda_j\gt\lambda}( f,e_j)\,e_j, $$

where

$(f, e_{j}) = \int_{Ω} f e_{j} d x, when f \in L^{2} (Ω), j \geq 1.$ $$ ( f,e_j)=\int_{\Omega} f\, e_j\,dx,\ \mbox{when}\ f\in L^2(\Omega),\ j\ge 1. $$

Throughout this paper the following notations are used:

$(f, g) = \int_{Ω} f g d x and ‖ f ‖_{L^{2} (Ω)} = {(f, f)}^{\frac{1}{2}} .$ $$ (f,g)=\int_{\Omega}fg\,dx\quad \mbox{and}\ \|f\|_{L^2(\Omega)}=\left(f,f\right)^{\frac 12}. $$

ν is the unit exterior normal vector to Ω. dσ is surface measure on ∂Ω. B_R(x₀) stands for the ball centered at x₀ in \ℝⁿ of radius R, Δ_R(x₀) denotes B_R(x₀)∩∂Ω, B_R=B_R(0) and Δ_R=Δ_R(0). For measurable sets ω⊂ℝⁿ and 𝒟 ⸦ ℝⁿ × (0,T), |ω| and |𝒟| stand for the Lebesgue measures of the sets. For each measurable set 𝒥 in ∂Ω × (0, T), 𝒥 in ∂ω × (0,T), |𝒥| denotes its surface measure on the lateral boundary of Ω × ℝ. {e^tΔ : t≥0} is the semigroup generated by Δ with zero Dirichlet boundary condition over ∂Ω. Consequently, e^tΔf is the solution of equation (1) with the initial state f in L²(Ω). The Lebeau-Robbiano spectral inequality is as follows:

For each 0 < R ≤ 1, there isN=N (Ω, R), such that the inequality

$‖ ℰ_{λ} f ‖_{L^{2} (Ω)} \leq N e^{N \sqrt{λ}} ‖ ℰ_{λ} f ‖_{L^{2} (B_{R} (x_{0}))}$ $$ \begin{equation} \|\mathscr E_\lambda f\|_{L^2(\Omega)}\le Ne^{N\sqrt\lambda}\|\mathscr E_\lambda f\|_{L^2(B_R(x_0))} \end{equation} $$(5)

holds, whenB_4R(x₀)⊂Ω, f ∈ L²(Ω) and λ > 0.

Observability inequalities

Our main results related to the observability inequalities are stated as follows, but, first, we will define the real-analyticity of the set Δ_4R(q₀).

Definition 1

Letq₀ ∈ ∂Ω and 0 < R ≤ 1. We say that Δ_4R(q₀) is real-analytic with constants ρ and δ if for each q ∈ Δ_4R(q₀), there are a new rectangular coordinate system where q=0, and a real-analytic function $ϕ : {B^{'}}_{ϱ} \subset ℝ^{n - 1} \to ℝ$ $\phi: B'_{\varrho}\subset\mathbb{R}^{n-1} \rightarrow \mathbb{R}$ verifying

${\begin{matrix} ϕ (0^{'}) = 0, | \partial^{α} ϕ (x^{'}) | \leq | α |! δ^{- | α | - 1}, \\ when x^{'} \in {B^{'}}_{ϱ}, α \in ℕ^{n - 1}, \\ B_{ϱ} \cap Ω = B_{ϱ} \cap {(x^{'}, x_{n}) : x^{'} \in {B^{'}}_{ϱ}, x_{n} > ϕ (x^{'})}, \\ B_{ϱ} \cap \partial Ω = B_{ϱ} \cap {(x^{'}, x_{n}) : x^{'} \in {B^{'}}_{ϱ}, x_{n} = ϕ (x^{'})} . \end{matrix}$ $$ \begin{equation} \left\{\begin{array}{l} \phi(0')=0,\;\;|\partial^{\alpha}\phi(x')|\leq |\alpha|!\delta^{-|\alpha|-1}, \\[1em] \mbox{when}\;\;x'\in B'_{\varrho},\;\alpha\in\mathbb{N}^{n-1}, \\[1em] B_\varrho\cap\Omega=B_\varrho\cap\{(x',x_n):x'\in B'_{\varrho},\;\; x_n\gt\phi(x')\}, \\[1em] \displaystyle B_\varrho\cap\partial\Omega=B_\varrho\cap\{(x',x_n):x'\in B'_{\varrho},\;\; x_n=\phi(x')\}. \end{array}\right. \end{equation} $$(6)

Here, B′_ρ denotes the open ball of radius ρ and with center at 0′ in ℝⁿ⁻¹.

In the next two theorems, we establish two observability inequalities for the heat equation over Ω × (0, T). In Theorem 1, the observation is from a subset of positive measure in Ω × (0, T), while in Theorem 2, the observation is from a subset of positive surface measure on ∂Ω × (0, T).

Theorem 1

Suppose that a bounded domain Ω verifies the condition (5) andT > 0. Letx₀ ∈ Ω andR ∈ (0, 1] be such thatB_4R(x₀)⊂Ω. Then, for each measurable set 𝒟 ⊂ B_R(x₀) × (0,T) with |𝒟| > 0, there is a positive constantB = B(Ω, T, R, 𝒟), such that

$‖ e^{T Δ} f ‖_{L^{2} (Ω)} \leq e^{B} \int_{D} | e^{t Δ} f (x) | d x d t,$ $$ \begin{equation} \|e^{T\Delta}f\|_{L^2(\Omega)}\le e^{B}\int_{\mathscr D}|e^{t\Delta}f(x)|\,dxdt, \end{equation} $$(7)

whenf ∈ L²(Ω).

Theorem 2

Suppose that a bounded Lipschitz domain Ω verifies the condition (5) andT > 0. Letq₀ ∈ ∂Ω andR ∈ (0, 1] be such that Δ_4R(q₀) is real-analytic. Then, for each measurable set 𝒥 ⊂ Δ_R(q₀)× (0,T) with |𝒥| > 0, there is a positive constantB = B(Ω, T, R, 𝒥, such that

$‖ e^{T Δ} f ‖_{L^{2} (Ω)} \leq e^{B} \int_{J} | \frac{\partial}{\partial ν} e^{t Δ} f (x) | d σ d t,$ $$ \begin{equation} \|e^{T\Delta}f\|_{L^2(\Omega)}\le e^{B}\int_{\mathscr J}|\frac{\partial}{\partial\nu}\, e^{t\Delta}f(x)|\,d\sigma dt, \end{equation} $$(8)

whenf ∈ L²(Ω).

Next, we will see some results that will be necessary in the proof of the previous Theorem 1.

Lemma 3

LetB_R(x₀)⊂Ω and 𝒟 ⊂ B_R(x₀) × (0,T) be a subset of positive measure. Set

$D_{t} = {x \in Ω : (x, t) \in D}, E = {t \in (0, T) : | D_{t} | \geq | D | / (2 T)}, t \in (0, T) .$ $$ \begin{equation} \mathscr D_t=\{x\in \Omega : (x,t)\in\mathscr D\},\ E=\{t\in (0,T): |\mathscr D_t |\ge |\mathscr D|/(2T)\},\ t\in (0,T). \end{equation} $$(9)

Then, 𝒟_t ⊂ Ω is measurable for a.e. t ∈ (0, T), Eis measurable in (0, T), |E| ≥ |𝒟|/2|B_R| and

$χ_{E} (t) χ_{D_{t}} (x) \leq χ_{D} (x, t), in Ω \times (0, T) .$ $$ \begin{equation} \chi_E(t)\chi_{\mathscr D_t}(x)\le \chi_{\mathscr D}(x,t),\ \text{in}\ \Omega\times (0,T). \end{equation} $$(10)

Proof

From Fubini’s theorem,

$| D | = \int_{0}^{T} | D_{t} | d t = \int_{E} | D_{t} | d t + \int_{[0, T] ∖ E} | D_{t} | d t \leq | B_{R} | | E | + | D | / 2.$ $$ \begin{equation*} |\mathscr D|=\int_0^T|\mathscr D_t|\,dt=\int_E|\mathscr D_t|\,dt+\int_{[0,T]\setminus E}|\mathscr D_t|\,dt\le|B_R||E|+|\mathscr D|/2. \end{equation*} $$

□

Theorem 4

Letx₀ ∈ Ω andR ∈ (0, 1] be such thatB_4R(x₀)⊂Ω. Let 𝒟⊂ B_R(x₀)× (0,T) be a measurable set with |𝒟| > 0. WriteE and 𝒟_tfor the sets associated to 𝒟 in Lemma 3. Then, for eachη ∈ (0, 1), there areN = N(Ω,R, |𝒟|/(T|B_R|),η) andθ = θ(Ω,R, |𝒟|/(T|B_R|), η) withθ ∈ (0, 1), such that

$‖ e^{t_{2} Δ} f ‖_{L^{2} (Ω)} \leq {(N e^{N / (t_{2} - t_{1})} \int_{t_{1}}^{t_{2}} χ_{E} (s) ‖ e^{s Δ} f ‖_{L^{1} (D_{s})} d s)}^{θ} ‖ e^{t_{1} Δ} f ‖_{L^{2} (Ω)}^{1 - θ},$ $$ \begin{equation} \|e^{t_2\Delta}f\|_{L^2(\Omega)} \le \left( N e^{N/(t_2-t_1)} \int_{t_1}^{t_2}\chi_E(s) \|e^{s\Delta}f\|_{L^1(\mathscr D_s)}\,ds \right)^{\theta}\|e^{t_1\Delta}f\|_{L^2(\Omega)}^{1-\theta}, \end{equation} $$(11)

when 0 ≤ t₁ < t₂ ≤ T, |E ∩ (t₁, t₂)| ≥ η(t₂−t₁) andf ∈ L²(Ω). Moreover,

$\begin{matrix} e^{- \frac{N + 1 - θ}{t_{2} - t_{1}}} ‖ e^{t_{2} Δ} f ‖_{L^{2} (Ω)} - e^{- \frac{N + 1 - θ}{q (t_{2} - t_{1})}} ‖ e^{t_{1} Δ} f ‖_{L^{2} (Ω)} \\ \leq N \int_{t_{1}}^{t_{2}} χ_{E} (s) ‖ e^{s Δ} f ‖_{L^{1} (D_{s})} d s, when q \geq (N + 1 - θ) / (N + 1) . \end{matrix}$ $$ \begin{equation} e^{-\frac{N+1-\theta}{t_2-t_1}}\|e^{t_2\Delta}f\|_{L^2(\Omega)}- e^{-\frac{N+1-\theta}{q\left(t_2-t_1\right)}}\|e^{t_1\Delta}f\|_{L^2(\Omega)} \\ \le N\int_{t_1}^{t_2}\chi_E(s) \|e^{s\Delta}f\|_{L^1(\mathscr D_s)}\,ds,\;\;\mbox{when}\;\;q\ge (N+1-\theta)/(N+1). \end{equation} $$(12)

The reader can find the proof of the following Lemma 2 in either [10, pp. 256-257] or [11, Proposition 2.1].

Lemma 5

Let E be a subset of positive measure in (0, T). Letlbe a density point of E. Then, for each z > 1, there isl₁=l₁(z, E) in (l, T) such that, the sequence {l_m} defined as

$l_{m + 1} = l + z^{- m} (l_{1} - l), m = 1, 2, \dots,$ $$ \begin{equation*} l_{m+1}= l+z^{-m}\left(l_1-l\right),\ m=1, 2,\cdots, \end{equation*} $$

verifies

$| E \cap (l_{m + 1}, l_{m}) | \geq \frac{1}{3} (l_{m} - l_{m + 1}), when m \geq 1.$ $$ \begin{equation} |E\cap (l_{m+1}, l_m)|\ge \frac 13 \left(l_m-l_{m+1}\right),\ \text{when}\ m\ge 1. \end{equation} $$(13)

Proof

[Theorem 1] LetE and 𝒟_t be the sets associated to 𝒟_t in Lemma 3 and l be a density point in E. For z > 1 to be fixed later, {l_m} denotes the sequence associated to l and z in Lemma 5. Because (13) holds, we may apply Theorem 4, with η=1/3, t₁=l_m+1 and t₂=l_m, for each m ≥ 1, to get that there are N = N(Ω,R, |𝒟|/(T|B_R|)) > 0 and θ = θ(Ω,R, |𝒟|/(T|B_{_R}|)), with θ ∈ (0, 1), such that

$\begin{matrix} e^{- \frac{N + 1 - θ}{l_{m} - l_{m + 1}}} ‖ e^{l_{m} Δ} f ‖_{L^{2} (Ω)} - e^{- \frac{N + 1 - θ}{q (l_{m} - l_{m + 1})}} ‖ e^{l_{m + 1} Δ} f ‖_{L^{2} (Ω)} \\ \leq N \int_{l_{m + 1}}^{l_{m}} χ_{E} (s) ‖ e^{s Δ} f ‖_{L^{1} (D_{s})} d s, when q \geq \frac{N + 1 - θ}{N + 1} and m \geq 1. \end{matrix}$ $$ \begin{equation} e^{-\frac{N+1-\theta}{l_m-l_{m+1}}}\|e^{l_m\Delta}f\|_{L^2(\Omega)}- e^{-\frac{N+1-\theta}{q\left(l_m-l_{m+1}\right)}}\|e^{l_{m+1}\Delta}f\|_{L^2(\Omega)} \\ \le N\int_{l_{m+1}}^{l_{m}}\chi_E(s) \|e^{s\Delta}f\|_{L^1(\mathscr D_s)}\,ds,\;\;\mbox{when}\;\;q\ge \frac{N+1-\theta}{N+1}\;\;\mbox{and}\;\;m\ge 1. \end{equation} $$(14)

Setting z=1/q in (14) (which leads to $1 < z \leq \frac{N + 1}{N + 1 - θ})$ $1\lt z\le \frac{N+1}{N+1-\theta})$ and

$γ_{z} (t) = e^{- \frac{N + 1 - θ}{(z - 1) (l_{1} - l) t}}, t > 0,$ $$ \begin{equation*} \gamma_z(t)=e^{-\frac{N+1-\theta}{\left(z-1\right)\left(l_1-l\right)t}},\ t\gt0, \end{equation*} $$

recalling that

$l_{m} - l_{m + 1} = z^{- m} (z - 1) (l_{1} - l), for m \geq 1,$ $$ \begin{equation*} l_m-l_{m+1}=z^{-m}\left(z-1\right)\left(l_1-l\right),\ \text{for}\ m\ge 1, \end{equation*} $$

we have

$\begin{matrix} γ_{z} (z^{- m}) ‖ e^{l_{m} Δ} f ‖_{L^{2} (Ω)} - γ_{z} (z^{- m - 1}) ‖ e^{l_{m + 1} Δ} f ‖_{L^{2} (Ω)} \\ \leq N \int_{l_{m + 1}}^{l_{m}} χ_{E} (s) ‖ e^{s Δ} f ‖_{L^{1} (D_{s})} d s, when m \geq 1. \end{matrix}$ $$ \begin{equation} \gamma_z(z^{-m})\|e^{l_m\Delta}f\|_{L^2(\Omega)}- \gamma_z(z^{-m-1})\|e^{l_{m+1}\Delta}f\|_{L^2(\Omega)} \\ \le N\int_{l_{m+1}}^{l_{m}}\chi_E(s) \|e^{s\Delta}f\|_{L^1(\mathscr D_s)}\,ds,\;\;\mbox{when}\;\;m\ge 1. \end{equation} $$(15)

Choose now

$z = \frac{1}{2} (1 + \frac{N + 1}{N + 1 - θ}) .$ $$ \begin{equation*} z=\frac 12\left(1+\frac{N+1}{N+1-\theta}\right). \end{equation*} $$

The choice of z and Lemma 5 determines l₁ in (l, T) and from (15)

$\begin{matrix} γ (z^{- m}) ‖ e^{l_{m} Δ} f ‖_{L^{2} (Ω)} - γ (z^{- m - 1}) ‖ e^{l_{m + 1} Δ} f ‖_{L^{2} (Ω)} \\ \leq N \int_{l_{m + 1}}^{l_{m}} χ_{E} (s) ‖ e^{s Δ} f ‖_{L^{1} (D_{s})} d s, when m \geq 1. \end{matrix}$ $$ \begin{equation} \begin{split} \gamma(z^{-m})\|e^{l_m\Delta}f\|_{L^2(\Omega)}- \gamma(z^{-m-1})\|e^{l_{m+1}\Delta}f\|_{L^2(\Omega)} \\ \le N\int_{l_{m+1}}^{l_{m}}\chi_E(s) \|e^{s\Delta}f\|_{L^1(\mathscr D_s)}\,ds,\;\;\mbox{when}\;\;m\ge 1. \end{split} \end{equation} $$(16)

with

$γ (t) = e^{- A / t} and A = A (Ω, R, E, | D | / (T | B_{R} |)) = \frac{2 {(N + 1 - θ)}^{2}}{θ (l_{1} - l)} .$ $$ \begin{equation*} \gamma(t)=e^{-A/t}\ \text{and}\ A=A(\Omega,R, E, |\mathscr D|/\left(T|B_R| \right))=\frac{2\left(N+1-\theta\right)^2}{\theta\left(l_1-l\right)}\, . \end{equation*} $$

Finally, because of

$‖ e^{T Δ} f ‖_{L^{2} (Ω)} \leq ‖ e^{l_{1} Δ} f ‖_{L^{2} (Ω)}, \sup_{t \geq 0} ‖ e^{t Δ} f ‖_{L^{2} (Ω)} < + \infty, \lim_{t \to 0 +} γ (t) = 0,$ $$ \begin{equation*} \|e^{T\Delta}f\|_{L^2(\Omega)}\le \|e^{l_1\Delta}f\|_{L^2(\Omega)},\ \sup_{t\ge 0}\|e^{t\Delta}f\|_{L^2(\Omega)}\lt+\infty,\ \lim_{t\to 0+}\gamma(t)=0, \end{equation*} $$

and (10), the addition of the telescoping series in (16) gives

$‖ e^{T Δ} f ‖_{L^{2} (Ω)} \leq N e^{z A} \int_{D \cap (Ω \times [l, l_{1}])} | e^{t Δ} f (x) | d x d t, for f \in L^{2} (Ω),$ $$ \begin{equation*} \|e^{T\Delta}f\|_{L^2(\Omega)}\le Ne^{zA}\int_{\mathscr D\cap(\Omega\times [l,l_1])}|e^{t\Delta}f(x)|\,dxdt,\;\;\mbox{for}\;\; f\in L^2(\Omega), \end{equation*} $$

which proves (7) with B=zA+logN. □

Remark 1

The constant B in Theorem 1 depends on E because the choice of l₁=l₁(z, E) in Lemma 5 depends on the possible complex structure of the measurable set E (See the proof of Lemma 5 in [Proposition 2.1]). When𝒟 = ω × (0,T), one may takel=T/2, l₁=T, z=2 and then,

$B = A (Ω, R, | ω | / | B_{R} |) / T .$ $$ \begin{equation*} B=A(\Omega, R, |\omega|/|B_R|)/T. \end{equation*} $$

Remark 2

The proof of Theorem 1 also implies the following observability estimate:

$\sup_{m \geq 0} \sup_{l_{m + 1} \leq t \leq l_{m}} e^{- z^{m + 1} A} ‖ e^{t Δ} f ‖_{L^{2} (Ω)} \leq N \int_{D \cap (Ω \times [l, l_{1}])} | e^{t Δ} f (x) | d x d t,$ $$ \begin{equation*} \sup_{m\ge 0}\sup_{l_{m+1}\le t\le l_m}e^{-z^{m+1}A}\|e^{t\Delta}f\|_{L^2(\Omega)}\le N\int_{\mathscr D\cap (\Omega\times [l,l_1])}|e^{t\Delta}f(x)|\,dxdt, \end{equation*} $$

forf in L²(Ω), and with z, N and A as defined along the proof of Theorem 1. Here, l₀=T.

Next, we will see some results that will be necessary in the proof of the previous Theorem 2.

Lemma 6

Let q₀ ∈ ∂Ω and 𝒥 ⊂ Δ_R(q₀) × (0,T) be a subset with |𝒥| > 0. Set

$J_{t} = {x \in \partial Ω : (x, t) \in J}, E = {t \in (0, T) : | J_{t} | \geq | J | / (2 T)}, t \in (0, T) .$ $$ \begin{equation*} \mathscr J_t=\{x\in \partial\Omega : (x,t)\in\mathscr J\},\ E=\{t\in (0,T): |\mathscr J_t|\ge |\mathscr J|/(2T)\},\ t\in (0,T). \end{equation*} $$

Then, 𝒥_t ⊂ Δ_R(q0) is measurable for a.e. t ∈ (0, T), Eis measurable in (0, T), |E| ≥ |𝒥|/(2|Δ_R(q₀)|) andχ_E(t)χ𝒥_t(x) ≤ χ𝒥(x,t) over ∂Ω × (0, T).

Proof

From Fubini’s theorem,

$| J | = \int_{0}^{T} | J_{t} | d t = \int_{E} | J_{t} | d t + \int_{[0, T] ∖ E} | J_{t} | d t \leq | △_{R} (x_{0}) | | E | + | J | / 2.$ $$ \begin{equation*} |\mathscr J|=\int_0^T|\mathscr J_t|\,dt=\int_E|\mathscr J_t|\,dt+\int_{[0,T]\setminus E}|\mathscr J_t|\,dt\le |\triangle_R(x_0)||E|+|\mathscr J|/2. \end{equation*} $$

□

Theorem 7

Suppose that Ω verifies the condition (5). Assume thatq₀ ∈ ∂Ω andR ∈ (0, 1] such that Δ_4R(q₀) is real-analytic. Let 𝒥 be a subset in Δ_R(q₀) × (0, T) of positive surface measure on ∂Ω × (0, T), E and 𝒥_tbe the measurable sets associated to 𝒥 in Lemma 6. Then, for eachη ∈ (0, 1), there areN = N(Ω,R, |𝒥|/(T|Δ_R(q₀)|),η) andθ = θ(Ω,R, |𝒥|/(T|Δ_R(q₀)|),η) withθ ∈ (0, 1), such that the inequality

$‖ e^{t_{2} Δ} f ‖_{L^{2} (Ω)} \leq {(N e^{N / (t_{2} - t_{1})} \int_{t_{1}}^{t_{2}} χ_{E} (t) ‖ \frac{\partial}{\partial ν} e^{t Δ} f ‖_{L^{1} (J_{t})} d t)}^{θ} ‖ e^{t_{1} Δ} f ‖_{L^{2} (Ω)}^{1 - θ},$ $$ \begin{equation} \|e^{t_2\Delta}f\|_{L^2(\Omega)} \le \left( N e^{N/(t_2-t_1)} \int_{t_1}^{t_2}\chi_E(t) \|\tfrac{\partial}{\partial\nu}e^{t\Delta}f\|_{L^1(\mathscr J_t)}\,dt \right)^{\theta}\|e^{t_1\Delta}f\|_{L^2(\Omega)}^{1-\theta}, \end{equation} $$(17)

holds, when 0 ≤ t₁ < t₂ ≤ Twitht₂−t₁ < 1, |E ∩ (t₁, t₂)| ≥ η(t₂−t₁) andf ∈ L²(Ω). Moreover,

$\begin{matrix} e^{- \frac{N + 1 - θ}{t_{2} - t_{1}}} ‖ e^{t_{2} Δ} f ‖_{L^{2} (Ω)} - e^{- \frac{N + 1 - θ}{q (t_{2} - t_{1})}} ‖ e^{t_{1} Δ} f ‖_{L^{2} (Ω)} \\ \leq N \int_{t_{1}}^{t_{2}} χ_{E} (t) ‖ \frac{\partial}{\partial ν} e^{t Δ} f ‖_{L^{1} (J_{t})} d t, when q \geq \frac{N + 1 - θ}{N + 1} . \end{matrix}$ $$ \begin{equation} e^{-\frac{N+1-\theta}{t_2-t_1}}\|e^{t_2\Delta}f\|_{L^2(\Omega)}- e^{-\frac{N+1-\theta}{q\left(t_2-t_1\right)}}\|e^{t_1\Delta}f\|_{L^2(\Omega)}\\ \le N\int_{t_1}^{t_2}\chi_E(t) \|\tfrac{\partial}{\partial\nu}\,e^{t\Delta}f\|_{L^1(\mathscr J_t)}\,dt,\;\;\mbox{when}\;\;q\ge \tfrac{N+1-\theta}{N+1}. \end{equation} $$(18)

Proof

[Theorem 2] Let E and 𝒥_t be the sets associated to 𝒥 in Lemma 6 and l be a density point in E. For z > 1 to be fixed later, {l_m} denotes the sequence associated to l and z in Lemma 5. Because of (13) and from Theorem 7 with η=1/3, t₁=l_m+1 and t₂=l_m, with m≥1, there are N = N(Ω,R, |𝒥|/(T|Δ_R(q₀)|)) > 0 and θ = θ(Ω,R, |𝒥|/(T|Δ_R(q₀)|)), with θ ∈ (0, 1), such that

$\begin{matrix} e^{- \frac{N + 1 - θ}{l_{m} - l_{m + 1}}} ‖ e^{l_{m} Δ} f ‖_{L^{2} (Ω)} - e^{- \frac{N + 1 - θ}{q (l_{m} - l_{m + 1})}} ‖ e^{l_{m + 1} Δ} f ‖_{L^{2} (Ω)} \\ \leq N \int_{l_{m + 1}}^{l_{m}} χ_{E} (s) ‖ \frac{\partial}{\partial ν} e^{s Δ} f ‖_{L^{1} (J_{s})} d s, when q \geq \frac{N + 1 - θ}{N + 1} and m \geq 1. \end{matrix}$ $$ \begin{equation*} \begin{split} e^{-\frac{N+1-\theta}{l_m-l_{m+1}}}\|e^{l_m\Delta}f\|_{L^2(\Omega)}- e^{-\frac{N+1-\theta}{q\left(l_m-l_{m+1}\right)}}\|e^{l_{m+1}\Delta}f\|_{L^2(\Omega)}\\ \le N\int_{l_{m+1}}^{l_{m}}\chi_E(s) \|\tfrac{\partial}{\partial\nu}\,e^{s\Delta}f\|_{L^1(\mathscr J_s)}\,ds,\;\;\mbox{when}\;\;q\ge \frac{N+1-\theta}{N+1}\;\;\mbox{and}\;\;m\ge 1. \end{split} \end{equation*} $$

Let

$z = \frac{1}{2} (1 + \frac{N + 1}{N + 1 - θ}) .$ $$ \begin{equation*} z=\frac{1}{2}\left(1+\frac{N+1}{N+1-\theta}\right). \end{equation*} $$

Then, we can use the same arguments as those in the proof of Theorem 1 to verify Theorem 2. □

Remark 3

The proof of Theorem 2 also implies the following observability estimate:

$\sup_{m \geq 0} \sup_{l_{m + 1} \leq t \leq l_{m}} e^{- z^{m + 1} A} ‖ e^{t Δ} f ‖_{L^{2} (Ω)} \leq N \int_{J \cap (\partial Ω \times [l, l_{1}])} | \frac{\partial}{\partial ν} e^{t Δ} f (x) | d σ d t,$ $$ \begin{equation*}%{yuanyuan6} \sup_{m\ge 0}\sup_{l_{m+1}\le t\le l_m}e^{-z^{m+1}A}\|e^{t\Delta}f\|_{L^2(\Omega)}\le N\int_{\mathscr J\cap (\partial\Omega\times [l,l_1])}\left|\tfrac{\partial}{\partial\nu}\,e^{t\Delta}f(x)\right|\,d\sigma dt, \end{equation*} $$

forfinL²(Ω), withA=2(N + 1−θ)²/[θ(l₁−l)] and withz, Nandθas given along the proof of Theorem 2. Here, l₀=T.

Remark 4

When 𝒥 = Γ × (0,T), Γ⊂Δ_R(q₀) is a measurable set, one may takel=T/2, l₁=T, z=2 and the constant Bin Theorem 2 becomes

$B = A (Ω, R, | Γ | / | △_{R} (q_{0}) |) / T .$ $$ \begin{equation*} B=A(\Omega, R, |\Gamma |/|\triangle_R(q_0)|)/T. \end{equation*} $$

Applications of observability inequalities

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 L^∞-norm control problems.

In this section we assume that T > 0 and that Ω is a bounded Lipschitz domain verifying the condition (5).

First of all, we will show that Theorems 1 and 2 imply the null controllability with controls restricted over measurable subsets in Ω × (0, T) and ∂Ω × (0, T) respectively. Let 𝒟 be a measurable subset with positive measure in B_R(x₀) × (0, T) with B_4R(x₀)⊂Ω. Let 𝒥 be a measurable subset with positive surface measure in Δ_R(q₀) × (0, T), where q₀ ∈ ∂Ω, R ∈ (0, 1] and Δ_4R(q₀) is real-analytic. Consider the following controlled heat equations:

${\begin{matrix} \partial_{t} u - Δ u = χ_{D} v, in Ω \times (0, T], \\ u = 0, on \partial Ω \times [0, T], \\ u (0) = u_{0}, in Ω, \end{matrix}$ $$ \begin{equation} \left\{\begin{array}{l} \partial_tu-\Delta u=\chi_{\mathscr{D}}v, \mbox{in}\ \Omega\times (0,T], \\[1em] u=0, \mbox{on}\ \partial\Omega\times [0,T], \\[1em] \displaystyle u(0)=u_0,\ \mbox{in}\ \Omega, \end{array}\right. \end{equation} $$(19)

and

${\begin{matrix} \partial_{t} u - Δ u = 0, in Ω \times (0, T], \\ 1 u = g χ_{J}, on \partial Ω \times [0, T], \\ u (0) = u_{0}, in Ω, \end{matrix}$ $$ \begin{equation} \left\{\begin{array}{l} \partial_tu-\Delta u=0,\ \mbox{in}\ \Omega\times (0,T], \\[1em] u=g\,\chi_{\mathscr J},\ \mbox{on}\ \partial\Omega\times [0,T], \\[1em] \displaystyle u(0)=u_0,\ \mbox{in}\ \Omega, \end{array}\right. \end{equation} $$(20)

where u₀ ∈ L²(Ω), v ∈ L^∞(Ω × (0, T)) and g ∈ L^∞(∂Ω × (0, T)) are controls. We say thatuis the solution to 20ifv ≡ u − e^tΔu₀is the unique solution defined in [6, Theorem 3.2] to

${\begin{matrix} \partial_{t} v - Δ v = 0, in Ω \times (0, T), \\ v = g χ_{J}, on \partial Ω \times (0, T), \\ v (0) = 0, in Ω, \end{matrix}$ $$ \begin{equation} \left\{\begin{array}{l} \partial_tv-\Delta v=0,\ \mbox{in}\ \Omega\times(0,T), \\[1em]v=g\chi_{\mathscr{J}},\ \mbox{on}\ \partial\Omega\times(0,T), \\[1em] \displaystyle v(0)=0,\ \mbox{in}\ \Omega, \end{array}\right. \end{equation} $$(21)

withg in L^p(∂Ω × (0, T)) for some 2 ≤ p ≤ ∞.

From now on, we always denote by u (· ; u₀, v) and u (· ; u₀, g) the solutions of equations (19) and (20) corresponding to v and g respectively.

Corollary 8

For eachu₀ ∈ L²(Ω), there are bounded control functionsvandgwith

$‖ v ‖_{L^{\infty} (Ω \times (0, T))} \leq C_{1} ‖ u_{0} ‖_{L^{2} (Ω)},$ $$ \|v\|_{L^\infty(\Omega\times (0,T))} \leq C_1\|u_0\|_{L^2(\Omega)}, $$

$‖ g ‖_{L^{\infty} (\partial Ω \times (0, T))} \leq C_{2} ‖ u_{0} ‖_{L^{2} (Ω)},$ $$ \begin{equation*} \displaystyle \|g\|_{L^\infty(\partial\Omega\times (0,T))}\leq C_2\|u_0\|_{L^2(\Omega)}, \end{equation*} $$

such thatu(T; u₀, v)=0 andu(T; u₀, g)=0. HereC₁ = C(Ω, T, R, 𝒟) andC₂ = C(Ω, T, R, 𝒥).

Proof

We only prove the boundary controllability. Let E be the measurable set associated to 𝒥 in Lemma 6. Write

$\tilde{J} = {(x, t) : (x, T - t) \in J} and \tilde{E} = {t : T - t \in E} .$ $$ \begin{equation*} \widetilde{\mathscr{J}}=\left\{(x,t) : (x, T-t)\in {\mathscr{J}}\right\}\;\;\mbox{and}\;\;\widetilde{E}=\left\{t : T-t\in {E}\right\}. \end{equation*} $$

Let l > 0 be a density point of $\tilde{E}$ $\widetilde{E}$ (Hence, T − l is a density point of E). We choose z, l₁ and the sequence {l_m} as in the proof of Theorem 2 but with 𝒥 and E accordingly replaced by $\tilde{J}$ $\widetilde {\mathscr{J}}$ and $\tilde{E}$ $\widetilde{E}$ . It is clear that

$0 < l < \dots < l_{m + 1} < l_{m} \dots < l_{1} < l_{0} = T, \lim_{m \to + \infty} l_{m} = l .$ $$ \begin{equation*} 0\lt l\lt\dots\lt l_{m+1}\lt l_m\dots\lt l_1\lt l_0=T,\ \lim_{m\to +\infty}l_m=l. \end{equation*} $$

We set

$ℳ = J \cap (\partial Ω \times [T - l_{1}, T - l]) \subset J .$ $$ \begin{equation*} \mathscr M=\mathscr J\cap\left(\partial\Omega\times [T-l_1,T-l]\right)\subset\mathscr J. \end{equation*} $$

It is clear that |𝓜| > 0. The proof of Theorem 2, the change of variables t=T−τ and Remark 3 show that the observability inequality

$‖ φ (0) ‖_{L^{2} (Ω)} \leq e^{B} \int_{ℳ} | \frac{\partial φ}{\partial ν} (p, t) | d σ d t,$ $$ \begin{equation} \|\varphi(0)\|_{L^2(\Omega)}\le e^B\int_{\mathscr M}|\tfrac{\partial\varphi}{\partial\nu}(p,t)|\,d\sigma dt, \end{equation} $$(22)

holds, when φ is the unique solution in $L^{\infty} ([0, T], L^{2} (Ω)) \cap L^{2} ([0, T], H_{0}^{1} (Ω))$ $L^\infty([0,T],L^2(\Omega))\cap L^2([0,T], H^1_0(\Omega))$ to

${\begin{matrix} \partial_{t} φ + Δ φ = 0, & in Ω \times [0, T), \\ φ = 0, & on \partial Ω \times [0, T), \\ φ (T) = φ_{T}, & in \partial Ω, \end{matrix}$ $$ \begin{equation} \begin{cases} \partial_t\varphi+\Delta\varphi=0,\ &\text{in}\ \Omega\times [0,T),\\ \varphi=0,\ &\text{on}\ \partial\Omega\times [0,T),\\ \varphi(T)=\varphi_T,\ &\text{in}\ \partial\Omega, \end{cases} \end{equation} $$(23)

for some φ_T in L²(Ω). Set

$X = {\frac{\partial φ}{\partial ν} |_{ℳ} : φ (t) = e^{(T - t) Δ} φ_{T}, for 0 \leq t \leq T, for some φ_{T} \in L^{2} (Ω)} .$ $$ \begin{equation*} X=\{\tfrac{\partial\varphi}{\partial\nu}|_{\mathscr M}: \varphi(t)=e^{(T-t)\Delta}\varphi_T,\ \text{for}\ 0\le t\le T,\ \text{for some}\ \varphi_T\in L^2(\Omega)\}. \end{equation*} $$

Since 𝓜 ⊂ ∂Ω ×[T − l₁,T − l], X is a subspace of L¹(𝓜) and from (22), the linear mapping Λ: X → ℝ, defined by

$Λ (\frac{\partial φ}{\partial ν} |_{ℳ}) = (u_{0}, φ (0)),$ $$ \begin{equation*} \Lambda(\tfrac{\partial\varphi}{\partial\nu}|_{\mathscr M})=(u_0,\varphi(0)), \end{equation*} $$

verifies

$| Λ (\frac{\partial φ}{\partial ν} |_{ℳ}) | \leq e^{B} ‖ u_{0} ‖_{L^{2} (Ω)} \int_{ℳ} | \frac{\partial φ}{\partial ν} (p, t) | d σ d t, when \frac{\partial φ}{\partial ν} |_{ℳ} \in X .$ $$ \begin{equation*} \left|\Lambda(\tfrac{\partial\varphi}{\partial\nu}|_{\mathscr M})\right|\le e^B\|u_0\|_{L^2(\Omega)}\int_{\mathscr M}|\tfrac{\partial\varphi}{\partial\nu}(p,t)|\,d\sigma dt,\ \text{when}\ \tfrac{\partial\varphi}{\partial\nu}|_{\mathscr M}\in X. \end{equation*} $$

From the Hahn-Banach theorem, there is a linear extension T : L¹(𝓜)→ ℝ of Λ, with

$\begin{matrix} T (\frac{\partial φ}{\partial ν} |_{ℳ}) = (u_{0}, φ (0)), when \frac{\partial φ}{\partial ν} |_{ℳ} \in X, \\ | T (f) | \leq e^{B} ‖ u_{0} ‖ ‖ f ‖_{L^{1} (ℳ)}, for all f \in L^{1} (ℳ) . \end{matrix}$ $$ \begin{equation*} \begin{split}{left} T(\tfrac{\partial\varphi}{\partial\nu}|_{\mathscr M})=(u_0,\varphi(0)),\ \text{when}\ \tfrac{\partial\varphi}{\partial\nu}|_{\mathscr M}\in X,\\ |T(f)|\le e^B\|u_0\|\|f\|_{L^1(\mathscr M)},\ \text{for all}\ f\in L^1(\mathscr M). \end{split} \end{equation*} $$

Thus, T is in L¹(𝓜)^* = L^∞(𝓜) and there is g in L^∞(𝓜) verifying

$T (f) = \int_{ℳ} f g d σ d t, for all f \in L^{1} (ℳ) and ‖ g ‖_{L^{\infty} (ℳ)} \leq e^{B} ‖ u_{0} ‖ .$ $$ \begin{equation*} T(f)=\int_{\mathscr M}fg\,d\sigma dt,\ \text{for all}\ f\in L^1(\mathscr M)\ \text{and}\ \|g\|_{L^\infty(\mathscr M)}\le e^B\|u_0\|. \end{equation*} $$

We extend g over ∂Ω × (0, T) by setting it to be zero outside 𝓜 and denote the extended function by g again. Then it holds that u(T;u₀, g)=0 provided that we know that

$\int_{Ω} u (T; u_{0}, g) φ_{T} d x = \int_{Ω} u_{0} φ (0) d x - \int_{ℳ} g \frac{\partial φ}{\partial ν} d σ d t, for all φ_{T} \in L^{2} (Ω) .$ $$ \begin{equation} \int_{\Omega}u(T;u_0, g)\varphi_T\,dx=\int_{\Omega}u_0\varphi(0)\,dx-\int_{\mathscr M}g\,\tfrac{\partial\varphi}{\partial\nu}\,d\sigma dt,\ \text{for all}\ \varphi_T\in L^2(\Omega). \end{equation} $$(24)

To prove (24), we first use the unique solvability for the problem

${\begin{matrix} \partial_{t} u - Δ u = 0, & in Ω \times (0, T], \\ u = γ, & on \partial Ω \times [0, T], \\ u (0) = 0 & in Ω, \end{matrix}$ $$ \begin{equation*} \begin{cases} \partial_tu-\Delta u=0,\ &\text{in}\ \Omega\times (0,T],\\ u=\gamma,\ &\text{on}\ \partial\Omega\times [0,T],\\ u(0)=0\ &\text{in}\ \Omega, \end{cases} \end{equation*} $$

with lateral Dirichlet data γ in L^p(∂Ω × (0, T)), 2 ≤ p ≤ ∞, stablished in [6, Theorem 3.2] (See also [3, Theorems 8.1 and 8.3]). Then, because gχ𝓜 is bounded and supported in ∂Ω × [T − l₁, T − l]⊂∂Ω × (2η, T − 2η) for some η > 0, the calculations leading to (24) can be justified via the regularization of gχ𝓜 and the approximation of Ω by smooth domains {Ω_j;j≥1} as in [3, Lemma 2.2]. For the sake of completeness we provide the detailed proof of this identity in the Appendix in Section 5. □

3.1

Definition of the Minimal Time Control Problems and Main Results

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 ω be a measurable subset with positive measure in B_R(x₀) and B_4R(x₀)⊂Ωm. Suppose that Δ_4R(q₀) is real-analytic for some q₀ ∈ ∂Ω and R ∈ (0, 1] and let Γ be a measurable subset with positive surface measure of Δ_R(x₀). For each M > 0, we define the following control constraint set:

$\begin{matrix} U_{M}^{1} = {v measurable on Ω \times R^{+} : | v (x, t) | \leq M for a.e. (x, t) \in Ω \times R^{+}} . \\ U_{M}^{1} = {v measurable on \partial Ω \times R^{+} : | v (x, t) | \leq M for a.e. (x, t) \in \partial Ω \times R^{+}} . \end{matrix}$ $$\eqalign{\mathscr{U}^1_M=\{v\;\;\text{measurable on}\;\; \Omega\times\mathbb{R}^+: \;\;|v(x,t)|\leq M\;\;\text{for a.e.}\;\;(x,t)\in\Omega\times\mathbb{R}^+\}.\\ \mathscr{U}^1_M=\{v\;\;\text{measurable on}\;\; \partial\Omega\times\mathbb{R}^+: \;\;|v(x,t)|\leq M\;\;\text{for a.e.}\;\;(x,t)\in \partial\Omega\times\mathbb{R}^+\}.}$$

Let u₀ ∈ L²(Ω)∖{0}. Consider the minimal time control problems:

$(T P)_{M}^{1} : T_{M}^{1} \equiv min_{v \in U_{M}^{1}} (t > 0 : e^{t Δ} u_{0} + \int_{0}^{t} e^{(t - s) Δ} (χ_{ω} v) d s = 0)$ $$ \begin{equation} (TP)^1_M:\;\; T^1_M\equiv \min\limits_{v\in\mathscr{U}^1_M} \left\{t\gt0:\;\;e^{t\Delta}u_0+\int_0^te^{(t-s)\Delta} ({\chi_{\omega}}v)\,ds=0\right\} \end{equation} $$

and

$(T P)_{M}^{2} : T_{M}^{2} \equiv min_{g \in U_{M}^{2}} (t > 0 : u (x, t; g) = 0 for a.e. x \in Ω),$ $$ \begin{equation} (TP)^2_M:\;\; T^2_M\equiv \min\limits_{g\in\mathscr{U}^2_M} \left\{t\gt0:\;\;u(x,t;g)=0\;\;\mbox{for a.e.}\;\; x\in \Omega\right\}, \end{equation} $$

where u (·, · ;g) is the solution to

${\begin{matrix} \partial_{t} u - Δ u = 0, & in Ω \times ℝ^{+}, \\ u = g χ_{Γ}, & on \partial Ω \times ℝ^{+}, \\ u (0) = u_{0}, & in Ω . \end{matrix}$ $$ \begin{equation} \begin{cases} \partial_t u- \Delta u=0,\ &\text{in}\ \Omega\times \mathscr {R}^+,\\ u=g\chi_{\Gamma},\ &\text{on}\ \partial\Omega\times\mathscr {R}^+,\\ u(0)=u_0,\ &\text{in}\ \Omega. \end{cases} \end{equation} $$(25)

Any solution of ${(T P)}_{M}^{i}$ $(TP)_{M}^i$ , i=1, 2, is called a minimal time control to this problem. According to Theorem 1 and Theorem 3.3 in [12], problem ${(T P)}_{M}^{1}$ $(TP)^1_M$ has solutions. By Theorem 2, using the same arguments as those in the proof of Theorem 3.3 in [12], we can verify that there is $g \in U_{M}^{2}$ $g\in\mathscr{U}^2_M$ such that for some t > 0, u(x, t; g)=0 for a.e. x ∈ Ω.

Lemma 9

Problem ${(T P)}_{M}^{2}$ $(TP)^2_M$ has solutions.

Proof

Let {t_n}_n≥1, with $t_{n} ↘ T_{M}^{2}$ $t_n\searrow T_M^2$ , and $g_{n} \in U_{M}^{2}$ $g_n\in \mathscr{U}^2_M$ be such that u(x, t_n; g_n)=0 over Ω. Hence, on a subsequence,

$g_{n} \to g^{*} weakly star in L^{\infty} (\partial Ω \times (0, t_{1})) .$ $$ \begin{equation} g_n\longrightarrow g^*\;\;\mbox{weakly star in}\;\; L^\infty(\partial\Omega\times(0,t_1)). \end{equation} $$(26)

It suffices to show that

$u_{n} (x, t_{n}) \equiv u (x, t_{n}; g_{n}) \to u^{*} (x, T_{M}^{2}) \equiv u (x, T_{M}^{2}; g^{*}), for all x \in Ω .$ $$ \begin{equation} u_n(x,t_n)\equiv u(x,t_n; g_n)\longrightarrow u^*(x,T_M^2)\equiv u(x, T_M^2; g^*),\ \mbox{for all}\ x\in \Omega. \end{equation} $$(27)

For this purpose, let G(x, y, t) be the Green’s function for Δ−∂_t in Ω × ℝ with zero lateral Dirichlet boundary condition. [6, Theorems 1.3 and 1.4] and [6, p. 643] show that for $g \in U_{M}^{2}$ $g\in \mathscr{U}^2_M$ and (x, t) ∈ Ω × (0, T),

$u (x, t; g) = e^{t △} u_{0} - \int_{0}^{t} \int_{\partial Ω} \frac{\partial G}{\partial ν_{q}} (x, q, t - s) χ_{Γ} (q, s) g (q, s) d σ_{q} d s$ $$ \begin{equation} u(x,t ;g)=e^{t\bigtriangleup}u_0- \int_{0}^{t}\int_{\partial\Omega}\tfrac{\partial G}{\partial\nu_q}(x,q,t-s)\,\chi_{\Gamma}(q, s)g(q,s)\,d\sigma_{q}ds \end{equation} $$(28)

and

$\int_{0}^{T} \int_{\partial Ω} | \frac{\partial G}{\partial ν_{q}} (x, q, τ) |^{2} d σ_{q} d τ < + \infty, when x \in Ω, T > 0.$ $$ \begin{equation} \int_0^T\int_{\partial\Omega}|\tfrac{\partial G}{\partial\nu_q}(x,q,\tau)|^2\,d\sigma_q d\tau\lt +\infty,\ \text{when}\ x\in\Omega,\ T\gt0. \end{equation} $$(29)

Also, by standard interior parabolic regularity there is N=N(n,ε) with

$| u (x, t; g) - u (x, s; g) | \leq N | t - s | (‖ g ‖_{L^{\infty} (\partial Ω \times (0, T))} + ‖ u_{0} ‖_{L^{2} (Ω)})$ $$ \begin{equation} |u(x,t;g)-u(x,s;g)|\le N|t-s|\left(\|g\|_{L^\infty(\partial\Omega\times (0,T))}+\|u_0\|_{L^2(\Omega)}\right) \end{equation} $$(30)

when d(x, ∂Ω) > $\sqrt{3}$ $\sqrt{\varepsilon}$ and t > s ≥ ε. Now, when x ∈ Ω with d(x, ∂Ω) > $\sqrt{3}$ $\sqrt{\varepsilon}$ , it holds that

$| u_{n} (x, t_{n}) - u^{*} (x, T_{M}^{2}) | \leq | u_{n} (x, t_{n}) - u_{n} (x, T_{M}^{2}) | + | u_{n} (x, T_{M}^{2}) - u^{*} (x, T_{M}^{2}) | .$ $$ \begin{equation}|u_n(x,t_n)-u^*(x,T_M^2)|\leq |u_n(x,t_n)-u_n(x,T_M^2)|+ |u_n(x,T_M^2)-u^*(x,T_M^2)|. \end{equation} $$

This, along with (26), (28), (29) and (30) indicates that (27) holds for all x ∈ Ω with d(x, ∂Ω) > $\sqrt{3}$ $\sqrt{\varepsilon}$ . Since ε > 0 is arbitrary, (27) follows at once. □

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.

Corollary 10

Problem ${(T P)}_{M}^{1}$ $(TP)^1_M$ has the bang-bang property: any minimal time controlvsatisfies that |v(x, t)|=Mfor a.e. $(x, t) \in ω \times (0, T_{M}^{1})$ $(x,t)\in \omega\times (0, T^1_M)$ . Consequently, this problem has a unique minimal time control.

Corollary 11

The problem ${(T P)}_{M}^{2}$ $(TP)^2_M$ has the bang-bang property: any minimal time boundary controlgsatisfies that |g(x, t)|=M for a.e. $(x, t) \in Γ \times (0, T_{M}^{2})$ $(x,t)\in \Gamma\times (0, T^2_M)$ . Consequently, this problem has a unique minimal time control.

3.2

Definition of the Time Optimal Control Problems and Main Results

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 L^∞-norm. These problems are usually called the second type of time optimal control problems and are stated as follows: Let T > 0 and M > 0. Write ω and Γ for the sets given in Problems ${(T P)}_{M}^{1}$ $(TP)_M^1$ and ${(T P)}_{M}^{2}$ $(TP)_M^2$ respectively. Consider the controlled heat equations:

${\begin{matrix} \partial_{t} u - Δ u = χ_{ω} χ_{(τ, T)} v, & in Ω \times (0, T], \\ u = 0, & on \partial Ω \times [0, T], \\ u (0) = u_{0}, & in Ω \end{matrix}$ $$ \begin{equation} \begin{cases} \partial_tu- \Delta u=\chi_{\omega}\chi_{(\tau,T)}v, &\text{in}\ \Omega\times (0,T],\\ u=0, &\text{on}\ \partial\Omega\times [0,T],\\ u(0)=u_0,\ &\text{in}\ \Omega \end{cases} \end{equation} $$(31)

and

${\begin{matrix} \partial_{t} u - Δ u = 0, & in Ω \times (0, T], \\ u = χ_{Γ} χ_{(τ, T)} g, & on \partial Ω \times [0, T], \\ u (0) = u_{0}, & in Ω, \end{matrix}$ $$ \begin{equation} \begin{cases} \partial_tu-\Delta u=0, &\text{in}\ \Omega\times (0,T],\\ u=\chi_{\Gamma}\chi_{(\tau,T)}g, &\text{on}\ \partial\Omega\times [0,T],\\ u(0)=u_0,\ &\text{in}\ \Omega, \end{cases} \end{equation} $$(32)

where u₀ ∈ L²(Ω). Write accordingly u (· ;χ_{(τ, T)}v) and u (· ;χ_{(τ, T)}g) for the solutions to equation (31) corresponding to χ_{(τ, T)}v, and to equation (32) corresponding to χ_{(τ, T)}g. Define the following control constraint sets:

$\begin{matrix} U_{M, T}^{1} = {v measurable on Ω \times (0, T) : | v (x, t) | \leq M for a .e . (x, t) \in Ω \times (0, T)} . \\ U_{M, T}^{2} = {g measurable on \partial Ω \times (0, T) : | g (x, t) | \leq M for a .e . (x, t) \in \partial Ω \times (0, T)} . \end{matrix}$ $$ \begin{equation*}\nonumber \mathscr{U}^1_{M,T}=\{v\ \text{measurable on}\ \Omega \times(0,T): |v(x,t)|\leq M\ \text{for a.e.}\ (x,t)\in\Omega\times(0,T)\}.\\ \mathscr{U}^2_{M,T}=\{g\ \text{measurable on}\ \partial\Omega \times(0,T): |g(x,t)|\leq M\ \text{for a.e.}\ (x,t)\in\partial\Omega\times(0,T)\}. \end{equation*} $$

Consider the time optimal control problems:

${(T P)}_{M, T}^{1} : τ_{M, T}^{1} \equiv \sup_{v \in U_{M, T}^{1}} {τ \in [0, T) : u (T; χ_{(τ, T)} v) = 0}$ $$ \begin{equation}(TP)^1_{M,T}:\ \tau^1_{M,T}\equiv \sup_{v\in\mathscr{U}^1_{M,T}} \left\{\tau\in[0,T):\;\;u(T; \chi_{(\tau,T)}v)=0\right\} \end{equation} $$

and

${(T P)}_{M, T}^{2} : τ_{M, T}^{2} \equiv \sup_{g \in U_{M, T}^{2}} {τ \in [0, T) : u (T; χ_{(τ, T)} g) = 0} .$ $$ \begin{equation}(TP)^2_{M,T}:\;\; \tau^2_{M,T}\equiv \sup_{g\in\mathscr{U}^2_{M,T}} \left\{\tau\in[0,T):\;\;u(T; \chi_{(\tau,T)}g)=0\right\}. \end{equation} $$

Any solution of ${(T P)}_{T, M}^{i}$ $(TP)_{T,M}^i$ , i=1, 2, is called an optimal control to the corresponding problem.

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:

Corollary 12

Any optimal controlv* to Problem ${(T P)}_{M, T}^{1}$ $(TP)^1_{M,T}$ , if it exists, satisfies the bang-bang property: |v*^(x, t)|=Mfor a.e. $(x, t) \in ω \times (τ_{M, T}^{1}, T)$ $(x,t)\in \omega\times(\tau^1_{M,T}, T)$ .

Corollary 13

Any optimal controlg* to Problem ${(T P)}_{M, T}^{2}$ $(TP)^2_{M,T}$ , if it exists, satisfies the bang-bang property: |g*^(x, t)|=Mfor a.e. $(x, t) \in Γ \times (τ_{M, T}^{2}, T)$ $(x,t)\in \Gamma\times(\tau^2_{M,T}, T)$ .

Remark 5

By Theorem 1 (See also Remark 1) and the energy decay property for the heat equation, one can easily prove the following: for a fixed M > 0, there is $v \in U_{M, T}^{1}$ $v\in \mathscr{U}^1_{M,T}$ such thatu(T;χ_(0,T)v)=0, whenTis large enough (such a controlvis called an admissible control); while for a fixedT > 0, the same holds whenMis large enough. The same can be said about Problem ${(T P)}_{M, T}^{2}$ $(TP)^2_{M,T}$ because of Theorem 2 (See also Remark 4). In the case where Problem ${(T P)}_{M, T}^{1}$ $(TP)^1_{M,T}$ has admissible controls, one can easily prove the existence of time optimal controls to this problem. In the case when Problem ${(T P)}_{M, T}^{2}$ $(TP)^2_{M,T}$ has admissible controls, one can make use of the similar method in the proof of Lemma 9 to verify the existence of time optimal controls for this problem.

3.3

Definition of the Minimal Norm Control Problems and Main Results

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 u₀ ∈ L²(Ω), we define two control constraint sets as follows:

$V_{D} = {v \in L^{\infty} (Ω \times (0, T)) : u (T; u_{0}, v) = 0}$ $$ \begin{equation*} \mathscr{V}_{\mathscr{D}}=\left\{v\in L^\infty (\Omega\times(0,T)): u(T; u_0, v)=0 \right\} \end{equation*} $$

and

$V_{J} = {g \in L^{\infty} (\partial Ω \times (0, T)) : u (T; u_{0}, g) = 0} .$ $$ \begin{equation*}\nonumber \mathscr{V}_{\mathscr{J}}=\left\{g\in L^\infty (\partial\Omega\times(0,T)):\;u(T;u_0,g)=0 \right\}. \end{equation*} $$

Consider the minimal norm control problems:

${(N P)}_{D} : M_{D} \equiv \min {‖ v ‖_{L^{\infty} (Ω \times (0, T))} : v \in V_{D}}$ $$ \begin{equation}(NP)_{\mathscr{D}}:\;\;\;\;M_{\mathscr{D}}\equiv \min\left\{\|v\|_{L^\infty(\Omega\times(0,T))}:\;\; v\in\mathscr{V}_{\mathscr{D}}\right\} \end{equation} $$

and

${(N P)}_{J} : M_{J} \equiv \min {‖ g ‖_{L^{\infty} (\partial Ω \times (0, T))} : g \in V_{J}} .$ $$ \begin{equation}(NP)_{\mathscr{J}}:\;\;\;\;M_{\mathscr{J}}\equiv \min\left\{\|g\|_{L^\infty(\partial\Omega\times(0,T))}:\;\; g\in\mathscr{V}_{\mathscr{J}}\right\}. \end{equation} $$

Any solution of (NP)_𝒟 (or NP)_𝒥 is called a minimal norm control to this problem. According to Corollary 8, the sets $V_{D}$ $\mathscr{V}_{\mathscr{D}}$ and $V_{J}$ $\mathscr{V}_{\mathscr{J}}$ are not empty. Since $V_{D}$ $\mathscr{V}_{\mathscr{D}}$ is not empty, it follows from the standard arguments that Problem (NP)_𝒟 has solutions. Because $V_{D}$ $\mathscr{V}_{\mathscr{D}}$ is not empty, by using the similar arguments as those in the proof of Lemma 9, we can justify that Problem (NP)_𝒥 has solutions.

We can use the same methods as those in [11] to get the following consequences of Theorem 1 and Theorem 2 respectively:

Corollary 14

Problem (NP)_𝒟has the bang-bang property: any minimal norm controlvsatisfies that |v(x,t)| = M_𝒟for a.e. (x,t) ∈ 𝒟. Consequently, this problem has a unique minimal norm control.

Corollary 15

The problem (NP)_𝒥has the bang-bang property: any minimal norm boundary-controlgsatisfies that |g(x,t)| = M_𝒥for a.e. (x,t) ∈ 𝒥. Consequently, this problem has a unique minimal norm control.

Open problems

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 ℝⁿ and consider the following heat equation,

${\begin{matrix} \partial_{t} u - Δ u = 0, in Ω \times (0, 1), \\ \frac{\partial}{\partial ν} u = 0, on \partial Ω \times (0, T), \\ u (0) = u_{0}, in Ω, \end{matrix}$ $$ \begin{equation} \left\{\begin{array}{l} \partial_tu-\Delta u=0,\quad\mbox{in $\Omega\times(0,1)$,} \\\frac{\partial}{\partial\nu}u=0, \quad\mbox{on $\partial\Omega\times(0,T)$,} \\\displaystyle u(0)=u_0, \quad\mbox{in $\Omega$,} \end{array}\right. \end{equation} $$(33)

with Neumann boundary condition and

${\begin{matrix} \partial_{t} u - Δ u = 0, in Ω \times (0, 1), \\ \frac{\partial}{\partial ν} u + α u = 0, on \partial Ω \times (0, T), \\ u (0) = u_{0}, in Ω, \end{matrix}$ $$ \begin{equation} \left\{\begin{array}{l} \partial_tu-\Delta u=0,\quad\mbox{in $\Omega\times(0,1)$,} \\\frac{\partial}{\partial\nu}u+\alpha u=0, \quad\mbox{on $\partial\Omega\times(0,T)$,} \\\displaystyle u(0)=u_0, \quad\mbox{in $\Omega$,} \end{array}\right. \end{equation} $$(34)

with Robin boundary condition, where α ∈ ℝ and u₀ in L²(Ω).

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 α is utilized. The heat transfer coefficient is determined by details of the interface structure (sharpness, geometry) between two media. This law describes the boundary between metals and gas quite well and is good for the convective heat transfer.

Appendix

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.

Definition 2

Let Ω be a bounded domain in ℝⁿ. øm is a Lipschitz domain (sometimes called strongly Lipschitz or Lipschitz graph domains) with constants m and ρ when for each point p on the boundary of Ω there is a rectangular coordinate system x=(x′, x_n) and a Lipschitz function φ: ℝⁿ⁻¹ → ℝ verifying

$ϕ (0^{'}) = 0, | ϕ (x_{1^{'}}) - ϕ (x_{2^{'}}) | \leq m | x_{1^{'}} - x_{2^{'}} |, for all x_{1^{'}}, x_{2^{'}} \in ℝ^{n - 1},$ $$ \begin{equation} \phi(0')=0,\quad |\phi(x_1')-\phi(x_2')|\le m|x_1'-x_2'|,\ \text{for all}\ x_1', x_2'\in\mathbb{R}^{n-1}, \end{equation} $$(35)

p=(0′, 0) on this coordinate system and

$\begin{matrix} Z_{m, ϱ} \cap Ω = {(x^{'}, x_{n}) : | x^{'} | < ϱ, ϕ (x^{'}) < x_{n} < 2 m ϱ}, \\ Z_{m, ϱ} \cap \partial Ω = {(x^{'}, ϕ (x^{'})) : | x^{'} | < ϱ}, \end{matrix}$ $$ \begin{equation} \begin{split} Z_{m,\varrho}\cap\Omega=\{(x',x_n) : |x'|\lt \varrho,\ \phi(x')\lt x_n \lt 2m\varrho \},\\ Z_{m,\varrho}\cap\partial\Omega= \{(x',\phi(x')) : |x'|\lt \varrho\}, \end{split} \end{equation} $$(36)

where Z_{m, ρ}=B′_ρ × (−2mρ, 2mρ).

Proof

(Proof of (24)] For each (p, τ) ∈ ∂Ω × ℝ and fixed ξ > 0, we define

$\begin{matrix} Γ (p) = {x \in Ω : | x - p | \leq (1 + ξ) d (x, \partial Ω)}, \\ Γ (p, τ) = {(x, t) \in Ω \times (0, T) : | x - p | + \sqrt{| t - τ |} \leq (1 + ξ) d (x, \partial Ω)} . \end{matrix}$ $$ \begin{equation*} \Gamma(p)=\{x\in\Omega: |x-p|\le\left(1+\xi\right)d(x,\partial\Omega)\},\\ \Gamma(p,\tau)=\{(x,t)\in\Omega\times (0,T): |x-p|+\sqrt{|t-\tau|}\le\left(1+\xi\right)d(x,\partial\Omega)\}. \end{equation*} $$

The later are called respectively elliptic and parabolic non-tangential approach regions from the interior of Ω × (0, T) to (p, τ). In particular,

$Γ (p) \times {τ} \subset Γ (p, τ), for all (p, τ) \in \partial Ω \times (0, T) .$ $$ \begin{equation*} \Gamma(p)\times\{\tau\}\subset\Gamma(p,\tau),\ \text{for all}\ (p,\tau)\in\partial\Omega\times (0,T). \end{equation*} $$

When u:Ω → ℝ or u:Ω× (0,T) → ℝ (or ℝⁿ, define the elliptic and parabolic non-tangential maximal function of u in ∂Ω × (0, T) as

$u^{*} (p) = \sup_{x \in Γ (p)} | u (x) |, u^{♯} (p, τ) = \sup_{(x, t) \in Γ (p, τ)} | u (x, t) |, when p \in \partial Ω and τ \in (0, T) .$ $$ \begin{equation*} u^\ast(p)=\sup_{x\in\Gamma(p)}|u(x)|,\quad u^\sharp(p,\tau)=\sup_{(x,t)\in\Gamma(p,\tau)}|u(x,t)|,\ \text{when}\ p\in\partial\Omega\ \text{and}\ \tau\in (0,T). \end{equation*} $$

Let η > 0 be fixed such that [T − l₁, T − l]⊂[2η, T − 2η], with l and l₁ as defined in Corollary 8. Denote by u the solution to

${\begin{matrix} \partial_{t} u - Δ u = 0, & in Ω \times (0, T), \\ u = g χ_{ℳ} \equiv γ, & on \partial Ω \times (0, T), \\ u (0) = u_{0}, & in Ω . \end{matrix}$ $$ \begin{equation*} \begin{cases} \partial_tu-\Delta u=0,\ &\text{in}\ \Omega\times(0,T),\\ u=g\chi_{\mathscr{M}}\equiv\gamma,\ &\text{on}\ \partial\Omega\times(0,T),\\ u(0)=u_0,\ &\text{in}\ \Omega. \end{cases} \end{equation*} $$

(See the beginning of Section 3 for the definition of the solution to this equation.)

Let γ^ε in $C_{0}^{1} (\partial Ω \times (0, T))$ $C^1_0(\partial\Omega\times(0,T))$ be a regularization of γ in ∂Ω × [0, T] such that

$\begin{matrix} ‖ γ^{ε} ‖_{L^{\infty} (\partial Ω \times [0, T])} + ε ‖ γ^{ε} ‖_{C^{1} (\partial Ω \times [0, T])} \leq ‖ γ ‖_{L^{\infty} (\partial Ω \times [0, T])}, \\ supp (γ^{ε}) \subset \partial Ω \times [η, T - η] \end{matrix}$ $$ \begin{equation*} \|\gamma^\varepsilon\|_{L^\infty(\partial\Omega\times[0,T])}+\varepsilon\,\|\gamma^\varepsilon\|_{C^1(\partial\Omega\times[0,T])} \leq \|\gamma\|_{L^\infty(\partial\Omega\times[0,T])},\\ \text{supp}(\gamma^\varepsilon)\subset\partial\Omega\times[\eta,T-\eta] \end{equation*} $$

and let v^ε be the solution to

${\begin{matrix} \partial_{t} v^{ε} - Δ v^{ε} = 0, & in Ω \times (0, T), \\ v^{ε} = γ^{ε}, & on \partial Ω \times (0, T), \\ v^{ε} (0) = 0, & in Ω . \end{matrix}$ $$ \begin{equation*} \begin{cases} \partial_tv^{\varepsilon}-\Delta v^{\varepsilon}=0,\ &\text{in}\ \Omega\times(0,T),\\ v^{\varepsilon}=\gamma^{\varepsilon},\ &\text{on}\ \partial\Omega\times(0,T),\\ v^{\varepsilon}(0)=0,\ &\text{in}\ \Omega. \end{cases} \end{equation*} $$

From [6, Theorem 3.2] [3, Theorem 6.1] or [4, Theorem 2.9]

$‖ v^{ε} ‖_{L^{\infty} (\partial Ω \times [0, T])} + ε ‖ {(\nabla v^{ε})}^{♯} ‖_{L^{2} (\partial Ω \times [0, T])} \leq ‖ γ ‖_{L^{\infty} (\partial Ω \times [0, T])},$ $$ \begin{equation} \|v^\varepsilon\|_{L^\infty(\partial\Omega\times[0,T])}+\epsilon \,\|\left(\nabla v^\epsilon\right)^\sharp\|_{L^2(\partial\Omega\times [0,T])}\le\|\gamma\|_{L^\infty(\partial\Omega\times[0,T])}, \end{equation} $$(37)

and the limits

$\lim_{\underset{(x, t) \in Γ (p, τ)}{(x, t) \to (p, τ)}} \nabla v^{ε} (x, t) = \nabla v^{ε} (p, τ)$ $$ \begin{equation*} \lim_{\underset{(x,t)\in\Gamma(p,\tau)}{(x,t)\rightarrow (p,\tau)}}\nabla v^\epsilon(x,t)= \nabla v^\epsilon(p,\tau) \end{equation*} $$

exist and are finite for a.e. (p, τ) in ∂Ω × (0, T). Also, v_ε ∈ C (Ω × [0, T])∩ C^∞(Ω × [0, T]), v^ε = 0 for t ≤ η, and v^ε = 0 on ∂Ω × (T−η, T]. Moreover, the Hölder regularity up to the boundary for bounded solutions to parabolic equations with zero local lateral Dirichlet data, shows that there are positive constants N = N(m, ρ, η) and α = α(m, ρ), with α ∈ (0, 1), such that

$| v^{ε} (x_{1}, t_{1}) - v^{ε} (x_{2}, t_{2}) | \leq N {[| x_{1} - x_{2} |^{2} + | t_{1} - t_{2} |]}^{α / 2} ‖ γ ‖_{L^{\infty} (\partial Ω \times [0, T])},$ $$ \begin{equation} \begin{split} &|v^\varepsilon(x_1,t_1)-v^\varepsilon(x_2,t_2)| \leq N\left[|x_1-x_2|^2+|t_1-t_2|\right]^{\alpha/2} \|\gamma\|_{L^\infty(\partial\Omega\times[0,T])}, \end{split} \end{equation} $$(38)

when $x_{1}, x_{2} \in \bar{Ω}, T - \frac{η}{2} \leq t_{1}, t_{2} \leq T$ $x_1,x_2\in\overline{\Omega},\;T-\frac\eta 2\leq t_1,t_2\leq T$ [9, Theorems 6.28 and 6.32].

Let φ(t) = e^(T−t)Δφ_T, t ∈ (0, T), where φ_T is in L²(Ω). From the regularity of caloric functions [5, Theorem 1.7]

$φ \in C ([0, T]; L^{2} (Ω)) \cap C^{\infty} (Ω \times [0, T)) \cap C (\bar{Ω} \times [0, T))$ $$ \begin{equation} \varphi\in C([0,T];L^2(\Omega))\cap C^\infty(\Omega\times[0,T))\cap C(\overline{\Omega}\times[0,T)) \end{equation} $$(39)

and from [6, Theorems 1.3 and 1.4] or the proof of (40) and (41) in this appendix, there are N = N(m, ρ) and ε = ε(m, ρ, n) > 0 such that

$∥ (\nabla φ)^{*} ∥_{L^{\infty} (0, T - δ; L^{2 + ε} (\partial Ω))} \leq N e^{1 / δ} ∥ φ_{T} ∥_{L^{2} (Ω)},$ $$ \begin{equation} \|(\nabla \varphi)^\ast\| _{L^\infty(0,T-\delta;\, L^{2+\varepsilon}(\partial\Omega))} \leq Ne^{1/\delta}\,\|\varphi_T\|_{L^2(\Omega)}, \end{equation} $$(40)

when 0 < δ < T and the limit

$\lim_{\underset{x \in Γ (p)}{x \to p}} \nabla φ (x, τ) = \nabla φ (p, τ),$ $$ \begin{equation} \lim_{\underset{x\in\Gamma(p)}{x\to p}}\nabla\varphi(x,\tau)= \nabla\varphi(p,\tau), \end{equation} $$(41)

exists and is finite for a.e. p ∈ ∂Ω and for all τ ∈ (0, T). Now, let Ω_j⊂Ω_j+1⊂Ω, j = 1, be a sequence of C^∞-domains approximating Ω as in [3, Lemma 2.2]. Set, u^ε = v^ε+ e^tΔu₀. By Green’s formula,

$\frac{d}{d t} \int_{Ω_{j}} u^{ε} (t) φ (t) d x = \int_{\partial Ω_{j}} \frac{\partial u^{ε}}{\partial ν_{j}} φ - \frac{\partial φ}{\partial ν_{j}} u^{ε} d σ_{j} .$ $$ \begin{equation*} \tfrac{d}{dt}\int_{\Omega_j}u^\varepsilon(t)\varphi(t)\,dx = \int_{\partial\Omega_j}\tfrac{\partial u^\varepsilon}{\partial\nu_j}\,\varphi-\tfrac{\partial \varphi}{\partial\nu_j}\, u^\varepsilon\,d\sigma_j. \end{equation*} $$

Integrating the above identity over [δ, T−δ] for a fixed $δ \in (0, \frac{η}{2})$ $\delta\in(0,\frac{\eta}{2})$ , we get

$\begin{matrix} \int_{Ω_{j}} u^{ε} (T - δ) φ (T - δ) d x - \int_{Ω_{j}} u^{ε} (δ) φ (δ) d x \\ = \int_{\partial Ω_{j} \times (δ, T - δ)} \frac{\partial u^{ε}}{\partial ν_{j}} φ - \frac{\partial φ}{\partial ν_{j}} u^{ε} d σ_{j} d t . \end{matrix}$ $$ \begin{equation} \begin{split} \int_{\Omega_j}u^\varepsilon(T-\delta)\varphi(T-\delta)\,dx -\int_{\Omega_j}u^\varepsilon(\delta)\varphi(\delta)\,dx \\ =\int_{\partial\Omega_j\times(\delta,T-\delta)}\tfrac{\partial u^\varepsilon}{\partial\nu_j}\,\varphi-\tfrac{\partial \varphi}{\partial\nu_j}\, u^\varepsilon\,d\sigma_jdt. \end{split} \end{equation} $$(42)

Recall that u^ε(δ) = e^δΔu₀ and let j → +∞ in (42) with ε and δ being fixed. Then, (37), (39), (41) and the dominated convergence theorem show that

$\int_{Ω} u^{ε} (T - δ) φ (T - δ) d x = \int_{Ω} (e^{δ Δ} u_{0}) φ (δ) d x - \int_{\partial Ω \times (δ, T - δ)} γ^{ε} \frac{\partial φ}{\partial ν} d σ d t .$ $$ \begin{equation*} \int_{\Omega}u^\varepsilon(T-\delta)\varphi(T-\delta)\,dx =\int_{\Omega}(e^{\delta\Delta}u_0)\varphi(\delta)\,dx -\int_{\partial\Omega\times(\delta,T-\delta)}\gamma^\varepsilon\tfrac{\partial \varphi}{\partial\nu}\,d\sigma dt. \end{equation*} $$

Because γ^ε is supported in [η, T−η], the later is the same as

$\int_{Ω} u^{ε} (T - δ) φ (T - δ) d x = \int_{Ω} (e^{δ Δ} u_{0}) φ (δ) d x - \int_{\partial Ω \times (η, T - η)} γ^{ε} \frac{\partial φ}{\partial ν} d σ d t,$ $$ \begin{equation} \int_{\Omega}u^\varepsilon(T-\delta)\varphi(T-\delta)\,dx =\int_{\Omega}(e^{\delta\Delta}u_0)\varphi(\delta)\,dx -\int_{\partial\Omega\times(\eta,T-\eta)}\gamma^\varepsilon\tfrac{\partial \varphi}{\partial\nu}\,d\sigma dt, \end{equation} $$(43)

when 0 < δ < η/8. Next, from (38),

$u^{ε} (T - δ) = v^{ε} (T - δ) + e^{(T - δ) Δ} u_{0} = v^{ε} (T) + e^{T Δ} u_{0} + O (δ^{α / 2}),$ $$ \begin{equation*} u^\varepsilon(T-\delta)=v^\varepsilon(T-\delta)+e^{(T-\delta) \Delta}u_0=v^\varepsilon(T)+e^{T\Delta}u_0+O(\delta^{\alpha/2}), \end{equation*} $$

uniformly for $x \in \bar{Ω}$ $x\in\overline{\Omega}$ , when 0 < δ < η/8. Hence, after letting δ → 0 in (43), we get

$\int_{Ω} (v^{ε} (T) + e^{T Δ} u_{0}) φ (T) d x = \int_{Ω} u_{0} φ (0) d x - \int_{\partial Ω \times (η, T - η)} γ^{ε} \frac{\partial φ}{\partial ν} d σ d t .$ $$ \begin{equation*} \int_{\Omega}(v^\varepsilon(T)+e^{T\Delta}u_0)\varphi(T)\,dx =\int_{\Omega}u_0\varphi(0)\,dx-\int_{\partial\Omega\times(\eta ,T-\eta)}\gamma^\varepsilon\tfrac{\partial\varphi}{\partial\nu}\, d\sigma dt. \end{equation*} $$

Also, from (37) and (38), v^ε converges uniformly over $\bar{Ω} \times [T - η / 2, T]$ $\overline{\Omega} \times[T-\eta/2,T]$ to some continuous function $\tilde{v}$ $\widetilde {v}$ as ε → 0. We claim that $\tilde{v} = v$ $\widetilde{v}=v$ . If it is the case, we get after letting ε → 0 in the last equality, that

$\int_{Ω} u (T) φ (T) d x = \int_{Ω} u_{0} φ (0) d x - \int_{\partial Ω \times (η, T - η)} γ \frac{\partial φ}{\partial ν} d σ d t,$ $$ \begin{equation*} \int_{\Omega}u(T)\varphi(T)\,dx =\int_{\Omega}u_0\varphi(0)\,dx-\int_{\partial\Omega\times(\eta ,T-\eta)}\gamma\tfrac{\partial\varphi}{\partial\nu}\, d\sigma dt, \end{equation*} $$

because γ^ε(p, τ) → γ(p, τ) for a.e. (p, τ) ∈ ∂Ω × (0, T), (39) and

$supp (γ^{ε}) \cup supp (γ) \subset \partial Ω \times [η, T - η] .$ $$ \begin{equation*} \text{supp}(\gamma^\varepsilon)\cup \text{supp}(\gamma)\subset \partial\Omega\times[\eta,T-\eta]. \end{equation*} $$

Recalling that γ = gχ_𝓜, we get

$\int_{Ω} u (T) φ (T) d x = \int_{Ω} u_{0} φ (0) d x - \int_{\partial Ω \times (0, T)} g χ_{ℳ} \frac{\partial φ}{\partial ν} d σ d t .$ $$ \begin{equation*} \int_{\Omega}u(T)\varphi(T)\,dx =\int_{\Omega}u_0\varphi(0)\,dx-\int_{\partial\Omega\times(0 ,T)}g\chi_{\mathscr{M}}\,\tfrac{\partial\varphi}{\partial\nu}\, d\sigma dt. \end{equation*} $$

Hence, (24) is proved.

To verify that $\tilde{v} = v$ $\widetilde{v}=v$ over $\bar{Ω} \times [0, T]$ $\overline{\Omega}\times[0,T]$ , observe that because v^ɛ−v is the unique solution to

${\begin{matrix} \partial_{t} u - Δ u = 0, & in Ω \times (0, T), \\ u = γ^{ε} - γ, & on \partial Ω \times (0, T), \\ u (0) = 0, & in Ω, \end{matrix}$ $$ \begin{equation*} \begin{cases} \partial_tu-\Delta u =0,\ &\text{in}\ \Omega\times(0,T),\\ u=\gamma^{\epsilon}-\gamma,\ &\text{on}\ \partial\Omega\times(0,T),\\ u(0)=0,\ &\text{in}\ \Omega, \end{cases} \end{equation*} $$

whose parabolic non-tangential maximal function is in L²(∂Ω × (0, T)) (See [6, Theorem 3.2]), it holds that

$‖ {(v^{ε} - v)}^{♯} ‖_{L^{2} (\partial Ω \times (0, T))} \leq N ‖ γ^{ε} - γ ‖_{L^{2} (\partial Ω \times (0, T))} .$ $$ \begin{equation} \|(v^\varepsilon-v)^\sharp\|_{L^2(\partial\Omega\times(0,T))} \leq N\|\gamma^\varepsilon-\gamma\|_{L^2(\partial\Omega\times(0,T))}. \end{equation} $$(44)

For fixed p in ∂Ω, we may assume that p = (0′, 0) and that near p,

$Ω \cap Z_{m, ϱ} = {(x^{'}, x_{n}) : ϕ (x^{'}) < x_{n} < 2 m ϱ, | x^{'} | \leq ϱ},$ $$ \begin{equation*} \Omega\cap Z_{m,\varrho}=\{(x',x_n):\phi(x')\lt x_n\lt 2m\varrho,\; |x'|\leq \varrho\}, \end{equation*} $$

with φ as in (35) and (36). Then,

$\begin{matrix} \int_{0}^{T} \int_{B_{ρ}^{'}} \int_{ϕ (y^{'})}^{ϕ (y^{'}) + m ρ} | F (y^{'}, y_{n}, t) |^{2} d y^{'} d y_{n} d t \\ \leq m ρ \int_{0}^{T} \int_{B_{ϱ}^{'}} F^{♯} (y^{'}, y_{n}, t)^{2} d y^{'} d t \leq m ρ \int_{\partial Ω \times (0, T)} F^{♯} (p, t)^{2} d σ d t, \end{matrix}$ $$ \begin{equation*} \begin{split} \int_0^T\int_{B'_{\rho}}\int_{\phi(y')}^{\phi(y')+m\rho} |F(y',y_n,t)|^2\,dy'dy_ndt \\ \leq m\rho\int_0^T\int_{B'_\varrho}F^\sharp(y',y_n,t)^2\,dy'dt \leq m\rho\int_{\partial\Omega\times(0,T)}F^\sharp(p,t)^2\,d\sigma dt, \end{split} \end{equation*} $$

for all functions F. The above estimate, a covering argument and (44) show that

$‖ v^{ε} - v ‖_{L^{2} (Ω_{m ϱ} \times (0, T))} \leq N ‖ γ^{ε} - γ ‖_{L^{2} (\partial Ω \times (0, T))},$ $$ \begin{equation} \|v^\varepsilon-v\|_{L^2(\Omega_{m\varrho}\times (0,T))}\le N\|\gamma^\varepsilon-\gamma\|_{L^2(\partial\Omega\times(0,T))}, \end{equation} $$(45)

with Ω_η = {x ∈ Ω: d(x, ∂Ω) ≤ η}. Recalling that v^ε = v = 0 for t ≤ η, the local boundedness properties of solutions to parabolic equations [9, Theorem 6.17] show that,

$| (v^{ε} - v) (x, τ) | \leq {(- \int_{B_{\frac{R}{20}} (x) \times [τ - \frac{R^{2}}{20^{2}}, τ]} | v^{ε} - v |^{2} d y d s)}^{1 / 2},$ $$ \begin{equation*} |(v^\varepsilon-v)(x,\tau)|\leq \left(-\!\!\!\!\!\!\!\!\! \int_{B_{\frac{R}{20}}(x) \times[\tau-\frac{R^2}{20^2},\tau]}|v^\varepsilon-v|^2\,dyds \right)^{1/2}, \end{equation*} $$

when x ∈ ∂Ω_R, 0 ≤ τ ≤ T, and taking $R < \frac{m ϱ}{20}$ $R\lt\frac{m\varrho}{20}$ above, we find from (45) that

$‖ v^{ε} - v ‖_{L^{\infty} (Ω^{R} \times {0} \cup \partial Ω^{R} \times [0, T])} \leq N_{R} {‖ γ^{ε} - γ ‖}_{L^{2} (\partial Ω \times (0, T))} .$ $$ \begin{equation*} \|v^\varepsilon-v\|_{L^\infty(\Omega^R\times\{0\}\cup\partial\Omega ^R\times[0,T])} \leq N_R\left\|\gamma^\varepsilon-\gamma\right\|_{L^2(\partial \Omega\times(0,T))}. \end{equation*} $$

By the maximum principle and the above estimate

$‖ v^{ε} - v ‖_{L^{\infty} (Ω^{R} \times [0, T])} \leq N_{R} {‖ γ^{ε} - γ ‖}_{L^{2} (\partial Ω \times (0, T))} \to 0, as ε \to 0,$ $$ \begin{equation*} \|v^\varepsilon-v\|_{L^\infty(\Omega^R\times[0,T])} \leq N_R\left\|\gamma^\varepsilon-\gamma\right\|_{L^2(\partial \Omega\times(0,T))}\longrightarrow 0,\;\;\text{as}\;\;\varepsilon\rightarrow 0, \end{equation*} $$

which shows that $\tilde{v} = v$ $\widetilde{v}=v$ in $\bar{Ω} \times [0, T]$ $\overline{\Omega}\times[0,T]$ . □

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Observability Inequalities for Parabolic Equations over Measurable Sets and Some Applications Related to the Bang-Bang Property for Control Problems

Published Online: Dec 15, 2017

Page range: 543 - 558

Received: Jun 01, 2017

Accepted: Dec 15, 2017

DOI: https://doi.org/10.21042/AMNS.2017.2.00045

KeywordsParabolic equations, control theory, controllability, observability inequalities, Bang-Bang properties

© 2017 J. Apraiz, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Keywords
Parabolic equations, control theory, controllability, observability inequalities, Bang-Bang properties