Nonnegative Boundary Control of 1D Linear Heat Equations

Abstract

We consider the controllability of a one dimensional heat equation with nonnegative boundary controls. Despite the controllability in any positive time of this system, the unilateral nonnegativity control constraint causes a positive minimal controllability time. In this article, it is proved that at the minimal time, there exists a nonnegative control in the space of Radon measures, which consists of a countable sum of Dirac impulses.

Appendices

Appendix A: Controllability with Nonnegative Controls

Let us first recall that, according to [7], for every y⁰ ∈ L²(0, 1), every time T > 0, there exists a control u ∈ C⁰([0, T]) such that the solution of (1.1) satisfies y(T,⋅) = 0. By duality there exists C_T > 0 (depending only on T) such that for every z¹ ∈ L²(0, 1), the solution z of

$$ \begin{array}{@{}rcl@{}} -\dot{z}(t,x) & =& \partial_{x}\left( p(x)\partial_{x} z(t,x)\right)z+q(x)z(t,x) \qquad (t\in(0,T), x\in(0,1)),\\ 0 & =& \alpha_{0}z(t,0)+\alpha_{1}\partial_{x}z(t,0) {\kern62pt}(t\in(0,T)),\\ 0 & =& \beta_{0}z(t,1)+\beta_{1}\partial_{x}z(t,1) {\kern62pt}(t\in(0,T)),\\ z(T,x) & =& z^{1} {\kern142pt}(x\in(0,1)), \end{array} $$

satisfies,

$$ \|z(0,\cdot)\|_{L^{2}(0,1)}\leq C_{T}\|z(t,1)\|_{L^{1}(0,T)}, $$

if β₁≠ 0, or

$$ \|z(0,\cdot)\|_{L^{2}(0,1)}\leq C_{T}\|\partial_{x} z(t,1)\|_{L^{1}(0,T)}, $$

if β₀≠ 0. By duality, this means that the control u steering y⁰ to 0 in time T can be chosen so that $\|u\|_{L^{\infty }(0,T)}\leq C_{T}\|y^{0}\|_{L^{2}(0,1)}$. It is then a trivial exercise to see that for every $y^{1}\in \mathcal {S}_{+}^{\ast }$, the control u steering y⁰ ∈ L²(0, 1) to y¹ can be chosen so that

$$ \|u-u^{1}\|_{L^{\infty}(0,T)}\leq C_{T}\|y^{0}-y^{1}\|_{L^{2}(0,1)}, $$

where u¹ > 0 is the control associated to the steady state y¹.

When L is dissipative

When L is dissipative, it is a classical exercise to show that for every τ > 0 and every $k\in \mathbb {N}^{\ast }$, we have $C_{k\tau }\leq \frac {C_{\tau }}{k}$. Meaning in particular, for every $y^{1}\in \mathcal {S}_{+}^{\ast }$ and every y⁰ ∈ L²(0, 1), that taking τ > 0 and taking $k\in \mathbb {N}^{\ast }$ sufficiently large, we have

$$ \|u-u^{1}\|_{L^{\infty}(0,T)}\leq \frac{C_{\tau}}{k}\|y^{0}-y^{1}\|_{L^{2}(0,1)}\leq u^{1} $$

(here again $u^{1}\in \mathbb {R}_{+}^{\ast }$ is the control associated to the steady state y¹).

This in particular means that, by taking k = k(y⁰, y¹, τ) sufficiently large, there exists a nonnegative control steering y⁰ to y¹ in time kτ.

General situation

When L is not dissipative, it is not possible to use the previous argument. However, we can use a quasi-static approach combined with a compactness argument. Indeed, given τ > 0, $y^{0},y^{1}\in \mathcal {S}_{+}^{\ast }$, with associated control u⁰ and u¹, for some $K\in \mathbb {N}^{\ast }$, we define, for every $k\in \{1,\dots ,K-1\}$, $y^{k/K}\in \mathcal {S}_{+}^{\ast }$ the steady state associated to the control $u^{0}+\frac {k}{K}u^{1}$. By taking K large enough such that $\frac {C_{\tau }}{K}\|y^{1}-y^{0}\|_{L^{2}(0,1)}\leq \min \limits \{u^{0},u^{1}\}$, there exists a nonnegative control u_k steering y^k/K to y^(k+ 1)/K in time τ. Then by concatenating these controls, we have found a nonnegative control steering y⁰ to y¹ in time Kτ.

Appendix B: Existence of a Nonnegative Minimal Time Control in the Space of Radon Measures

The proof of this fact follows the one of [11, Theorems 2.1 and 3.1] or [12, Proposition 5.1.7]. Indeed, if y⁰ and y¹ ∈ L²(0, 1) are such that $\underline {T}(y^{0},y^{1})<\infty$ (i.e., y¹ is reachable form y⁰ with nonnegative controls), then there exists a nonincreasing sequence $(T_{k})_{k\in \mathbb {N}}\in [\underline {T}(y^{0},y^{1}),\infty )^{\mathbb {N}}$ such that $T_{k}\to \underline {T}(y^{0},y^{1})$ as $k\to \infty$ and such that for every $k\in \mathbb {N}$, there exists a nonnegative control u_k ∈ L²(0, T_k) steering the solution y_k of (1.1) from y⁰ to y¹ in time T_k. In addition, since the operator L defined by (1.2) is symmetric and diagonalizable, we pick an eigenvalue $\lambda \in \mathbb {R}$ of L and $\varphi \in \mathcal {D}(L)\setminus \{0\}$ an associated eigenfunction. Let us define for every i ∈{0, 1}, $Y^{i}=\langle y^{i},\varphi \rangle _{L^{2}(0,1),L^{2}(0,1)}$. Since u_k is a control steering y⁰ to y¹ in time T_k, we deduce that, u_k shall satisfy (2.2), with y(T,⋅) = y¹, i.e.,

$$ \frac{Y^{1}-e^{\lambda T_{n}}Y^{0}}{\gamma}={\int}_{0}^{T_{k}} e^{\lambda(T_{k}-t)}u_{k}(t) \mathrm{d} t\qquad (k\in\mathbb{N}), $$

with $\gamma =\left \{\begin {array}{ll} \frac {p(1)}{\beta _{1}}\varphi (1)&\quad \text {if } \beta _{1}\neq 0,\\ -\frac {p(1)}{\beta _{0}}\partial _{x}\varphi (1) &\quad \text {if } \beta _{0}\neq 0 \end {array}\right .$ (recall that in both situations, we have γ≠ 0).

This ensures that (since u_k ≥ 0 and $e^{\lambda (T_{k}-t)}\geq e^{-|\lambda |T_{k}}$ for every t ∈ [0, T_k]),

$$ {\int}_{0}^{T_{k}} u_{k}(t) \mathrm{d} t\leq e^{|\lambda|T_{k}}\frac{|Y^{1}|+e^{|\lambda|T_{k}}|Y^{0}|}{|\gamma|}. $$

Finally, since (T_k)_k is a nonincreasing sequence, we obtain (by extending u_k by 0 on (T_k, T₀)) that,

$$ \|u_{k}\|_{L^{1}(0,T_{0})}\leq e^{|\lambda|T_{0}}\frac{e^{|\lambda|T_{0}}|Y^{0}|+|Y^{1}|}{|\gamma|}\qquad (k\in\mathbb{N}), $$

that is to say that the sequence (u_k)_k is uniformly bounded in L¹(0, T₀) and hence, is up to the extraction of a subsequence, vaguely convergent to some Radon measure $\underline {u}\in {\mathscr{M}}([0,T_{0}])$ (see e.g., [3, Corollary 31.3, p. 206]). Since $u_{k}\geqslant 0$ and suppu_k ⊂ [0, T_k] for every $k\in \mathbb {N}$, we easily deduce that we necessarily have $\underline {u}\geq 0$ and $\text {supp} \underline {u}\subset [0,\underline {T}(y^{0},y^{1})]$.

It remains to check that $\underline {u}$ is indeed a control steering y⁰ to y¹ in time $\underline {T}(y^{0},y^{1})$. This can be deduced by taking the limit $k\to \infty$ in (2.1) or (2.2) (with T = T_k and $\mathrm {d}\underline {u}(t)=u_{k}(t)\mathrm {d} t$).

Appendix C: No Gap Situation

Let y⁰ ∈ L²(0, 1) and $y^{1}\in \mathcal {S}_{+}^{\ast }$ and assume that $\underline {T}(y^{0},y^{1})<\infty$, i.e., y⁰ can be steered to y¹ with a nonnegative L² control. Recall that $\underline {T}(y^{0},y^{1})$ is defined by (1.3), i.e.,

$$ \begin{array}{@{}rcl@{}} \underline{T}(y^{0},y^{1})=\inf\{T>0&|&\exists u\in L^{2}(0,T)~\text{ s.t. }~u\geq0 \\ &&\text{ and the solution \textit{y} of (1.1) satisfies $y(T,\cdot)=y^{1}$}\}. \end{array} $$

The aim of this paragraph is to show that we have $\underline {T}(y^{0},y^{1})=\underline {T}_{{\mathscr{M}}}(y^{0},y^{1})$, where we have set

$$ \begin{array}{@{}rcl@{}} \underline{T}_{\mathcal{M}}(y^{0},y^{1})=\inf\{T>0&|& \exists \underline{u}\in \mathcal{M}([0,T])~\text{ s.t. }~\underline{u}\geq0 \\ &&\text{ and the solution \textit{y} of (1.1) satisfies $y(T,\cdot)=y^{1}$}\}. \end{array} $$

Obviously, we always have $0\leq \underline {T}_{{\mathscr{M}}}(y^{0},y^{1})\leq \underline {T}(y^{0},y^{1})$. Arguing as in [12, Proposition 5.1.11] and using comments contained in Appendix A, we can prove that, given a time T ≥ 0 and a nonnegative control $\underline {u}\in {\mathscr{M}}([0,T])$ steering y⁰ to y¹ in time T, and given any ε > 0, there exists a control u ∈ L²(0, T + ε) steering y⁰ to y¹ in time T + ε. This fact shows that $\underline {T}(y^{0},y^{1})=\underline {T}_{{\mathscr{M}}}(y^{0},y^{1})$.

Remark 5

When y¹ is not a positive steady state, it has been shown (on some finite dimensional systems) in [12] that an infimum gap could occur (i.e., $\underline {T}(y^{0},y^{1})<\underline {T}_{{\mathscr{M}}}(y^{0},y^{1})$). Let us refer to [13] for general no gap condition for finite dimensional control systems.