Time-varying Bang-bang Property of Minimal Controls for Approximately Null-controllable Heat Equations

TL;DR

This paper investigates the time-varying bang-bang property of minimal controls in heat equations that are approximately null-controllable, introducing new boundary conditions for control variables and establishing key equivalence results.

Contribution

It extends existing control theory by analyzing time-varying control boundaries and proving a bang-bang property for optimal controls in heat equations.

Findings

Established the time-varying bang-bang property for optimal controls.

Proved an equivalence theorem between optimal control and target problems.

Extended control boundary conditions from constants to functions.

Abstract

In this paper, optimal time control problems and optimal target control problems are studied for the approximately null-controllable heat equations. Compared with the existed results on these problems, the boundary of control variables are not constants but time varying functions. The time-varying bang-bang property for optimal time control problem, and an equivalence theorem for optimal control problem and optimal target problem are obtained.

Equations

$$\{\begin{array}{ll}{\partial }_{t}y(x,t)-\Delta y(x,t)={\chi }_{\omega }{\chi }_{(\tau ,T)}u(x,t),& \text{mbox}in\Omega \times (0,T),\\ y(x,t)=0,& \text{mbox}on\partial \Omega \times (0,T),\\ y(x,0)=y_0(x),& \text{mbox}in\Omega .\end{array}$$

$$\mathcal{U}_{M}\equiv\left\{v\in L^{\infty}(0,T;L^{2}(\Omega))\bigm{|}\|v(t)\|\leq M(t)\mbox{ for a.e. }t\in(\tau,T)\right\},$$

$$\mathcal{R}(y_{0},\tau,T)=\left\{y(T;\chi_{(\tau,T)}u,y_{0})\bigm{|}u\in\mathcal{U}_{M}\right\}$$

$$\mathcal{R}(y_0,T)=\bigcup _{\tau \in (0,T)}\mathcal{R}(y_0,\tau ,T).$$

$$\tau(\varepsilon)=\sup_{\tau\in[0,T]}\left\{\tau\bigm{|}y(T;\chi_{(\tau,T)}u,y_{0})\in\bar{B}(0,\varepsilon),u\in\mathcal{U}_{M}\right\}.$$

$$u^{\ast }$$

$$\mathcal{U}_{M_{0}}=\left\{v\in L^{\infty}(0,T;L^{2}(\Omega))\bigm{|}\|v(t)\|\leq M_{0}\mbox{ for a.e. }t\in(\tau,T)\right\}.$$

$$\Vert u^{\ast }(t)\Vert =M_0\text{mbox}fora.e.t\in (\tau (0),T).$$

$$\Vert u^{\ast }(t)\Vert =M(t)\text{mbox}fora.e.t\in (\tau (\epsilon ),T).$$

$$\varepsilon(\tau)=\inf\left\{\|y(T;\chi_{(\tau,T)}u,y_{0})\|\bigm{|}u\in\mathcal{U}_{M}\right\}.$$

$${\epsilon }_T=\Vert y(T;{\chi }_{(0,T)}0,y_0)\Vert .$$

$$\epsilon =\epsilon (\tau (\epsilon )),\epsilon \in [\epsilon (0),{\epsilon }_T),\text{mbox}and\tau =\tau (\epsilon (\tau )),\tau \in [0,T).$$

$$y(T;{\chi }_{(\tau (\epsilon ),T)}{u}^{\ast },y_0)\in \bar{B}(0,\epsilon ).$$

$$y(T;{\chi }_{({\tau }_n,T)}{u}_n,y_0)\in \bar{B}(0,\epsilon ).$$

$${\tilde{u}}_n(t)=\{\begin{array}{ll}0,& \text{mbox}ift\in (0,{\tau }_n],\\ u_n(t),& \text{mbox}ift\in ({\tau }_n,T).\end{array}$$

$${\tilde{u}}_n\to u^{\ast }\text{mbox}{weakly}^{\ast }\text{mbox}in{L}^{\infty }(0,T;L^2(\Omega )).$$

$$v(x,t)={\chi }_{(t_0-\lambda ,t_0+\lambda )}(t)\zeta (x).$$

$$\begin{array}{rl}{\int }_{t_0-\lambda }^{t_0+\lambda }⟨u^{\ast }(t),\zeta ⟩dt& ={\int }_0^T⟨u^{\ast }(t),{\chi }_{(t_0-\lambda ,t_0+\lambda )}\zeta ⟩dt\\ & ={\int }_0^T⟨u^{\ast }(t),v(t)⟩dt\\ & =\underset{n\to \infty }{\lim }{\int }_0^T⟨{\tilde{u}}_n(t),v(t)⟩dt\\ & =0.\end{array}$$

$$⟨u^{\ast }(t_0),\zeta ⟩=\underset{\lambda \to 0}{\lim }\frac{1}{2\lambda }{\int }_{t_0-\lambda }^{t_0+\lambda }⟨u^{\ast }(t),\zeta ⟩dt=0.$$

$$u^{\ast }{|}_{(0,\tau (\epsilon ))}=0.$$

$$\{\begin{array}{ll}{\partial }_t{y}^{\ast }(x,t)-\Delta y^{\ast }(x,t)={\chi }_{\omega }{\chi }_{(\tau (\epsilon ),T)}{u}^{\ast }(x,t),& \text{mbox}in\Omega \times (0,T),\\ y^{\ast }(x,t)=0,& \text{mbox}on\partial \Omega \times (0,T),\\ y^{\ast }(x,0)=y_0(x),& \text{mbox}in\Omega ,\end{array}$$

$$\begin{array}{rl}{y}_n\to y^{\ast }& \text{mbox}weaklyin{L}^2(0,T;H_0^1(\Omega ))\cap H^1(0,T;L^2(\Omega )),\\ & \text{mbox}stronglyinC([\delta ,T];L^2(\Omega ))\text{mbox}asn\to \infty ,\end{array}$$

$$\{\begin{array}{ll}{\partial }_t{y}_n(x,t)-\Delta y_n(x,t)={\chi }_{\omega }{\chi }_{({\tau }_n,T)}{\tilde{u}}_n(x,t),& \text{mbox}in\Omega \times (0,T),\\ y_n(x,t)=0,& \text{mbox}on\partial \Omega \times (0,T),\\ y_n(x,0)=y_0(x),& \text{mbox}in\Omega .\end{array}$$

$$\Vert u^{\ast }(t)\Vert >M(t)+{\epsilon }_0,\forall t\in E_0,$$

$$\zeta (x,t)={\chi }_{{}_{E_0}}\frac{u^{\ast }(t)}{\Vert u^{\ast }(t)\Vert }.$$

$$\Vert \zeta {\Vert }_{L^1(\tau (\epsilon ),T;L^2(\Omega ))}={\int }_{\tau (\epsilon )}^T\Vert \zeta (t)\Vert dt={\int }_{\tau (\epsilon )}^T\Vert {\chi }_{{}_{E_0}}\frac{u^{\ast }(t)}{\Vert u^{\ast }(t)\Vert }\Vert dt=|E_0|<\infty ,$$

$$\left|{\int }_{\tau (\epsilon )}^T⟨u_n-u^{\ast },\zeta ⟩dt\right|<ϵ.$$

$$\Vert \zeta (t)\Vert =\Vert {\chi }_{{}_{E_0}}\frac{u^{\ast }(t)}{\Vert u^{\ast }(t)\Vert }\Vert ={\chi }_{{}_{E_0}},$$

$$\begin{array}{rl}\left|{\int }_{\tau (\epsilon )}^T⟨u_n-u^{\ast },\zeta ⟩dt\right|=& \left|{\int }_{\tau (\epsilon )}^T⟨u_n,\zeta ⟩dt-{\int }_{\tau (\epsilon )}^T⟨u^{\ast },\zeta ⟩dt\right|\\ \ge & \left|{\int }_{\tau (\epsilon )}^T⟨u^{\ast },\zeta ⟩dt\right|-\left|{\int }_{\tau (\epsilon )}^T⟨u_n,\zeta ⟩dt\right|\\ \ge & {\int }_{\tau (\epsilon )}^T⟨u^{\ast },{\chi }_{E_0}\frac{u^{\ast }(t)}{\Vert u^{\ast }(t)\Vert }⟩dt-{\int }_{\tau (\epsilon )}^T\Vert u_n(t)\Vert \Vert \zeta (t)\Vert dt\\ =& {\int }_{E_0}\Vert u^{\ast }(t)\Vert dt-{\int }_{E_0}\Vert u_n(t)\Vert dt\\ \ge & {\int }_{E_0}(M(t)+{\epsilon }_0)dt-{\int }_{E_0}M(t)dt\\ =& |E_0|{\epsilon }_0.\end{array}$$

$$y_1\equiv y(T;{\chi }_{(\tau (\epsilon ),T)}{u}_1^{\ast },y_0),y_2\equiv y(T;{\chi }_{(\tau (\epsilon ),T)}{u}_2^{\ast },y_0)\in \bar{B}(0,\epsilon )$$

Full text

**Keywords: **Heat equation, bang-bang property, optimal time control problem, optimal target control problem

**AMS subject classification: **35K05, 49J20

1 Introduction

Let $T$ be a positive number (can be taken $\infty$) and $\Omega$ be an open bounded domain with smooth boundary in $\mathbb{R}^{N},\,N\geq 1$. Let $\omega$ be an open set of $\Omega$. Consider the following controlled system:

[TABLE]

Here $y_{0}\in L^{2}(\Omega)$ is a given initial data, $u\in L^{\infty}(0,T;L^{2}(\Omega))$, $\tau\in[0,T)$, $\chi_{\omega}$ and $\chi_{(\tau,T)}$ stand for the characteristic functions of $\omega$ and $(\tau,T)$, respectively. We denote the solution to (1.1) by $y(\cdot;\chi_{(\tau,T)}u,y_{0})$ with initial data $y_{0}$ and control $u$.

In this paper, denote by $L^{\infty}_{+}(0,T)$ the subset of $L^{\infty}(0,T)$, whose element is almost surely positive. Denote by $\|\cdot\|$ and $\langle\cdot,\cdot\rangle$ the norm and inner product of $L^{2}(\Omega)$, respectively, and $B(0,r)$/$\bar{B}(0,r)$ the open/closed ball of $L^{2}(\Omega)$ with center 0 and radius $r>0$.

The null and approximate controllability of system (1.1) has been studied in many works (see, e.g. [2, 5]). Especially, for each $\varepsilon>0$, since the system (1.1) is energy decaying, taking $u=0$, we have $\|y(t;0,y_{0})\|\leq\varepsilon$, when $t$ is large enough. By this, we can easily see that the system (1.1) is approximately null-controllable for large $T$. The reader can also refer to [1, 7, 8, 14, 15] for more discussions on controlled heat equations.

Three kinds of optimal control problems: the optimal time, target and norm control problems are important and interesting branches of optimization. For the deterministic systems, the reader can refer to [4] to obtain the recent results and open problems. The reader can also refer to [3, 9, 10, 11, 12, 14] for the optimal time control problems. For the stochastic ones, the optimal norm control problems were considered in [18, 19, 21] for controlled stochastic ordinary differential equations, and in [20] for controlled stochastic heat equations. The reader can also refer to [6, 15, 17] for the work on equivalence relation between these three optimal control problems.

For a given function $M(\cdot)\in L^{\infty}_{+}(0,T)$, we can define the admissible control set of controlled system (1.1):

[TABLE]

and can denote the reachable set of system (1.1) with $u\in\mathcal{U}_{M}$ by

[TABLE]

and

[TABLE]

By above discussion, without loss of generality, we can assume that $\mathcal{R}(y_{0},T)\cap\bar{B}(0,\varepsilon)\neq\emptyset$ for $\varepsilon>0$ and $T>0$. We need note that $y(T;\chi_{(\tau,T)}0,y_{0})$ may not be in $\bar{B}(0,\varepsilon)$.

Consider the following optimal time control problem

[TABLE]

If the optimal time control problem (1.2) is solvable, i.e., there exist at least one $u^{*}\in\mathcal{U}_{M}$ such that $y(T;\chi_{(\tau(\varepsilon),T)}u^{*},y_{0})\in\bar{B}(0,\varepsilon)$, we call that $u^{*}$ an optimal time control. By by choosing the minimal sequence and applying the classical variational method, we can prove that the optimal time control problem (1.2) has a solution $u^{*}$ (see Lemma 2.1). What we are interested in is the following problem:

[TABLE]

The (time-invariant) bang-bang property is a classical problem in control theory. There are many works on this topic (see, e.g. [12, 13, 14]). In [14], the author obtained the following the bang-bang property of a null-controlled heat equation. For a given positive constant $M_{0}$, define

[TABLE]

Then, if $u^{*}\in\mathcal{U}_{M_{0}}$ is an optima time control respect to (1.2), then the following time-invariant bang-bang property holds:

[TABLE]

[16] owns many interesting results, but the bang-bang property is also depend on the positive constant $M_{0}$. This work is inspired by [14, 16]. In this work, we study the time-varying bang-bang property of an approximately null-controllable heat equation. Compared with the problem studied in [14], the boundary is not a constant $M_{0}$ but a function $M(\cdot)$. Hence, the method used in [14] is not workable any more. Until now, to our best knowledge, there does not exist any work on this kind time-varying bang-bang property. The following is our first main result.

**    Theorem 1.1****.**

Suppose that $M(\cdot)\in L^{\infty}_{+}(0,T)$, $\varepsilon>0$, and $\mathcal{R}(y_{0},T)\cap\bar{B}(0,\varepsilon)\neq\emptyset$. Then there exists a unique optimal time control $u^{*}\in\mathcal{U}_{M}$ such that the optimal time control problem (1.2) is solvable. Moreover, the optimal control $u^{*}$ satisfies the following time-varying bang-bang property:

[TABLE]

Now, we consider the following optimal target control problem:

[TABLE]

Define

[TABLE]

It is obviously that $0\leq\varepsilon(\tau)\leq\varepsilon_{T}$ and $0\leq\tau(\varepsilon)\leq T$. As an application of the time-varying bang-bang property, we shall give our second main result: a kind of equivalence related to $\varepsilon(\tau)$ and $\tau(\varepsilon)$.

**    Theorem 1.2****.**

Let $M(\cdot)\in L^{\infty}_{+}(0,T)$. Then the map $\tau\mapsto\varepsilon(\tau)$ is strictly monotonically increasing and continuous from $[0,T)$ onto $[\varepsilon(0),\varepsilon_{T})$. Furthermore, it holds that

[TABLE]

Consequently, the maps $\tau\mapsto\varepsilon(\tau)$ and $\varepsilon\mapsto\tau(\varepsilon)$ are inverse of each other.

When $M(\cdot)\equiv M_{0}$ ($M_{0}>0$ is a constant), a kind of equivalence theorem of optimal time and target control problems has been discussed in [15]. In our work, for the time variant function $M(\cdot)$, we can also obtain that equivalence result.

We organize this paper as follows. In Section 2, we prove the time-varying bang-bang property (Theorem 1.1). In Section 3, we prove the equivalence theorem of optimal time and target control problems (Theorem 1.2).

2 Proof of Theorem 1.1

The following lemma is crucial in the proof of Theorem 1.1.

**    Lemma 2.1****.**

Under the assumption of Theorem 1.1, let $\tau(\varepsilon)$ be defined as (1.2). Then there exists an optimal control $u^{*}\in\mathcal{U}_{M}$, such that the optimal time problem (1.2) is solvable, i.e.,

[TABLE]

Proof.

Since $\mathcal{R}(y_{0},T)\cap\bar{B}(0,\varepsilon)\neq\emptyset$, there exist $\tau_{0}\in(0,T)$ and $u\in\mathcal{U}_{M}$, such that $y(T;\chi_{(\tau_{0},T)}u,y_{0})\in\bar{B}(0,\varepsilon)$. Let $\{\tau_{n}\}_{n=1}^{\infty}$ be a monotonically increasing sequence such that $\tau_{n}\rightarrow\tau(\varepsilon)$. Then, for each $n\in\mathbb{N}$, there exists $u_{n}\in\mathcal{U}_{M}$ such that

[TABLE]

Set

[TABLE]

Since $\|\tilde{u}_{n}\|_{L^{\infty}(0,T;L^{2}(\Omega))}\leq\|M\|_{L^{\infty}(0,T)}$, there exist a subsequence of $\{\tilde{u}_{n}\}$, still denoted by itself, and $u^{*}\in L^{\infty}(0,T;L^{2}(\Omega))$ such that

[TABLE]

Take $t_{0}\in(0,\tau(\varepsilon))$ to be the Lebesgue point of $u^{*}$, and $\lambda\in(0,\frac{\tau(\varepsilon)-t_{0}}{2})$. Then there exists $N_{0}\in\mathbb{N}$ such that $t_{0}+\lambda\leq\tau_{n}$ for all $n\geq N_{0}$. For any $\zeta\in L^{2}(\Omega)$, set

[TABLE]

Then $\|v\|_{L^{1}(0,T;L^{2}(\Omega))}=2\lambda\|\zeta\|<\infty$, and

[TABLE]

Hence

[TABLE]

By the arbitrary of $\zeta\in L^{2}(\Omega)$, we get $\tilde{u}^{*}(t_{0})=0$. Since the Lebesgue measure of the set of $u^{*}$’s Lebesgue points in $(0,\tau(\varepsilon))$ is equal to $\tau(\varepsilon)$, we have

[TABLE]

On the other side, by (2.1), the solution $y^{*}(\cdot;\chi_{(\tau(\varepsilon),T)}u^{*},y_{0})$ to

[TABLE]

satisfies

[TABLE]

for any $0<\delta<T$. Here $y_{n}\equiv y_{n}(\cdot;\chi_{(\tau_{n},T)}u_{n},y_{0})$ is the solution to the system

[TABLE]

Since $y_{n}(T;\chi_{(\tau_{n},T)}u_{n},y_{0})\in\bar{B}(0,\varepsilon)$, we get $y^{*}(T;\chi_{(\tau(\varepsilon),T)}u^{*},y_{0})\in\bar{B}(0,\varepsilon)$ by (2.3), which implies the optimal time $\tau(\varepsilon)$ is attainable and the optimal control $u^{*}$ exits.

We claim that $\|u^{*}(t)\|\leq M(t)$ for a.e. $t\in(\tau(\varepsilon),T)$.

Indeed, if there exists $\varepsilon_{0}>0$ and $E_{0}\subset[\tau(\varepsilon),T]$ with $|E_{0}|>0$ such that

[TABLE]

where $|E_{0}|$ represents the Lebesgue measure of $E_{0}$. Define

[TABLE]

It is obviously that $\zeta(x,t)$ is well-defined since $\|u^{*}(\cdot)\|>M(\cdot)+\varepsilon_{0}>0$ in $E_{0}$, and

[TABLE]

i.e., $\zeta\in L^{1}(\tau(\varepsilon),T;L^{2}(\Omega))$.

Since $\tilde{u}_{n}\rightarrow\tilde{u}^{*}$ weakly∗ in $L^{\infty}(0,T;L^{2}(\Omega))$, we get $u_{n}\rightarrow u^{*}$ weakly∗ in $L^{\infty}(\tau(\varepsilon),T;L^{2}(\Omega))$. For any $\epsilon\in(0,\frac{|E_{0}|\varepsilon_{0}}{2})$, there exists $N_{0}>0$, such that, for any $n\geq N_{0}$,

[TABLE]

Noting

[TABLE]

we obtain

[TABLE]

This contradicts (2.4). That proves our claim, and completes the proof.

Now we can prove Theorem 1.1.

Proof of Theorem 1.1.

The proof is long, we separate it to two steps.

Step 1. By Lemma 2.1, one knows that $\mathcal{R}(y_{0},\tau(\varepsilon),T)\cap\bar{B}(0,\varepsilon)\neq\emptyset$. We now show that $\mathcal{R}(y_{0},\tau(\varepsilon),T)\cap\bar{B}(0,\varepsilon)$ has only one point.

Otherwise, there exists at least two different $u_{1}^{*},u_{2}^{*}\in\mathcal{U}_{M}$ such that

[TABLE]

and

[TABLE]

Note that $\displaystyle\hat{u}=\frac{u_{1}^{*}}{2}+\frac{u_{2}^{*}}{2}\in\mathcal{U}_{M}$ and $\displaystyle\hat{y}(\cdot)=\frac{1}{2}y_{1}(\cdot;\chi_{(\tau(\varepsilon),T)}u_{1}^{*},y_{0})+\frac{1}{2}y(\cdot;\chi_{(\tau(\varepsilon),T)}u_{2}^{*},y_{0})$ is the solution to the system

[TABLE]

One can easily get $\displaystyle\hat{y}(T)=\frac{y_{1}}{2}+\frac{y_{2}}{2}\in\bar{B}(0,\varepsilon)$. Since $\bar{B}(0,\varepsilon)\subset L^{2}(\Omega)$ is strictly convex, we get $\hat{y}(T)$ is an inner point of $\bar{B}(0,\varepsilon)$. Hence there exists $\gamma>0$ such that $B(\hat{y}(T),\gamma)\subset B(0,\varepsilon)$. Let $\tilde{y}$ be the solution to the following system

[TABLE]

Then

[TABLE]

Choosing $\xi>0$ small enough such that $\|h_{\xi}\|<\gamma$, one has $\tilde{y}(T)\in\bar{B}(0,\varepsilon)$. That implies $\tau(\varepsilon)\geq\tau(\varepsilon)+\xi$, which is impossible. That completes the proof of Step 1.

Step 2. The optimal time control $u^{*}$ has the time-varying bang-bang property.

Since $\mathcal{R}(y_{0},\tau(\varepsilon),T)\cap\bar{B}(0,\varepsilon)$ has only one point (denoted by $y^{*}=y(T;u^{*},y_{0})$), and $\mathcal{R}(y_{0},\tau(\varepsilon),T)$ and $\bar{B}(0,\varepsilon)$ are two convex sets, by hyperplane separation theorem, there exists $\eta^{*}\in L^{2}(\Omega)$ such that

[TABLE]

Notice that the element in $\mathcal{R}(y_{0},\tau(\varepsilon),T)$ can be written by

[TABLE]

Then by (2.5), one can get

[TABLE]

i.e.,

[TABLE]

Here

[TABLE]

and

[TABLE]

Let $E_{0}$ be the Lebesgue points of $\bar{u}^{*}(\cdot)$ in $[\tau(\varepsilon),T]$. For given $t_{0}\in E_{0}$, choosing

[TABLE]

where $\zeta\in L^{2}(\Omega)$ with $\|\zeta\|\leq 1$, and $\lambda\in(0,\min\{t_{0}-\tau(\varepsilon),T-t_{0}\})$. Setting $A=(\tau(\varepsilon),T)\setminus(t_{0}-\lambda,t_{0}+\lambda)$, by (2.6) we have

[TABLE]

i.e.,

[TABLE]

Hence

[TABLE]

By the arbitrary of $\zeta$, we get

[TABLE]

i.e.,

[TABLE]

This implies that

[TABLE]

By $\bar{u}^{*}\in\mathcal{U}_{1}$ we get $\|\bar{u}^{*}(t_{0})\|=1$. (2.8), together with (2.7) and $|E_{0}|=\hat{\tau}(\varepsilon)$, yields

[TABLE]

From above, we get the time optimal control $u^{*}$ satisfies the time-varying bang-bang property. That completes the proof.

3 Proof of Theorem 1.2

In order to show Theorem 1.2, we need a lemma in the following:

**    Lemma 3.1****.**

Let $\varepsilon(\tau)$ be defined as (1.4). Then there exists $u^{*}\in\mathcal{U}_{M}$ such that

[TABLE]

Proof.

Let $\{u_{n}\}_{n=1}^{\infty}\subset\mathcal{U}_{M}$ be the minimal sequence of (1.4), i.e.,

[TABLE]

Here $y(\cdot;\chi_{(\tau,T)}u_{n},y_{0})$ is the solution to the system

[TABLE]

Since

[TABLE]

there exist a subsequence of $\{u_{n}\}_{n=1}^{\infty}$, still denoted by itself, and $u^{*}\in L^{\infty}(\tau,T;L^{2}(\Omega))$ such that

[TABLE]

Therefore, we have

[TABLE]

for any $\delta\in(0,T]$. Here $y(\cdot;\chi_{(\tau,T)}u^{*},y_{0})$ is the solution to the system

[TABLE]

Hence,

[TABLE]

Now, we shall show that $u^{*}\in\mathcal{U}_{M}$.

By contradiction. We assume there exist $\epsilon_{0}>0$ and $E_{0}\subset(\tau,T)$ with $|E_{0}|>0$ such that

[TABLE]

Then

[TABLE]

Taking $\zeta(x,t)=\chi_{{}_{E_{0}}}\frac{u^{*}(t)}{\|u^{*}(t)\|}$, by the assumption of $u^{*}$, we get

[TABLE]

i.e., $\zeta\in L^{1}(0,T;L^{2}(\Omega))$. Hence, by $u_{n}\rightarrow u^{*}$ weakly∗ in $L^{\infty}(\tau,T;L^{2}(\Omega))$, one has

[TABLE]

as $n\rightarrow\infty$. On the other hand, since $u_{n}\rightarrow u^{*}$ weakly∗ in $L^{\infty}(\tau,T;L^{2}(\Omega))$ and $E_{0}\subset(\tau,T)$, we get $u_{n}\rightarrow u^{*}$ weakly∗ in $L^{\infty}(E_{0};L^{2}(\Omega))$. Therefore, by $\|\zeta(t)\|=\left\|\chi_{{}_{E_{0}}}\frac{u^{*}(t)}{\|u^{*}(t)\|}\right\|=1$ and $\|u_{n}(t)\|\leq M(t)$ for a.e. $t\in(\tau,T)$, one gets

[TABLE]

which is contradict with (3.1). That completes the proof.

Now, we are in the position to prove Theorem 1.2.

Proof of Theorem 1.2.

We carry out the proof by three steps as follows.

Step 1. We shall show that $\tau\mapsto\varepsilon(\tau)$ is strictly monotonically increasing.

Let $0\leq\tau_{1}<\tau_{2}<T$. For $\tau_{2}$, by Theorem 1.1, there exists a unique $u_{2}\in\mathcal{U}_{M}$ such that

[TABLE]

Now, take

[TABLE]

Then

[TABLE]

By the definition of $\varepsilon(\tau)$ we get

[TABLE]

In other words, $\tau\mapsto\varepsilon(\tau)$ is a monotonically increasing function.

Now, we show that $\tau\mapsto\varepsilon(\tau)$ is strictly monotonically increasing. If not, suppose that $\varepsilon(\tau_{1})=\varepsilon(\tau_{2})$. Then, there exist $u_{1},u_{2}\in\mathcal{U}_{M}$, such that

[TABLE]

Taking

[TABLE]

one can easily check that

[TABLE]

By Theorem 1.1, the optimal control is unique. Hence, we get

[TABLE]

By the bang-bang property of minimal control, we have

[TABLE]

That is impossible, since $\hat{u}_{2}(t)=0$ for a.e. $t\in(\tau_{1},\tau_{2})$ and $M(t)>0$ for a.e. $t\in(0,T)$. Therefore, $\tau\mapsto\varepsilon(\tau)$ is a strictly monotonically increasing function.

Step 2. We shall show that $\tau\mapsto\varepsilon(\tau)$ is continuous.

For $\tau,\hat{\tau}\in[0,T)$ with $\tau<\hat{\tau}$, and for each $u\in\mathcal{U}_{M}$, the solutions to (1.1) are

[TABLE]

and

[TABLE]

respectively. Then

[TABLE]

where $C$ is a constant independent of $\tau,\hat{\tau}$. Since $u\in\mathcal{U}_{M}$ (i.e., $\|u(t)\|\leq M(t)\leq M_{T}$ for a.e. $t\in[0,T]$), by (3.2), we get \mbox{dist}\big{(}\mathcal{R}(y_{0},\tau,T),\mathcal{R}(y_{0},\hat{\tau},T)\big{)}<C|\tau-\hat{\tau}|. Here dist is the distance of two reachable sets $\mathcal{R}(y_{0},\tau,T)$ and $\mathcal{R}(y_{0},\hat{\tau},T)$. Hence $\tau\mapsto\varepsilon(\tau)$ is a continuous function.

Step 3. We shall prove (1.5).

(1) We show that $\varepsilon=\varepsilon(\tau(\varepsilon))$ for $\varepsilon\in[\varepsilon(0),\varepsilon_{T})$.

Let $\varepsilon\in[\varepsilon(0),\varepsilon_{T})$. By Step 1 in the proof of Theorem 1.1, there exist $\tau(\varepsilon)\in[0,T)$ and $u\in\mathcal{U}_{M}$ such that

[TABLE]

For such $\tau(\varepsilon)\in[0,T)$, denote $\tau^{*}=\tau(\varepsilon)$. We consider the following problem

[TABLE]

By (3.3), we can obtain

[TABLE]

By lemma 3.1, there exists a control $u^{*}\in\mathcal{U}_{M}$ such that

[TABLE]

Now, taking

[TABLE]

we have

[TABLE]

By the definition of $\tau(\varepsilon)$ and (3.3) we get

[TABLE]

which, together with (3.4), yields

[TABLE]

(2) We show that $\tau=\tau(\varepsilon(\tau))$ for $\tau\in[0,T)$.

Let $\tau\in[0,T)$. By Lemma 3.1, there exists $u\in\mathcal{U}_{M}$ such that

[TABLE]

For the given $\varepsilon(\tau)\in[\varepsilon(0),\varepsilon_{T})$, denote $\varepsilon^{*}=\varepsilon(\tau)$. Consider the following problem

[TABLE]

Then we have

[TABLE]

By Lemma 2.1, there exists a control $u^{*}\in\mathcal{U}_{M}$ such that

[TABLE]

Then by the definition of $\varepsilon(\tau)$ and (3.5) we get

[TABLE]

which, together with (3.6), yields $\tau=\tau(\varepsilon(\tau))$ for $\tau\in[0,T)$. That completes the proof.

Acknowledgement

The third author gratefully acknowledges Dr. Yubiao Zhang, Wuhan University, for his helpful discussion during this work.