- Research
- Open access
- Published:
Some optimal control problems of heat equations with weighted controls
Boundary Value Problems volume 2017, Article number: 148 (2017)
Abstract
In this paper, the time and norm optimal control problems of controlled heat equations with a weight function are considered. For the time optimal problems, we study the following two cases: one is for equations with multi-domain control under null controllability, and the other is for equations under approximate null controllability. We prove the solvability, and obtain the bang-bang principle of the time optimal controls for aforementioned both cases. For the norm optimal control problems, we focus on equations with multi-time and multi-domain control, and present the solvability of these problems.
1 Introduction
Let T be a positive number and Ω be an open bounded domain with smooth boundary in \(\mathbb{R}^{N}\), \(N\geq1\). Let \(K\in \mathbb{Z}^{+}\), \(\{E_{i}\}\equiv\{E_{i}\}_{i=1}^{K}\) be a sequence of Lebesgue measurable subsets of \((0,T)\) and \(\{\omega_{i}\}\equiv\{\omega_{i} \}_{i=1}^{K}\) be a sequence of positive Lebesgue measurable subsets of Ω with \(\omega_{i}\cap\omega_{j}=\emptyset\), for all \(i,j\in\{1,2,\ldots,K\}\) and \(i\neq j\). Denote by \(\chi_{E_{i}}\), \(\chi_{\omega_{i}}\) the characteristic function of \(E_{i}\), \(\omega_{i}\), respectively, for each \(i\in\{1,2,\ldots,K\}\). Consider the following controlled heat equation with a weight function:
where \(\rho\in L^{2}(\Omega)\) is a weight function satisfying \(0<\rho(x)\leq1\) for a.e. \(x\in\Omega\), and \(0\neq y_{0}\in L^{2}(\Omega)\) is a given function. We denote the solution to (1.1) by \(y(\cdot, \cdot; \{ \chi_{{E_{i}}}\chi_{\omega_{i}}u_{i}\}, y_{0})\). For simplicity, when \(E_{i}=(0,T)\) for all \(i\in\{1,2,\ldots ,K\}\), we write \(y(\cdot, \cdot; \{\chi_{\omega_{i}}u_{i}\}, y_{0})\) for \(y(\cdot, \cdot; \{ \chi_{{E_{i}}}\chi_{\omega_{i}}u_{i}\}, y_{0})\); furthermore, when \(K=1\), write ω, \(y(\cdot ,\cdot ;\chi_{\omega}u,y_{0})\) for \(\omega_{i}\), \(y(\cdot ,\cdot ;\{\chi_{\omega_{i}}u_{i}\}, y_{0})\), respectively.
The weight function ρ in equation (1.1) is meaningful, which stands for the different influence of the control function in different location.
As is well known, optimization is one of the most important problems in control theory and there exist some work on this topic (see, e.g., [1–3]). Roughly speaking, the goal of optimization is to improve a variable in order to maximize a benefit (or minimize a cost). The time and norm optimal control problems are important and interesting branches of optimization. For the deterministic systems, the reader can refer [4] to obtain recent results and find open problems. The reader can also refer [5–10] for controlled heat equations. For the stochastic ones, the norm optimal control problems were considered in [11, 12] for controlled stochastic ordinary differential equations, and in [13] for controlled stochastic heat equations.
In this paper, we shall consider the time and norm optimal control problems of heat equations with a weight function. In Section 2, we consider two kind time optimal control problems: one is for equations with multi-domain control under null controllability, and the other is for equations under approximate null controllability. We obtain the bang-bang principle of the time optimal controls for these two problems. In Section 3, we consider the norm optimal problems with multi-time and multi-domain control, and we obtain the solvability of these problems.
2 Time optimal control problems
In this section, we first state two time optimal control problems, and then study the solvability of these problems, obtain the bang-bang property of the time optimal controls. Throughout this section, for all \(i\in\{1,2,\ldots ,K\}\), \(E_{i}=(0,T)\), and
When \(K=1\), for simplicity, we write \(\mathcal{U}_{\mathrm{ad}}\) for \(\mathcal{U}_{\mathrm{ad}}^{i}\).
In the following, we consider the following two time optimal control problems subject to (1.1):
Problem (TP1)
Problem (TP2)
For \(K=1\),
Here and in what follows, we denote by \(B(u, r)\) the open ball in \(L^{2}(\Omega)\) with center \(u\in L^{2}(\Omega)\) and radius \(r>0\), and by \(\bar{B}(u,r)\) the closed ball in \(L^{2}(\Omega)\) with center \(u\in L^{2}(\Omega)\) and radius \(r>0\).
In order to obtain the solvability of Problem (TP1), we assume that there exists a constant \(M>0\) such that
Notice that the hypothesis (2.2) is reasonable: for a single control system
its optimal time
as \(M\rightarrow\infty\).
It is obvious that Problem (TP1) is related to null controllable problem of (1.1), while Problem (TP2) is related to approximately controllable problem of (1.1). It is well known that, when \(K=1\) and \(\rho\equiv1\), the system (1.1) is null controllable for the measurable control domain ω (see [5]), even if the characteristic function \(\chi_{\omega}\) can be relaxed by a measurable function \(\beta\in L^{2}(\Omega)\) with \(0\leq\beta\leq1\) for a.e. \(x\in\Omega\) and \(\int_{\Omega}\beta^{2}(x)\,dx=\alpha\vert \Omega\vert\) (see [6]). Here \(\alpha\in(0,1)\) is a given constant and \(\vert\Omega\vert\) is the Lebesgue measure of Ω. It is natural that there exist a positive constant T and a control \(u\in L^{\infty}(0,T; L^{2}(\Omega))\) such that \(y(x, T;u,y_{0})=0\) (see [5, 6]).
The following result is related to the solvability of Problem (TP1).
Theorem 2.1
Let \(\{M_{i}\}\) be a given positive real number sequence satisfying (2.2). Then there exists \(T^{*}>0\), such that \(T^{*}\) is the solution to Problem (TP1). Moreover, for each \(i=1,2,\ldots ,K\), there exists a unique \(u_{i}^{*}\in L^{\infty}(0, T^{*}; L^{2}( \Omega))\), such that
i.e., the time optimal controls sequence of Problem (TP1) has the bang-bang property.
The following lemma is needed in proving Theorem 2.1, which comes from [5, 9].
Lemma 2.2
Let \(E\subset[0,T]\) and \(\omega\subset\Omega\) be two positive measurable sets. Then, for each \(y_{0}\in L^{2}(\Omega)\), there is a bounded control function \(u(\cdot)\in L^{\infty}(0,T; L^{2}(\Omega ))\) with
such that the solution to the equation
satisfies \(y(\cdot, T; u, y_{0})=0\). Here \(C=C(\Omega, T, \vert E\vert, \vert \omega\vert)\) is a constant.
We are now in the position to prove Theorem 2.1.
Proof of Theorem 2.1
Since the proof is long, we separate it into two steps.
Step 1. For fixed \(i_{0}\in\{1,2,\ldots ,K\}\), consider the following system:
By Lemma 2.1 of [6], we know that there exist a control \(u_{i_{0}}\in\mathcal{U}_{\mathrm{ad}}^{i_{0}}\) and T, such that \(y(\cdot ,T,\chi_{\omega_{i_{0}}}u_{i_{0}},y_{0})=0\). we also know that \(0\neq y_{0}\in L^{2}(\Omega)\) is a given function. Therefore,
Hence, there exists a sequence \(\{T_{n}\}\), such that \(\{T_{n}\}\) is a monotone decreasing sequence with \(y(\cdot, T_{n}; \{\chi_{\omega_{i}}u _{i}^{n}\}, y_{0})=0\) and
Without loss of generality, we assume that \(T_{n}\leq T^{*}+1\) for all \(n\in\mathbb{N}\). Then \(y^{n}\equiv y(\cdot, \cdot; \{\chi_{\omega _{i}}u_{i}^{n}\}, y_{0})\) is a solution to the following equation:
Now, denote
Then
solves the following system:
Moreover, by the definition of \(\tilde{y}^{n}\), it is easy to see that \(\tilde{y}^{n}\) solves the following system:
Note that \(\Vert \tilde{u}_{1}^{n}\Vert _{L^{2}(\Omega)}\leq M_{1}\) for all \(n\in\mathbb{N}\). Then there exist a subsequence \(\{\tilde{u}_{1} ^{n_{1}}\}\subset L^{\infty}(0,T^{*}+1; L^{2}(\Omega))\) of \(\{\tilde{u}_{1}^{n}\}\) and \(\tilde{u}_{1}^{0}\in L^{\infty}(0,T^{*}+1; L^{2}(\Omega))\) such that
Similarly, since \(\Vert u_{2}^{n_{1}}\Vert _{L^{2}(\Omega)}\leq M_{2}\) for all \(n_{1}\in\mathbb{N}\), there exist a subsequence \(\{\tilde{u}_{2}^{n _{2}}\}\) of \(\{\tilde{u}_{2}^{n_{1}}\}\) and \(\tilde{u}_{2}^{0}\in L ^{\infty}(0,T^{*}+1; L^{2}(\Omega))\) such that
By inductive argument, for each \(i\in\{1,2,\ldots ,K\}\), there exist a subsequence \(\{\tilde{u}_{i}^{n_{i}}\}\) of \(\{\tilde{u}_{i}^{n_{i-1}} \}\) and \(\tilde{u}_{i}^{0}\in L^{\infty}(0,T^{*}+1; L^{2}(\Omega))\) such that
By the diagram argument, for all \(i\in\{1,2,\ldots ,K\}\), we can abstract a subsequence \(\{\tilde{u}_{i}^{n_{n}}\}\) of \(\{\tilde{u}_{i}^{n}\}\) such that
Since \(\omega_{i}\cap\omega_{j}=\emptyset\) for all \(i, j\in\{ 1,2,\ldots ,K \}\) with \(i\neq j\), one can get
On the other hand, \(\tilde{y}^{n_{n}}\) is the solution to the following system:
Then there exist a subsequence of \(\{\tilde{y}^{n_{n}}\}\), still so denoted, and \(\tilde{y}^{0}\) such that
and
Note that \(\tilde{y}^{n_{n}}=0\) in \(\Omega\times[T_{n_{n}}, T^{*}+1)\). Since for \(n\in\mathbb{N}\), one has \(\tilde{y}^{n_{n}}=0\) in \(\Omega\times[T_{n_{n}}, T^{*}+1)\),
We get \(\tilde{y}^{0}=0\) in \(\Omega\times[T_{n_{n}}, T^{*}+1)\). We get \(\tilde{y}^{0}=0\) in \([T^{*}, T^{*}+1)\) since \(T_{n_{n}}\rightarrow T^{*}\) as \(n\rightarrow\infty\). Take
By the fact that \(\tilde{y}^{0} \in C((0,T]; L^{2}(\Omega))\), \(y^{0}\) is the solution to the following system:
which implies that \(\{u_{i}^{0}\}\) are the desired controls respect to the optimal time \(T^{*}\).
Step 2. In the following, we shall show that the time optimal control of Problem (TP1) has the bang-bang property. Otherwise, we suppose that there exist \(i_{0}\in\{1,2,\ldots ,K\}\) and a subset \(E^{0}\subset[\alpha, T^{*}-\alpha]\) with positive measure for some \(\alpha>0\) and a positive number \(\varepsilon _{0}\), such that
where \(u_{i_{0}}^{*}\) is the time optimal control respect to \(T^{*}\). It is obvious that the solution to (1.1) satisfies
and, for each \(t\in E^{0}\), \(B(u_{i_{0}}^{*}(t), \frac{\varepsilon _{0}}{2}) \subset B(0, M_{i_{0}})\).
We denote by \(e^{\Delta t}\) the semigroup generated by Δ with the Dirichlet boundary condition. Set
Considering the following system:
where \(E_{\delta}^{0}\) is the set \(\{t\mid t+\delta\in E^{0}\}\). By Lemma 2.2, there exist positive constants \(\delta_{1}\) and \(L=L(\Omega, T, \vert E_{0}\vert, \vert\omega_{i_{0}}\vert)\), such that, for each δ with \(0<\delta\leq\delta_{1}\), there is a control \(w_{\delta}\) in the space \(L^{\infty}(0,T^{*}-\delta; L^{2}(\Omega ))\) with the estimate
and the solution to (2.5) satisfies
On the other hand, by (2.4), there exists a positive number \(\delta_{2}\) such that, for each positive number δ with \(\delta\leq\delta_{2}\), one has
Therefore, for each \(\delta\leq\delta_{0}\equiv\min\{\delta_{1}, \delta_{2}\}\), there exists a control \(w_{\delta}\) satisfying
and the corresponding solution to (2.5) satisfies (2.6).
Set
It is obvious that \(v_{i}\in\mathcal{U}_{\mathrm{ad}}^{i}\) for all \(i\in\{1,2,\ldots ,K\}\). Consider the following system:
For any \(0<\delta<\min\{\frac{T^{*}}{2}, \delta_{0}\}\), it is easy to check that the solution to above equation satisfies
which shows that \(\{v_{i}\}\) are the desired controls such that \(y(\cdot, T^{*}-\delta; \{\chi_{\omega_{i}}v_{i}\}, y_{0})=0\). It contradicts the definition of \(T^{*}\) and we have proved the theorem. □
Remark 2.3
There are some relations between the optimal control problem of (1.1) and shape design problem. For more about shape design problem see [8, 14–16].
Before stating the results on Problem (TP2), we define the following reachable set:
for each \(T\in(0,+\infty)\).
Theorem 2.4
For any given positive constant ε, Problem (TP2) has a solution \(T_{\varepsilon }^{*}\), and \(\mathcal{R}(T_{\varepsilon }^{*})\cap\bar{B}(0,\varepsilon )\) has only one point belonging to the boundary of \(B(0,\varepsilon )\). Moreover, the corresponding time optimal control \(u_{\varepsilon }^{*}\) is unique and has the bang-bang property.
Proof
Since the proof is long, we separate it into the following several steps.
Step 1. We shall show that there exists at least one time optimal control, i.e., there exists at least one \(u^{*}\in \mathcal{U}_{\mathrm{ad}}\) such that \(y(\cdot, T_{\varepsilon }^{*}; u^{*}, y_{0})\in\bar{B}(0,\varepsilon )\).
Let \(\{T_{n}\}\) be a monotone decreasing sequence such that \(T_{n}\rightarrow T_{\varepsilon }^{*}\) as \(n\rightarrow+\infty\), then there exists a sequence \(\{u_{n}\}\subset\mathcal{U}_{\mathrm{ad}}\) such that \(y(\cdot, T_{n}; \chi_{\omega}u_{n}, y_{0})\in \bar{B}(0,\varepsilon )\). Set
Since \(M\in L^{\infty}(0,T_{1}; L^{2}(\Omega))\), \(\{\tilde{u}_{n}\}\) is a bounded sequence in \(L^{\infty}(0,T_{1}; L^{2}(\Omega))\). Then there exist \(\tilde{u}^{*}\in L^{\infty}(0,T_{1}; L^{2}(\Omega))\) and a subsequence of \(\{u_{n}\}\), still so denoted, such that \(u_{n} \rightarrow u^{*}\) weakly∗ in \(L^{\infty }(0,T_{1};L^{2}(\Omega))\). Moreover,
Therefore the solution \(y_{n}(\cdot, \cdot ; \chi_{\omega}\tilde {u}_{n}, y _{0})\) to the following system:
satisfies \(y_{n}(\cdot, t; \chi_{\omega} \tilde{u}_{n}, y_{0})\in \bar{B}(0,\varepsilon )\) for \(t\geq T_{n}\). Denote by \(y^{*}\) the solution to the following system:
Then
for any \(\delta>0\). Since \(y_{n}(\cdot, t; \chi_{\omega} \tilde{u}^{n}, y_{0})\in\bar{B}(0,\varepsilon )\) for \(t\geq T_{n}\), \(y^{*}(\cdot, t; \chi _{\omega} \tilde{u}^{*}, y_{0})\in\bar{B}(0,\varepsilon )\) for all \(t\geq T_{n}\) and \(n\in\mathbb{N}\). Hence \(y^{*}(\cdot, T_{\varepsilon }^{*}; \chi _{\omega} \tilde{u}^{*}, y_{0})\in\bar{B}(0,\varepsilon )\). Set
Then \(y^{*}\) satisfies the following system:
and \(y^{*}(\cdot, T_{\varepsilon }^{*}; \chi_{\omega} u^{*}, y_{0})\in\bar{B}(0,\varepsilon )\).
Claim: \(\Vert u^{*}(t)\Vert _{L^{2}(\Omega)}\leq M(t)\) for a.e. \(t\in[0,T _{\varepsilon }^{*}]\). Indeed, let \(\{\zeta_{k}\}_{k\in\mathbb{N}}\) be the countable density subset of \(L^{2}(\Omega)\). Denote by \(\mathcal{L}\) the Lebesgue point of \(\langle u_{n}(t), \zeta_{k}\rangle\), \(t\in[0, T_{\varepsilon }^{*}]\), where \(\langle\cdot, \cdot\rangle\) is the inner product of \(u_{n}(t)\) and \(\zeta_{k}\) in \(L^{2}(\Omega)\). Since \(\langle u_{n}(t), \zeta_{k}\rangle, \langle u^{*}(t), \zeta_{k} \rangle\in L^{\infty}(0,T_{\varepsilon }^{*})\), for each
we have
By virtue of
\(u_{n}\rightarrow u^{*}\) weakly∗ in \(L^{\infty}(0,T_{\varepsilon }^{*}; L ^{2}(\Omega))\), and the arbitrary of \(\delta>0\), we get
Since \(\{\zeta_{k}\}\) is dense in \(L^{2}(\Omega)\), we have
for each \(\zeta\in L^{2}(\Omega)\). That implies \(u_{n}(t_{0})\rightarrow u^{*}(t_{0})\) weakly in \(L^{2}(\Omega)\), and hence \(\Vert u^{*}(t _{0})\Vert _{L^{2}(\Omega)}\leq\liminf_{n\rightarrow\infty }\Vert u_{n}(t _{0})\Vert _{L^{2}(\Omega)}\leq M \). Applying
we obtain \(\Vert u^{*}(t)\Vert _{L^{2}(\Omega)} \leq M \) for a.e. \(t\in[0,T_{\varepsilon }^{*}]\). That proves the claim.
Step 2. We show that \(\mathcal{R}(T_{\varepsilon }^{*})\cap\bar {B}(0,\varepsilon )\) has only one point. If so, it is obvious that this point belongs to the boundary of \(B(0,\varepsilon )\).
In Step 1 we get \(\mathcal{R}(T_{\varepsilon }^{*})\cap\bar{B}(0,\varepsilon )\neq \emptyset\). By contradiction, we assume that \(\mathcal{R}(T_{\varepsilon } ^{*})\cap\bar{B}(0,\varepsilon )\) has at least two points, i.e., there exist \(y_{1}\equiv y(\cdot, T_{\varepsilon }^{*}; \chi_{\omega} u_{1}^{*}, y_{0})\), \(y_{2}\equiv y(\cdot, T_{\varepsilon }^{*};\chi_{\omega} u_{2}^{*}, y_{0})\in \mathcal{R}(T_{\varepsilon }^{*})\cap\bar{B}(0,\varepsilon )\) with \(y_{1}\neq y_{2}\). It is obvious that \(u_{1}^{*}\neq u_{2}^{*}\) in \(\mathcal{U}_{\mathrm{ad}}\). Define
Since \(\hat{u}\in\mathcal{U}_{\mathrm{ad}}\), and \(B(0,\varepsilon )\) is strongly convex in \(L^{2}(\Omega)\), we get \(\hat{y}\equiv y(\cdot, T_{\varepsilon }^{*}; \chi_{\omega} \hat{u}, y_{0})= \frac{1}{2}y_{1}+\frac{1}{2}y_{2}\) is an inner point of \(B(0,\varepsilon )\), i.e., there exists \(\gamma>0\) such that \(B(\hat{y}, \gamma )\subset B(0,\varepsilon )\).
For any \(\xi>0\), define
It is easy to check that
Hence, ξ can be chosen small enough such that \(\Vert h_{\xi }\Vert _{L ^{2}(\Omega)}\leq\gamma\). Therefore, we can get \(y(\cdot, T_{\varepsilon } ^{*}-\xi; \chi_{\omega} \hat{u}, y_{0})\in\bar{B}(0,\varepsilon )\). This contradicts the optimal time \(T_{\varepsilon }^{*}\). Subsequently, the set \(\mathcal{R}(T_{\varepsilon }^{*})\cap\bar{B}(0,\varepsilon )\) has only one point, and this point belongs to the boundary of \(B(0,\varepsilon )\).
Step 3. The time optimal control \(u^{*}\) has the bang-bang property.
Since \(\mathcal{R}(T_{\varepsilon }^{*})\cap\bar{B}(0,\varepsilon )\) has only one point (denote this point by \(y^{*}=y(\cdot, T_{\varepsilon }^{*}; \chi_{\omega} u^{*}, y_{0})\)), and \(\mathcal{R}(T_{\varepsilon }^{*})\) and \(\bar{B}(0,\varepsilon )\) are two convex sets, by hyperplane separation theorem, there exists \(\eta^{*}\in L^{2}(\Omega)\) such that
Notice that \(y\in\mathcal{R}(T_{\varepsilon }^{*})\) can be written by
Then (2.11) can be written as
Here
and
Hence, we have
For given \(t_{0}\in E_{0}\), choosing
where \(\zeta\in L^{2}(\Omega )\), we get
i.e.,
This implies that
by \(\bar{u}^{*}\in\mathcal{U}_{1}\). Equation (2.13), together with (2.12) and \(\vert E_{0}\vert=T_{\varepsilon }^{*}\), yields
From the above, we get the time optimal control’s bang-bang property. That completes the proof. □
3 Norm optimal control problems with multi-time and multi-domain controls
In this section, let \(T\in \mathbb{R}^{+}\), \(K\in \mathbb{Z}^{+}\) be given finite constants, and \(\Pi^{K}\) be time partition of \([0, T]\) defined by
For any \(i\in\{1,2,\ldots ,K\}\), set \(I_{i}=(t_{i-1}, t_{i}]\). Taking \(E_{i}=I_{i}\), we can rewrite (1.1) as
It is obvious that the system is null controllable (see [5, 17]). For any given partition \(\Pi^{K}\), by standard minimizing sequence method, there exists a solution to the following norm optimal control problem:
We are interested in the partition’s existence of the following norm optimal control problem:
Problem (NP)
We have the following solvability result on Problem (NP).
Theorem 3.1
For any \(K>1\), there exists at least one solution to Problem (NP).
Proof
It is obviously that \(N_{K}^{*}<\infty\). Let \(\{\Pi_{K}^{n}\}\) be the partition sequence such that
Then there exists a control sequence \(\{u_{1}^{n},u_{2}^{n},\ldots ,u _{K}^{n}\}_{n=1}^{\infty}\), such that
with
Since \(\{t_{1}^{n}\}\subset[0,T]\) is a bounded sequence, there exist a subsequence of \(\{t_{1}^{n}\}\), still so denoted, and \(t_{1}^{0}\) such that \(t_{1}^{n}\rightarrow t_{1}^{0}\) as \(n\rightarrow\infty\). Also, there exist a subsequence of \(\{t_{2}^{n}\}\), still so denoted, and \(t_{2}^{0}\) satisfying \(t_{2}^{n}\rightarrow t_{2}^{0}\) as \(n\rightarrow \infty\). Moreover, since \(t_{1}^{n}\leq t_{2}^{n}\) for all \(n\in\mathbb{N}\), we have \(t_{1}^{0}\leq t_{2}^{0}\). By the same line, we can get \(\{t_{i}^{0}\}_{i=1}^{K}\) satisfying
Denote the above partition by \(\Pi_{K}^{0}\). We also can define \(I_{i}^{0}\) for each \(i\in\{1,2,\ldots ,K\}\). Naturally, one has
Here and in what follows, we define \(I_{i}^{n}\triangle I_{i}^{0} \equiv(I_{i}^{n}-I_{i}^{0})\cup(I_{i}^{0}-I_{i}^{n})\) for all i, n.
On the other hand, since \(\{u_{1}^{n},u_{2}^{n},\ldots ,u_{K}^{n}\}_{n=1} ^{\infty}\) is bounded, \(\{u_{i}^{n}\}_{n=1}^{\infty}\) is also a bounded sequence in \(L^{\infty}(0,T;L^{2}(\Omega))\) for all \(i=1,\ldots, K\). Then, for each \(i=1,\ldots, K\), there exist a subsequence of \(\{u_{i}^{n}\}_{n=1}^{\infty}\), still so denoted, and \(u_{i}^{0}\in L^{\infty}(0,T; L^{2}(\Omega))\) such that
For each \(v\in L^{1}(0,T;L^{2}(\Omega))\), we have
By (3.6), we get
By (3.5) and the absolutely continuity of \(v\in L^{1}(0,T; L ^{2}(\Omega))\), we have
Equation (3.7), together with (3.8) and (3.9), yields
That implies, for all \(i\in\{1,\ldots, K\}\),
Therefore, one gets
and
Let \(y^{0}\) be the solution to the following system:
By (3.10) we get
for every \(\delta>0\). Since \(y^{n}(T)=0\), immediately we have
Combining (3.12), (3.13) and (3.11), we complete the proof. □
Now, let us consider an alteration of system (3.2):
where
By the aforementioned discussion, this system is null controllable, and for given partition \(\Pi_{K}\) and map σ, there exists at least a solution to the following norm optimal control problem:
Let us consider the solvability of the following norm optimal control problem:
Since the number of σ is finite, it is obvious that there exists \(\sigma ^{*}\) satisfying \(N^{*}=N(\sigma ^{*})\). Hence, we obtain the following result.
Proposition 3.2
Let \(\{I_{i}\}_{i=1}^{K}\), \(\{\omega_{k}\}_{k=1}^{K}\) be defined as before. Then there exists at least a solution to the problem (3.15).
References
Fattorini, HO: Infinite-Dimensional Optimization and Control Theory. Encyclopedia of Mathematics and Its Applications, vol. 62. Cambridge University Press, Cambridge (1999)
Lü, Q, Zhang, X: General Pontryagin-Type Stochastic Maximum Principle and Backward Stochastic Evolution Equations in Infinite Dimensions. SpringerBriefs in Mathematics. Springer, Cham (2014)
Yong, J, Zhou, XY: Stochastic Controls: Hamiltonian Systems and HJB Equations. Applications of Mathematics (New York), vol. 43. Springer, New York (1999)
Fattorini, HO: Time and norm optimal controls: a survey of recent results and open problems. Acta Math. Sci. Ser. B Engl. Ed. 31, 2203-2218 (2011)
Apraiz, J, Escauriaza, L, Wang, G, Zhang, C: Observability inequalities and measurable sets. J. Eur. Math. Soc. 16, 2433-2475 (2014)
Guo, B-Z, Xu, Y, Yang, D-H: Optimal actuator location of minimum norm controls for heat equation with general controlled domain. J. Differ. Equ. 261, 3588-3614 (2016)
Guo, B-Z, Yang, D-H, Zhang, L: On optimal location of diffusion and related optimal control for null controllable heat equation. J. Math. Anal. Appl. 433, 1333-1349 (2016)
Guo, B-Z, Yang, D-H: Optimal actuator location for time and norm optimal control of null controllable heat equation. Math. Control Signals Syst. 27, 23-48 (2015)
Wang, G: \(L^{\infty}\)-null controllability for the heat equation and its consequences for the time optimal control problem. SIAM J. Control Optim. 47, 1701-1720 (2008)
Wang, G, Xu, Y, Zhang, Y: Attainable subspaces and the bang-bang property of time optimal controls for heat equations. SIAM J. Control Optim. 53, 592-621 (2015)
Wang, Y, Yang, D-H, Yong, J, Yu, Z: Exact controllability of linear stochastic differential equations and related problems. Math. Control Relat. Fields 7(2), 305-345 (2017)
Wang, Y, Zhang, C: The norm optimal control problem for stochastic linear control systems. ESAIM Control Optim. Calc. Var. 21, 399-413 (2015)
Yang, D-H, Zhong, J: Observability inequality of backward stochastic heat equations for measurable sets and its applications. SIAM J. Control Optim. 54, 1157-1175 (2016)
Guo, B-Z, Yang, D-H: On convergence of boundary Hausdorff measure and application to a boundary shape optimization problem. SIAM J. Control Optim. 51, 253-273 (2013)
Guo, B-Z, Yang, D-H: Some compact classes of open sets under Hausdorff distance and application to shape optimization. SIAM J. Control Optim. 50, 222-242 (2012)
Yang, D-H: Shape optimization of stationary Navier-Stokes equation overclasses of convex domains. Nonlinear Anal. 71, 6202-6211 (2009)
Zuazua, E: Controllability of partial differential equations. Manuscript (2006)
Acknowledgements
The authors are grateful to the anonymous referees for helpful comments and suggestions, which greatly improved the presentation of this paper.
Author information
Authors and Affiliations
Contributions
All authors read and approved the final manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors declare that there are no competing interests regarding the publication of this article.
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Liu, S., Liu, D. & Wang, G. Some optimal control problems of heat equations with weighted controls. Bound Value Probl 2017, 148 (2017). https://doi.org/10.1186/s13661-017-0880-z
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13661-017-0880-z