1 Introduction
Optimal control problems appear frequently in the operation of physical, social, economic processes, and other fields, and the numerical solutions of optimal control problems are extremely important for better performance of those fields. Therefore, one needs some efficient numerical methods to approximate the solutions of optimal control problems. Finite element methods seem to be the most popular used numerical methods in solving optimal control problems. Meanwhile, other numerical methods, such as the spectral method, the mixed finite element method and the finite volume method have also been applied to approximate some optimal control problems. For example, there has been done much work on the finite element method for optimal control problems [11, 13‐15, 17], the spectral method for optimal control problems [4, 8, 9], the mixed finite element method for optimal control problems [3, 5‐7, 18‐20, 23, 24], and the finite volume method for optimal control problems [21, 22].
The spectral method has two important features: it enjoys the great superiority of fast convergence rate and provides very accurate approximations with a relatively small number of unknowns when the solutions are smooth. Let us briefly review the current literature. In [9], Ghanem and Sissaoui derived a posteriori error estimates by a spectral method of a linear–quadratic elliptic optimal control problem without inequality constraints. In [8], the authors studied the Legendre Galerkin spectral approximation of optimal control problems governed by elliptic equations and obtained a priori and a posteriori error estimates. A posteriori error estimates of a Legendre Galerkin spectral approximation of optimal control problems governed by parabolic equations were derived in [4]. To the best of our knowledge, a posteriori error estimates of the spectral method for nonlinear optimal control problems have never been studied. Nonlinear optimal control problems appear frequently in real life such as economics, chemical engineering, robotics and aeronautics, and the spectral method has several attractive features. Therefore, it is necessary to study a posteriori error estimates of the spectral method for nonlinear parabolic optimal control problems.
Advertisement
The purpose of this work is to derive a posteriori error estimates for the spectral approximation of an optimal control problem governed by nonlinear parabolic equations. We present a fully discrete scheme which uses the backward Euler scheme in time and uses the spectral approximation in space, and we obtain a posteriori error estimates of the spectral approximation solution for both the state and the control.
The outline of this paper is as follows. In Sect. 2, we shall construct spectral approximation scheme for nonlinear parabolic optimal control problem. In Sect. 3, by using orthogonal projection operator, \(L^{2}(H^{1})-L^{2}(L^{2})\) a posteriori error estimates of the spectral approximation solutions for optimal control problem solution are derived. In Sect. 4, by using two auxiliary equations, \(L^{2}(L^{2})-L ^{2}(L^{2})\) a posteriori error estimates of the spectral approximation solutions for optimal control problem are derived. In the last section, we briefly give conclusions and some possible future work.
Let Ω be bounded open sets in \(\mathbb{R}^{2} \) with a Lipschitz boundary ∂Ω. In this paper, we adopt the standard notation \(W^{m,q}(\Omega )\) for Sobolev spaces on Ω with the norm \(\Vert \cdot \Vert _{W^{m,q}(\Omega )}\) and the semi-norm \(\vert \cdot \vert _{W ^{m,q}(\Omega )}\). We set \(W_{0}^{m,q}(\Omega )\equiv \{w\in W^{m,q}( \Omega ): \partial^{\alpha }w|_{\partial \Omega } =0, \vert \alpha \vert \leq m-1\}\). We denote \(W^{m,2}(\Omega )\) (\(W_{0}^{m,2}(\Omega )\)) by \(H^{m}(\Omega )\) (\(H_{0}^{m}(\Omega )\)). We denote by \(L^{s}(0,T;W ^{m,q}(\Omega ))\) the Banach space of all \(L^{s}\) integrable functions from \((0,T)\) into \(W^{m,q}(\Omega )\) with norm \(\Vert v \Vert _{L^{s}(0,T;W ^{m,q}(\Omega ))}=(\int_{0}^{T} \Vert v \Vert ^{s}_{W^{m,q}(\Omega )}\,dt)^{ \frac{1}{s}}\) for \(s\in [0,\infty )\) and the standard modification for \(s=\infty \). Similarly, we define the spaces \(H^{1}(0,T;W^{m,q}( \Omega ))\) and \(C^{l}(0,T;W^{m,q}(\Omega ))\), the details can be found in [12].
2 Spectral approximation of nonlinear parabolic optimal control
In this section, we shall state the spectral approximation scheme and its optimality conditions for the optimal control problem governed by nonlinear parabolic equations. Now, we set the state space \(W=L^{2}(0,T;H _{0}^{1}(\Omega ))\), the control space \(X=L^{2}(0,T;L^{2}(\Omega ))\), and \(V=H_{0}^{1}(\Omega )\). We will study the following nonlinear parabolic optimal control problem:
where K is a set defined by
and \(f,y_{d}\in L^{2}(0,T;L^{2}(\Omega ))\), \(y_{0}\in H_{0}^{1}( \Omega )\), let B be a linear continuous operator from X to \(L^{2}(0,T;V^{\prime })\), \(\phi (\cdot )\in W^{2,\infty }(-R,-R)\) for any \(R>0\), \(\phi '(y)\in L^{2}(\Omega )\) for any \(y\in H^{1}(\Omega )\), \(\phi '\geq 0\), and the matrix \(A(\cdot )=(a_{i,j}(\cdot ))_{2\times 2}\in (C^{\infty }(\bar{\Omega }))^{2\times 2}\), such that there is a constant \(c>0\) satisfying
$$\begin{aligned}& \min_{u\in X; u(t)\in K} \biggl\{ \frac{1}{2} \int_{0}^{T}\bigl( \Vert y-y_{d} \Vert ^{2}_{L^{2}(\Omega )}+ \Vert u \Vert ^{2}_{L^{2}( \Omega )} \bigr)\,dt \biggr\} , \end{aligned}$$
(2.1)
$$\begin{aligned}& y_{t}- \operatorname{div} (A\nabla y)+\phi (y)=f+Bu, \quad x\in \Omega ,t\in (0,T], \end{aligned}$$
(2.2)
$$\begin{aligned}& y| _{\partial \Omega }=0, \quad t\in [0,T], \end{aligned}$$
(2.3)
$$\begin{aligned}& y(x,0)=y_{0}(x), \quad x\in \Omega , \end{aligned}$$
(2.4)
$$ K= \biggl\{ v\in X: \int_{0}^{T} \int_{\Omega }v\,dx\,dt\geq 0 \biggr\} , $$
$$ \xi^{t}A\xi \geq c \Vert \xi \Vert ^{2}, \quad \xi \in \mathbb{R}^{2}. $$
Advertisement
Let
It is well known that there are constants c and \(C>0\) such that
Then a weak formula of the optimal control problem can be obtained:
where \(y\in W\), \(u\in X\), \(u(t)\in K\) subject to
It is well known (see, e.g., [16]) that the optimal control problem (2.5)–(2.7) has at least one solution \((y,p,u)\), and that if a triplet \((y,p,u)\) is the solution of (2.5)–(2.7), then there is a co-state \(p\in W\) such that \((y,p,u)\) satisfies the following optimality conditions:
where \(B^{*}\) is the adjoint operator of B.
$$\begin{aligned}& a(y,w)= \int_{\Omega }(A\nabla y)\cdot \nabla w \,dx, \quad \forall\ y,w \in H_{0}^{1}(\Omega ), \\& (f_{1},f_{2})= \int_{\Omega } f_{1}f_{2}\,dx, \quad \forall\ f_{1},f_{2} \in L^{2}(\Omega ), \\& (u,v)= \int_{\Omega }uv\,dx, \quad \forall\ u,v\in L^{2}(\Omega ). \end{aligned}$$
$$ a(v,v)\geq c \Vert v \Vert _{1, \Omega }^{2},\qquad \bigl\vert a(v,w) \bigr\vert \leq C \vert v \vert _{1, \Omega } \vert w \vert _{1, \Omega }, \quad \forall\ v,w\in H_{0}^{1}(\Omega ). $$
$$ \min_{u(t)\in K} \biggl\{ \frac{1}{2} \int_{0}^{T}\bigl( \Vert y-y_{d} \Vert ^{2}_{L ^{2}(\Omega )}+ \Vert u \Vert ^{2}_{L^{2}(\Omega )} \bigr)\,dt \biggr\} , $$
(2.5)
$$\begin{aligned}& (y_{t},w)+ a(y,w)+\bigl(\phi (y),w\bigr)=(f+Bu,w), \quad \forall\ w \in V, t\in (0,T], \end{aligned}$$
(2.6)
$$\begin{aligned}& y(x,0)=y_{0}(x), \quad x\in \Omega . \end{aligned}$$
(2.7)
$$\begin{aligned}& (y_{t},w)+ a(y,w)+\bigl(\phi (y),w\bigr)=(f+Bu,w), \quad \forall\ w \in V, y(0)=y_{0}, \end{aligned}$$
(2.8)
$$\begin{aligned}& -(p_{t},q)+ a(q,p)+\bigl(\phi '(y)p,q \bigr)=(y-y_{d}, q), \quad \forall\ q \in V, p(T)=0, \end{aligned}$$
(2.9)
$$\begin{aligned}& \int_{0}^{T}\bigl(u+B^{*}p, v-u\bigr) \,dt \geq 0, \quad \forall\ v(t) \in K, v\in X, \end{aligned}$$
(2.10)
Lemma 2.1
Let
\((y,p,u)\)
be the solution of (2.8)–(2.10), we have
where
denotes the integral average on
\(\Omega \times [0,T]\)
of the function
\(B^{*}p\).
$$ u=\max\bigl\{ 0,\overline{B^{*}p}\bigr\} -B^{*}p, $$
$$ \overline{B^{*}p}=\frac{\int_{0}^{T}\int_{\Omega }B^{*}p\,dx\,dt}{\int _{0}^{T}\int_{\Omega }1\,dx\,dt} $$
Proof
For any function \(p\in W\), we have
which satisfies the following variational inequality:
If \(\overline{B^{*}p}>0\), then it is easy to see \(u=\overline{B^{*}p}-B ^{*}p\) and
If \(\overline{B^{*}p}\leq 0\), then it is clear that \(u=-B^{*}p\) and
So we have \(u=\max\{0,\overline{B^{*}p}\}-B^{*}p\). □
$$ u=\max\bigl\{ 0,\overline{B^{*}p}\bigr\} -B^{*}p, $$
$$ \int_{0}^{T}\bigl(u+B^{*}p, v-u\bigr) \,dt \geq 0, \quad \forall\ v(t) \in K. $$
$$\begin{aligned} \int_{0}^{T}\bigl(u+B^{*}p, v-u\bigr) \,dt &= \int_{0}^{T} \int_{\Omega }\bigl(u+B^{*}p\bigr) ( v-u)\,dx\,dt \\ &= \int_{0}^{T} \int_{\Omega }\overline{B^{*}p}\bigl( v- \overline{B^{*}p}+B ^{*}p\bigr)\,dx\,dt \\ &=\overline{B^{*}p} \int_{0}^{T} \int_{\Omega }v \,dx\,dt\geq 0, \quad \forall\ v(t) \in K. \end{aligned}$$
$$ \int_{0}^{T}\bigl(u+B^{*}p, v-u\bigr) \,dt \geq 0. $$
Next, we will use the Legendre Galerkin spectral method to investigate the spectral approximation of the nonlinear parabolic optimal control problem (2.8)–(2.10). From now on, we assume that \(\Omega =(-1,1)^{2}\). Firstly, let us introduce some basic notations which will be used in the sequel. For \(x_{i}\), \(i=1,2\), we denote by \(L_{r}(x_{i})\) the rth degree Legendre polynomial in the variable \(x_{i}\), and we set
where \(N\geq 0\) is an integer. We define a product space such as
We introduce the finite dimensional spaces \(V^{N}=X_{N}\bigcap V\) and \(K^{N}=X_{N} \bigcap K\). In addition, C denotes a general positive constant independent of N, the order of the spectral approximation.
$$ X^{i}_{N}=\operatorname{span}\bigl\{ L_{0}(x_{i}), L_{1}(x_{i}),\ldots, L _{N}(x_{i}) \bigr\} , $$
$$ X_{N}=\prod_{i=1}^{2} X_{N}^{i}. $$
Then the Legendre Galerkin spectral approximation for the nonlinear parabolic optimal control problem is
where \(y_{N}\in H^{1}(0,T;V^{N})\) and \(y_{0}^{N}\in V^{N}\) is an approximation of \(y_{0}\).
$$\begin{aligned}& \min_{u_{N}(t)\in K^{N}} \biggl\{ \frac{1}{2} \int_{0}^{T}\bigl( \Vert y_{N}-y _{d} \Vert ^{2}_{L^{2}(\Omega )}+ \Vert u_{N} \Vert ^{2}_{L^{2}(\Omega )}\bigr)\,dt \biggr\} , \end{aligned}$$
(2.11)
$$\begin{aligned}& (y_{Nt},w_{N})+ a(y_{N},w_{N})+ \bigl(\phi (y_{N}),w_{N}\bigr)=(f+Bu_{N},w_{N}), \quad \forall\ w_{N} \in V^{N}, \end{aligned}$$
(2.12)
$$\begin{aligned}& y_{N}(x,0)=y_{0}^{N}(x), \quad x\in \Omega , \end{aligned}$$
(2.13)
It follows that the optimal control problem (2.11)–(2.13) has at least one solution \((y_{N},u_{N})\), and that if a pair \((y_{N},u_{N})\) is the solution of (2.11)–(2.13), then there is a co-state \(p_{N}\) such that \((y_{N},p_{N} ,u_{N})\) satisfies the following optimality conditions:
$$\begin{aligned}& (y_{Nt},w_{N})+a(y_{N},w_{N})+ \bigl(\phi (y_{N}),w_{N}\bigr)=(f+Bu_{N},w_{N}), \quad \forall\ w_{N} \in V^{N}, \end{aligned}$$
(2.14)
$$\begin{aligned}& y_{N}(x,0)=y_{0}^{N}(x), \quad x\in \Omega , \\& {} -(p_{Nt},q_{N})+a(q_{N},p_{N})+ \bigl(\phi '(y_{N})p_{N},q_{N} \bigr)=(y_{N}-y _{d},q_{N}), \quad \forall\ q_{N} \in V^{N}, \end{aligned}$$
(2.15)
$$\begin{aligned}& p_{N}(x,T)=0, \quad x\in \Omega , \\& \bigl(u_{N}+B^{*}p_{N}, v_{N}-u_{N}\bigr)\geq 0, \quad u_{N}(t)\in K^{N}, \forall\ v_{N}\in K^{N}. \end{aligned}$$
(2.16)
Now, we shall construct the fully discrete approximation scheme for the above semi-discrete problem. Let \(0=t_{0}< t_{1}<\cdots <t_{M-1}<t_{M}=T\), \(k_{i}=t_{i}-t_{i-1}\), \(i=1,2,\ldots,M\), \(k=\max_{1\leq i\leq N} \{k_{i}\}\). For \(i=1,2,\ldots,M\), construct the approximation spaces \(V_{i}^{N}\subset H_{0}^{1}(\Omega )\) (similar as \(V^{N}\)) on the ith time step. Similarly, construct the approximation spaces \(K_{i}^{N}\subset L^{2}(\Omega )\) (similar to \(K^{N}\)) on the ith time step. The fully discrete approximation scheme of (2.11)–(2.13) is to find \((y_{N}^{i},u_{N}^{i})\in V_{i} ^{N}\times K_{i}^{N}\), \(i=1,2,\ldots,M\), such that
It follows that the optimal control problem (2.17)–(2.19) has at least one solution \((Y_{N}^{i},U_{N}^{i})\), \(i=1,2,\ldots,M\), and that if a pair \((Y_{N}^{i},U_{N}^{i})\in V_{i}^{N}\times K_{i}^{N}\), \(i=1,2,\ldots,M\), is the solution of (2.17)–(2.19), then there is a co-state \(P_{N}^{i-1}\in V_{i}^{N}\), \(i=M,\ldots,2,1\), such that the triplet \((Y_{N}^{i},P_{N}^{i-1},U_{N}^{i})\in V_{i}^{N}\times V _{i}^{N}\times K_{i}^{N}\), \(i=1,2,\ldots,M\), satisfies the following optimality conditions:
$$\begin{aligned}& \min_{u_{N}^{i}\in K_{i}^{N}} \Biggl\{ \frac{1}{2}\sum _{i=1}^{N}k _{i} \bigl( \bigl\Vert y_{N}^{i}-y_{d}(x,t_{i}) \bigr\Vert ^{2}_{L^{2}( \Omega )}+ \bigl\Vert u_{N}^{i} \bigr\Vert ^{2}_{L^{2}(\Omega )} \bigr) \Biggr\} , \end{aligned}$$
(2.17)
$$\begin{aligned}& \begin{aligned}[b] &\biggl( \frac{y_{N}^{i}-y_{N}^{i-1}}{k_{i}},w_{N} \biggr) +a \bigl(y_{N}^{i},w _{N}\bigr)+\bigl(\phi \bigl(y^{i}_{N}\bigr),w_{N}\bigr)= \bigl(f(x,t_{i})+ Bu_{N}^{i},w_{N} \bigr), \\ &\quad \forall\ w_{N}\in V_{i}^{N}, i=1,2,\ldots,M, \end{aligned} \end{aligned}$$
(2.18)
$$\begin{aligned}& y_{N}^{0}(x)=y_{0}^{N}(x), \quad x\in \Omega . \end{aligned}$$
(2.19)
$$\begin{aligned}& \begin{aligned}[b] & \biggl( \frac{Y_{N}^{i}-Y_{N}^{i-1}}{k_{i}},w_{N} \biggr) +a \bigl(Y_{N} ^{i},w_{N}\bigr)+\bigl(\phi \bigl(Y_{N}^{i}\bigr),w_{N}\bigr)= \bigl(f(x,t_{i})+ BU_{N}^{i},w_{N} \bigr), \\ & \quad \forall\ w_{N}\in V_{i}^{N}, i=1,2,\ldots,M, \end{aligned} \end{aligned}$$
(2.20)
$$\begin{aligned}& Y_{N}^{0}(x)=y_{0}^{N}(x),\quad x \in \Omega , \end{aligned}$$
(2.21)
$$\begin{aligned}& \begin{aligned}[b]& \biggl( \frac{P_{N}^{i-1}-P_{N}^{i}}{k_{i}},q_{N} \biggr) +a \bigl(q_{N},P _{N}^{i-1}\bigr)+\bigl(\phi '\bigl(Y_{N}^{i-1}\bigr)p_{N}^{i-1},q_{N} \bigr)=\bigl(Y_{N}^{i}-y_{d}(x,t _{i}),q_{N}\bigr), \\ & \quad \forall\ q_{N}\in V_{i}^{N}, i=M,\ldots,2,1, \end{aligned} \end{aligned}$$
(2.22)
$$\begin{aligned}& P_{N}^{M}(x) = 0,\quad x\in \Omega , \end{aligned}$$
(2.23)
$$\begin{aligned}& \bigl(U_{N}^{i}+B^{*}P_{N}^{i-1}, v_{N}-U_{N}^{i}\bigr)\geq 0, \quad \forall\ v_{N}\in K_{i}^{N}, i=1,2,\ldots,M. \end{aligned}$$
(2.24)
For \(i=1,2,\ldots,M\), let
For any function \(w\in C(0,T;L^{2}(\Omega ))\), let \(\hat{w}(x,t)| _{t \in (t_{i-1},t_{i}] }= w(x,t_{i})\), \(\tilde{w}(x,t)| _{t\in (t_{i-1},t _{i}]}=w(x,t_{i-1})\). Then the optimality conditions (2.20)–(2.24) can be restated:
$$\begin{aligned}& Y_{N} | _{(t_{i-1},t_{i}]}=\bigl((t_{i}-t) Y_{N}^{i-1}+(t-t_{i-1})Y_{N} ^{i}\bigr)/k_{i}, \\& P_{N} | _{(t_{i-1},t_{i}]}=\bigl((t_{i}-t) P_{N}^{i-1}+(t-t_{i-1})P_{N} ^{i}\bigr)/k_{i}, \\& U_{N}| _{(t_{i-1},t_{i}]}=U_{N}^{i}. \end{aligned}$$
$$\begin{aligned}& \begin{aligned}[b]& (Y_{Nt},w_{N})+a(\hat{Y}_{N},w_{N})+ \bigl(\phi (\hat{Y}_{N}),w_{N}\bigr)=( \hat{f}+BU_{N},w_{N}), \\ & \quad \forall\ w_{N}\in V_{i}^{N}, t\in (t_{i-1},t_{i}], i=1,2,\ldots,M, \end{aligned} \end{aligned}$$
(2.25)
$$\begin{aligned}& Y_{N}(x,0) = y_{0}^{N}(x),\quad x\in \Omega , \end{aligned}$$
(2.26)
$$\begin{aligned}& \begin{aligned}[b]& {} -(P_{Nt},q_{N})+a(q_{N}, \tilde{P}_{N})+\bigl(\phi '(\tilde{Y}_{N}) \tilde{P}_{N},q_{N}\bigr)=(\hat{Y}_{N}- \hat{y}_{d},q_{N}), \\ & \quad \forall\ q_{N}\in V_{i}^{N}, t\in (t_{i-1},t_{i}], i=M,\ldots,2,1, \end{aligned} \end{aligned}$$
(2.27)
$$\begin{aligned}& P_{N}(x,T) = 0,\quad x\in \Omega , \end{aligned}$$
(2.28)
$$\begin{aligned}& \begin{aligned}[b]& \bigl(U_{N}+B^{*}\tilde{P}_{N}, v_{N}-U_{N}\bigr)\geq 0,\quad U_{N}\in K_{i} ^{N}, \\ & \quad \forall\ v_{N}\in K_{i}^{N}, t\in (t_{i-1},t_{i}], i=1,2,\ldots,M. \end{aligned} \end{aligned}$$
(2.29)
For any \(u \in L^{2}(\Omega )\), we define the orthogonal projection operator \(P_{N}: L^{2}(\Omega )\rightarrow K^{N} \) which satisfies
It can be shown (see [2]) that
where \(k=(k_{1},k_{2})\) and \(\phi_{k}(x)=L_{k_{1}}(x_{1}) L_{k_{2}}(x _{2})\).
$$ (P_{N}u-u, w_{N})=0, \quad \forall\ w_{N} \in K^{N}. $$
$$ P_{N}u=\sum_{\max _{1\leq i \leq 2} \{k_{i}\} \leq N} \hat{u}_{k} \phi_{k}, \quad \hat{u}_{k}=\prod _{i=1}^{2} \biggl( k _{i}+\frac{1}{2} \biggr) \int_{\Omega }u(x)\phi_{k}(x)\,dx, $$
For any \(u \in H_{0}^{1}(\Omega )\), \(P_{1,N}^{0} :H_{0}^{1}(\Omega )\rightarrow V^{N}\) is defined by
The following lemma will play a very important role in a posteriori error estimates. It can be found in the reference book of Ref. [2].
$$ \int_{\Omega }\nabla \bigl(u-P_{1,N}^{0}u\bigr) \cdot \nabla w\,dx =0, \quad \forall\ w\in V^{N}. $$
Lemma 2.2
For all
\(u\in H^{m}(\Omega )\) (\(m\geq 0\)), we have
where
\(\sigma (l)=0\)
if
\(l=0\), and
\(\sigma (l)=2l-\frac{1}{2}\)
for
\(l>0\). If
\(u\in H_{0}^{1}(\Omega )\cap H^{m}(\Omega )\), \(m \geq 1\), then we have
$$ \Vert u-P_{N}u \Vert _{H^{l}(\Omega )}\leq CN^{\sigma (l)-m} \Vert u \Vert _{H^{m}( \Omega )}, \quad 0\leq l \leq m, $$
$$ \bigl\Vert u-P_{1,N}^{0}u \bigr\Vert _{H^{\mu }(\Omega )} \leq C N^{\mu -m} \Vert u \Vert _{H^{m}(\Omega )}, \quad 0\leq \mu \leq 1. $$
For \(\varphi \in V^{N}\), we shall write
where
are bounded functions in Ω̄, more details can be found in [7].
$$\begin{aligned}& \phi (\varphi )-\phi (\rho )=-\tilde{\phi }^{\prime }( \varphi ) ( \rho -\varphi )=-\phi^{\prime }(\rho ) (\rho -\varphi ) +\tilde{ \phi } ^{\prime \prime }(\varphi ) (\rho -\varphi )^{2}, \end{aligned}$$
(2.30)
$$ \tilde{\phi }^{\prime }(\varphi )= \int_{0}^{1}\phi^{\prime }\bigl(\varphi +s(\rho -\varphi )\bigr)\,ds,\quad\quad \tilde{\phi }^{\prime \prime }(\varphi )= \int_{0}^{1}(1-s)\phi^{\prime \prime }\bigl(\rho +s( \varphi -\rho )\bigr)\,ds $$
3 \(L^{2}(H^{1})-L^{2}(L^{2})\) a posteriori error estimates
In this section, we shall derive a \(L^{2}(H^{1})-L^{2}(L^{2})\) posteriori error estimates for the spectral approximation of the optimal control problem governed by nonlinear parabolic equations. Set
It can be shown that (see [12])
where \(p(U_{N})\) is the solution of the auxiliary equations:
We assume that the cost function J is strictly convex near the solution u, i.e., for the solution u there exists a neighborhood of u in \(L^{2}\) such that J is convex in the sense that there is a constant \(c>0\) satisfying
for all v in this neighborhood of u. The convexity of \(J(\cdot )\) is closely related to the second order sufficient optimality conditions of optimal control problems, which are assumed in many studies on numerical methods of the problem. For instance, in many references, the authors assume the following second order sufficiently optimality condition (see [13]): there is \(c >0\) such that \(J^{\prime \prime }(u)v^{2}\geq c \Vert v \Vert _{0}^{2}\).
$$\begin{aligned}& J(u)=\frac{1}{2} \int_{0}^{T} \bigl( \Vert y-y_{d} \Vert ^{2}_{L^{2}(\Omega )}+ \Vert u \Vert ^{2}_{L^{2}(\Omega )} \bigr) \,dt, \\& J_{N}(U_{N})=\frac{1}{2} \int_{0}^{T} \bigl( \Vert Y_{N}-y_{d} \Vert ^{2} _{L^{2}(\Omega )}+ \Vert U_{N} \Vert ^{2}_{L^{2}(\Omega )} \bigr) \,dt. \end{aligned}$$
$$\begin{aligned}& \bigl(J^{\prime }(u),v\bigr)=\bigl(u+B^{*}p,v \bigr), \end{aligned}$$
(3.1)
$$\begin{aligned}& \bigl(J_{N}^{\prime }(U_{N}),v\bigr)= \bigl(U_{N}+B^{*}\tilde{P}_{N},v\bigr), \end{aligned}$$
(3.2)
$$\begin{aligned}& \bigl(J^{\prime }(U_{N}),v\bigr)= \bigl(U_{N}+B^{*}p(U_{N}),v\bigr), \end{aligned}$$
(3.3)
$$\begin{aligned}& \bigl(y_{t}(U_{N}),w\bigr)+a \bigl(y(U_{N}),w\bigr)+\bigl(\phi \bigl(y(U_{N})\bigr),w \bigr)=(f+BU_{N},w), \quad \forall\ w\in V, \end{aligned}$$
(3.4)
$$\begin{aligned}& y(U_{N}) (x,0)=y_{0}(x), \quad x\in \Omega , \end{aligned}$$
(3.5)
$$\begin{aligned}& \begin{aligned}[b]& {} -\bigl(p_{t}(U_{N} ),q\bigr)+a \bigl(q,p(U_{N})\bigr)+\bigl(\phi '\bigl(y(U_{N}) \bigr)p(U_{N}),q\bigr) \\ &\quad =\bigl(y(U_{N}) -y_{d},q \bigr), \quad \forall\ q\in V,\end{aligned} \end{aligned}$$
(3.6)
$$\begin{aligned}& p(U_{N}) (x,T)=0, \quad x\in \Omega . \end{aligned}$$
(3.7)
$$\begin{aligned}& \bigl(J^{\prime }(u)-J^{\prime }(v), u-v\bigr)\geq c \Vert u-v \Vert ^{2}, \end{aligned}$$
(3.8)
From the assumption (3.8), by the proof contained in [1], there is a \(c>0\) independent of N, such that
where u and \(u_{N}\) are the solutions of (2.8)–(2.10) and (2.14)–(2.16), respectively.
$$ \int_{0}^{T}\bigl(J'(u)-J'(u_{N}),u-u_{N} \bigr)\,dt\geq c \Vert u-u_{N} \Vert _{L^{2}(0,T;L ^{2}(\Omega ))}^{2}, $$
(3.9)
Theorem 3.1
Let
\((y,p,u)\)
and
\((Y_{N},P_{N},U_{N})\)
be the solutions of (2.8)–(2.10) and (2.25)–(2.29), respectively. Then we have
where
\(p(U_{N})\)
is defined by (3.5) and
$$ \Vert u-U_{N} \Vert ^{2}_{L^{2}(0,T;L^{2}(\Omega ))} \leq C\eta_{1}^{2}+C \bigl\Vert p(U _{N})- \tilde{P}_{N} \bigr\Vert ^{2}_{L^{2}(0,T;L^{2}(\Omega ))}, $$
(3.10)
$$ \eta_{1}^{2}=\sum_{i=1}^{M} \int_{t_{i-1}}^{t_{i}} \bigl\Vert U_{N}+B^{*} \tilde{P}_{N} \bigr\Vert _{L^{2}(\Omega )}^{2}\,dt. $$
Proof
According to (3.9), we have
Note that \(U_{N}\in K^{N}\subset K\). It follows from (2.10) that we have
Therefore, we can get
Firstly, we can easily estimate the first term \(E_{1}\) of (3.11) as follows:
For the second term \(E_{2}\), we can also easily obtain
Finally, for any sufficiently small positive number δ, and from (3.11)–(3.13), we have
This proves (3.10). □
$$\begin{aligned} c \Vert u-U_{N} \Vert ^{2}_{L^{2}(0,T;L^{2}(\Omega ))} &\leq \int_{0}^{T}\bigl(J'(u)-J'(U _{N}), u-U_{N}\bigr)\,dt \\ & = \int_{0}^{T}\bigl(u+B^{*}p, u-U_{N}\bigr)\,dt+ \int_{0}^{T}\bigl(U_{N}+B^{*}P(U _{N}), U_{N}-u\bigr)\,dt \\ & = \int_{0}^{T}\bigl(u+B^{*}p, u-U_{N}\bigr)\,dt+ \int_{0}^{T}\bigl(U_{N}+B^{*} \tilde{P}_{N}, U_{N}-u\bigr)\,dt \\ &\quad {} + \int_{0}^{T}\bigl(B^{*}\bigl( \tilde{P}_{N}-p(U_{N})\bigr), u-U_{N}\bigr)\,dt. \end{aligned}$$
$$ \int_{0}^{T}\bigl(u+B^{*}p, u-U_{N}\bigr)\,dt\leq 0. $$
$$\begin{aligned} \begin{aligned}[b] c \Vert u-U_{N} \Vert ^{2}_{L^{2}(0,T;L^{2}(\Omega ))}&\leq \int_{0}^{T}\bigl(U_{N}+B ^{*} \tilde{P}_{N}, U_{N}-u\bigr)\,dt \\ & \quad {} + \int_{0}^{T}\bigl(B^{*}\bigl( \tilde{P}_{N}-p(U _{N})\bigr), u-U_{N}\bigr)\,dt \\ &\equiv E_{1}+E_{2}. \end{aligned} \end{aligned}$$
(3.11)
$$\begin{aligned} E_{1}&= \int_{0}^{T}\bigl(U_{N}+B^{*} \tilde{P}_{N}, U_{N}-u\bigr)\,dt=\sum _{i=1}^{M} \int_{t_{i-1}}^{t_{i}}\bigl(U_{N}+B^{*} \tilde{P}_{N}, U_{N}-u\bigr)\,dt \\ &\leq C(\delta )\sum_{i=1}^{M} \int_{t_{i-1}}^{t_{i}} \bigl\Vert U_{N}+B ^{*}\tilde{P}_{N} \bigr\Vert _{L^{2}(\Omega )}^{2}\,dt +\delta \sum_{i=1} ^{M} \int_{t_{i-1}}^{t_{i}} \Vert U_{N}-u \Vert _{L^{2}(\Omega )}^{2}\,dt \\ &\leq C( \delta )\eta_{1}^{2}+\delta \Vert u-U_{N} \Vert ^{2}_{L^{2}(0,T;L^{2}(\Omega ))}. \end{aligned}$$
(3.12)
$$\begin{aligned} \begin{aligned}[b] E_{2}&= \int_{0}^{T}\bigl(B^{*}\bigl( \tilde{P}_{N}-p(U_{N})\bigr), u-U_{N}\bigr)\,dt \\ &\leq C(\delta ) \int_{0}^{T} \bigl\Vert B^{*}( \tilde{P}_{N}-p(U_{N}) \bigr\Vert ^{2}_{L ^{2}(\Omega )}\,dt+ \delta \int_{0}^{T} \Vert U_{N}-u \Vert _{L^{2}(\Omega )}^{2}\,dt \\ &\leq C \bigl\Vert \tilde{P}_{N}-p(U_{N}) \bigr\Vert ^{2}_{L^{2}(0,T;L^{2}(\Omega ))}+ \delta \Vert u-U_{N} \Vert ^{2}_{L^{2}(0,T;L^{2}(\Omega ))}. \end{aligned} \end{aligned}$$
(3.13)
$$ \Vert u-U_{N} \Vert ^{2}_{L^{2}(0,T;L^{2}(\Omega ))}\leq C \eta_{1}^{2}+C \bigl\Vert p(U _{N})- \tilde{P}_{N} \bigr\Vert ^{2}_{L^{2}(0,T;L^{2}(\Omega ))}. $$
Theorem 3.2
Let
\((Y_{N},P_{N},U_{N})\)
be the solutions of (2.25)–(2.29), let
\((y(U_{N}),p(U_{N}))\)
be defined by (3.4)–(3.7). Then we have
where
$$ \bigl\Vert Y_{N}-y(U_{N}) \bigr\Vert ^{2}_{L^{2}(0,T;H^{1}(\Omega ))}+ \bigl\Vert P_{N}-p(U_{N}) \bigr\Vert ^{2}_{L^{2}(0,T;H^{1}(\Omega ))}\leq C\sum_{i=2}^{10} \eta_{i} ^{2}, $$
(3.14)
$$\begin{aligned}& \eta_{2}^{2} = \int_{0}^{T}N^{-2} \int_{\Omega }\bigl(\hat{Y}_{N}-\hat{y} _{d}+ \operatorname{div}\bigl(A^{*}\nabla \tilde{P}_{N}\bigr)-\phi '(\tilde{Y}_{N}) \tilde{P}_{N}+P_{Nt} \bigr)^{2}\,dx\,dt, \\& \eta_{3}^{2} = \int_{0}^{T} \int_{\Omega } \bigl\vert A^{*}\nabla (\tilde{P} _{N}-P_{N}) \bigr\vert ^{2}\,dx\,dt, \\& \eta_{4}^{2} = \Vert y_{d}- \hat{y}_{d} \Vert _{L^{2}(0,T;L^{2}(\Omega ))} ^{2}, \\& \eta_{5}^{2} = \Vert Y_{N}- \tilde{Y}_{N} \Vert _{L^{2}(0,T;L^{2}(\Omega ))} ^{2}, \\& \eta_{6}^{2} = \Vert Y_{N}- \hat{Y}_{N} \Vert _{L^{2}(0,T;L^{2}(\Omega ))} ^{2}, \\& \eta_{7}^{2} = \int_{0}^{T}N^{-2} \int_{\Omega }\bigl(\hat{f}+BU_{N}+ \operatorname{div}(A\nabla \hat{Y}_{N})-\phi (\hat{Y}_{N})-Y_{Nt} \bigr)^{2}\,dx\,dt, \\& \eta_{8}^{2} = \int_{0}^{T} \int_{\Omega } \bigl\vert A\nabla (\hat{Y}_{N}-Y _{N}) \bigr\vert ^{2}\,dx\,dt, \\& \eta_{9}^{2} = \Vert f-\hat{f} \Vert _{L^{2}(0,T;L^{2}(\Omega ))}^{2}, \\& \eta_{10}^{2} = \bigl\Vert Y_{N}(x,0)-y_{0}(x) \bigr\Vert _{L^{2}(\Omega )}^{2}. \end{aligned}$$
Proof
Part I. Let \(e^{p}=p(U_{N})-P_{N}\) and \(e_{I}^{p}=P^{0}_{1,N}e ^{p}\in V^{N}\), where \(P^{0}_{1,N}\) is the orthogonal projection operator defined as in Lemma 2.2. Note that \((p(U_{N})-P _{N})(x,T)=0\), hence
By using Eqs. (2.27), (3.6), and (3.15), we have
From (3.16), we can get
Thanks to \(e^{p}-e_{I}^{p}\in V=H_{0}^{1}(\Omega )\), from Eqs. (2.27), (3.6), and (3.17), we can obtain
Then we have
$$\begin{aligned} & \int_{0}^{T}-\bigl(p_{t}(U_{N})-P_{Nt},e^{p} \bigr)\,dt \\ &\quad =- \int_{0}^{T} \frac{1}{2}\frac{d}{d t} \bigl\Vert p(U_{N})-P_{N} \bigr\Vert _{L^{2}(\Omega )}^{2}\,dt \\ &\quad=- \frac{1}{2} \bigl( \bigl\Vert \bigl(p(U_{N})-P_{N} \bigr) (x,T) \bigr\Vert _{L^{2}(\Omega )}^{2}- \bigl\Vert \bigl(p(U _{N})-P_{N}\bigr) (x,0) \bigr\Vert _{L^{2}(\Omega )}^{2} \bigr) \\ &\quad=\frac{1}{2} \bigl\Vert \bigl(p(U _{N})-P_{N} \bigr) (x,0) \bigr\Vert _{L^{2}(\Omega )}^{2}\geq 0. \end{aligned}$$
(3.15)
$$\begin{aligned} \begin{aligned}[b] & c \bigl\Vert e^{p} \bigr\Vert _{L^{2}(0,T;H^{1}(\Omega ))}^{2} \\ &\quad\leq \int_{0}^{T}a\bigl(e ^{p},p(U_{N})-P_{N} \bigr)\,dt+ \int_{0}^{T}\bigl(\phi ' \bigl(y(U_{N})\bigr) \bigl(p(U_{N})-P_{N}\bigr),e ^{p}\bigr)\,dt \\ &\quad\leq \int_{0}^{T}\bigl(\nabla e^{p},A^{*} \nabla \bigl(p(U_{N})-P_{N}\bigr)\bigr)\,dt- \int_{0}^{T}\bigl(p_{t}(U_{N})-P_{Nt},e^{p} \bigr)\,dt \\ &\quad\quad {} + \int_{0}^{T}\bigl(\phi '\bigl(y(U _{N})\bigr) \bigl(p(U_{N})-\tilde{P}_{N} \bigr),e^{p}\bigr)\,dt \\ &\quad\quad {} + \int_{0}^{T}\bigl(\phi '\bigl(y(U _{N})\bigr) (\tilde{P}_{N}-P_{N}),e^{p} \bigr)\,dt \\ &\quad= \int_{0}^{T}\bigl(\nabla e^{p},A ^{*}\nabla \bigl(p(U_{N})-\tilde{P}_{N}\bigr) \bigr)\,dt- \int_{0}^{T}\bigl(p_{t}(U_{N})-P _{Nt},e^{p}\bigr)\,dt \\ &\quad\quad {} + \int_{0}^{T}\bigl(\nabla e^{p},A^{*} \nabla (\tilde{P} _{N}-P_{N})\bigr)\,dt \\ &\quad\quad {} + \int_{0}^{T}\bigl(\phi ' \bigl(y(U_{N})\bigr)p(U_{N})-\phi '( \tilde{Y}_{N})\tilde{P}_{N},e^{p}\bigr)\,dt \\ &\quad\quad {} + \int_{0}^{T}\bigl(\tilde{\phi }''( \tilde{Y}_{N}) \bigl(\tilde{Y}_{N}-y(U_{N}) \bigr)\tilde{P}_{N},e^{p}\bigr)\,dt \\ &\quad\quad {} + \int _{0}^{T}\bigl(\phi ' \bigl(y(U_{N})\bigr) (\tilde{P}_{N}-P_{N}),e^{p} \bigr)\,dt. \end{aligned} \end{aligned}$$
(3.16)
$$\begin{aligned} \begin{aligned}[b] & c \bigl\Vert e^{p} \bigr\Vert _{L^{2}(0,T;H^{1}(\Omega ))}^{2} \\ &\quad\leq \int_{0}^{T}\bigl( \nabla \bigl(e^{p}-e_{I}^{p} \bigr),A^{*}\nabla \bigl(p(U_{N})-\tilde{P}_{N} \bigr)\bigr)\,dt- \int _{0}^{T}\bigl(p_{t}(U_{N})-P_{Nt},e^{p}-e_{I}^{p} \bigr)\,dt \\ &\quad\quad {} + \int_{0}^{T}\bigl( \phi ' \bigl(y(U_{N})\bigr)p(U_{N})-\phi '( \tilde{Y}_{N})\tilde{P}_{N},e^{p}-e _{I}^{p}\bigr)\,dt \\ &\quad \quad {} + \int_{0}^{T}\bigl(\nabla e_{I}^{p},A^{*} \nabla \bigl(p(U_{N})- \tilde{P}_{N}\bigr)\bigr)\,dt- \int_{0}^{T}\bigl(p_{t}(U_{N})-P_{Nt},e_{I}^{p} \bigr)\,dt \\ &\quad \quad {} + \int_{0}^{T}\bigl(\phi ' \bigl(y(U_{N})\bigr)p(U_{N})-\phi '( \tilde{Y}_{N})\tilde{P} _{N},e_{I}^{p} \bigr)\,dt \\ &\quad \quad {} + \int_{0}^{T}\bigl(\tilde{\phi }''( \tilde{Y}_{N}) \bigl( \tilde{Y}_{N}-y(U_{N}) \bigr)\tilde{P}_{N},e^{p}\bigr)\,dt \\ &\quad\quad {} + \int_{0}^{T}\bigl(\phi '\bigl(y(U _{N})\bigr) (\tilde{P}_{N}-P_{N}),e^{p} \bigr)\,dt+ \int_{0}^{T}\bigl(\nabla e^{p},A^{*} \nabla (\tilde{P}_{N}-P_{N})\bigr)\,dt. \end{aligned} \end{aligned}$$
(3.17)
$$\begin{aligned} \begin{aligned}[b] & c \bigl\Vert e^{p} \bigr\Vert _{L^{2}(0,T;H^{1}(\Omega ))}^{2} \\ &\quad= \int_{0}^{T}\bigl(y(U_{N})-y _{d}+\operatorname{div}\bigl(A^{*}\nabla \tilde{P}_{N} \bigr)-\phi '(\tilde{Y}_{N}) \tilde{P}_{N}+P_{Nt},e^{p}-e_{I}^{p} \bigr)\,dt \\ &\quad\quad {} + \int_{0}^{T}\bigl(\tilde{\phi }''( \tilde{Y}_{N}) \bigl(\tilde{Y}_{N}-y(U_{N}) \bigr)\tilde{P}_{N},p(U_{N})-P_{N}\bigr)\,dt \\ &\quad\quad {} + \int_{0}^{T}\bigl(y(U_{N})- \hat{Y}_{N},e_{I}^{p}\bigr)\,dt+ \int_{0}^{T}\bigl( \hat{y}_{d}-y_{d},e_{I}^{p} \bigr)\,dt \\ &\quad\quad {} + \int_{0}^{T}\bigl(\phi ' \bigl(y(U_{N})\bigr) ( \tilde{P}_{N}-P_{N}),p(U_{N})-P_{N} \bigr)\,dt \\ &\quad\quad {}+ \int_{0}^{T}\bigl(\nabla e^{p},A ^{*}\nabla (\tilde{P}_{N}-P_{N})\bigr)\,dt. \end{aligned} \end{aligned}$$
(3.18)
$$\begin{aligned} \begin{aligned}[b] & c \bigl\Vert e^{p} \bigr\Vert _{L^{2}(0,T;H^{1}(\Omega ))}^{2} \\ &\quad\leq \int_{0}^{T}\bigl( \hat{Y}_{N}- \hat{y}_{d}+\operatorname{div}\bigl(A^{*}\nabla \tilde{P}_{N}\bigr)-\phi '( \tilde{Y}_{N}) \tilde{P}_{N}+P_{Nt},e^{p}-e_{I}^{p} \bigr)\,dt \\ &\quad\quad {} + \int_{0} ^{T}\bigl(\phi ' \bigl(y(U_{N})\bigr) (\tilde{P}_{N}-P_{N}),p(U_{N})-P_{N} \bigr)\,dt+ \int_{0} ^{T}\bigl(\nabla e^{p},A^{*} \nabla (\tilde{P}_{N}-P_{N})\bigr)\,dt \\ &\quad\quad {} + \int_{0} ^{T}\bigl(\hat{y}_{d}-y_{d},e^{p} \bigr)\,dt+ \int_{0}^{T}\bigl(\tilde{\phi }''( \tilde{Y}_{N}) \bigl(\tilde{Y}_{N}-y(U_{N}) \bigr)\tilde{P}_{N},P(U_{N})-P_{N}\bigr)\,dt \\ &\quad\quad {} + \int_{0}^{T}\bigl(y(U_{N})- \hat{Y}_{N},e^{p}\bigr)\,dt \\ &\quad \equiv \sum_{i=1}^{6}I_{i}. \end{aligned} \end{aligned}$$
(3.19)
By using Lemma 2.2, we can estimate the first term \(I_{1}\) as follows:
where δ is an arbitrary positive number, \(C(\delta )\) is a constant dependent on δ. Note that \(\phi '(\cdot )\geq 0\), we can obtain
For the third term \(I_{3}\), we can derive
Similarly, for \(I_{4}\) we have the following estimate:
Due to \(\phi (\cdot )\in W^{2,\infty }(-R,-R)\), we can get
Then we can estimate the last term \(I_{6}\) of (3.19) as follows:
Therefore, let δ be small enough, from (3.16)–(3.25), we obtain
$$\begin{aligned} \begin{aligned}[b] I_{1}&= \int_{0}^{T}\bigl(\hat{Y}_{N}- \hat{y}_{d}+\operatorname{div}\bigl(A^{*}\nabla \tilde{P}_{N}\bigr)-\phi '(\tilde{Y}_{N}) \tilde{P}_{N}+P_{Nt},e^{p}-e_{I} ^{p}\bigr)\,dt \\ &\leq C(\delta ) \int_{0}^{T}N^{-2} \int_{\Omega }\bigl(\hat{Y} _{N}-\hat{y}_{d}+ \operatorname{div}\bigl(A^{*}\nabla \tilde{P}_{N}\bigr)-\phi '( \tilde{Y}_{N})\tilde{P}_{N}+P_{Nt} \bigr)^{2}\,dx\,dt \\ &\quad {} +\delta \int_{0}^{T} \bigl\Vert e ^{p} \bigr\Vert _{H^{1}(\Omega )}^{2}\,dt \\ &\leq C(\delta )\eta_{2}^{2}+\delta \bigl\Vert p(U_{N})-P_{N} \bigr\Vert _{L^{2}(0,T;H^{1}(\Omega ))}^{2}, \end{aligned} \end{aligned}$$
(3.20)
$$\begin{aligned} I_{2}&= \int_{0}^{T}\bigl(\phi ' \bigl(y(U_{N})\bigr) (\tilde{P}_{N}-P_{N}),p(U_{N})-P _{N}\bigr)\,dt \\ &\leq C(\delta ) \Vert \tilde{P}_{N}-P_{N} \Vert _{L^{2}(0,T;L^{2}( \Omega ))}^{2}+\delta \bigl\Vert p(U_{N})-P_{N} \bigr\Vert _{L^{2}(0,T;L^{2}(\Omega ))} ^{2} \\ &\leq C(\delta ) \int_{0}^{T} \int_{\Omega } \bigl\vert A^{*}\nabla ( \tilde{P}_{N}-P_{N}) \bigr\vert ^{2}\,dx\,dt+\delta \bigl\Vert p(U_{N})-P_{N} \bigr\Vert _{L^{2}(0,T;H ^{1}(\Omega ))}^{2} \\ &\leq C(\delta )\eta_{3}^{2}+\delta \bigl\Vert p(U_{N})-P _{N} \bigr\Vert _{L^{2}(0,T;H^{1}(\Omega ))}^{2}. \end{aligned}$$
(3.21)
$$\begin{aligned} \begin{aligned}[b] I_{3}&= \int_{0}^{T}\bigl(\nabla e^{p},A^{*} \nabla (\tilde{P}_{N}-P_{N})\bigr)\,dt \\ &\leq C(\delta ) \int_{0}^{T} \int_{\Omega } \bigl\vert A^{*}\nabla (\tilde{P} _{N}-P_{N}) \bigr\vert ^{2}\,dx\,dt+\delta \int_{0}^{T} \int_{\Omega } \bigl\vert \nabla e^{p} \bigr\vert ^{2}\,dx\,dt \\ &\leq C(\delta )\eta_{3}^{2}+\delta \bigl\Vert p(U_{N})-P_{N} \bigr\Vert _{L^{2}(0,T;H ^{1}(\Omega ))}^{2}. \end{aligned} \end{aligned}$$
(3.22)
$$\begin{aligned} \begin{aligned}[b] I_{4}&= \int_{0}^{T}\bigl(\hat{y}_{d}-y_{d},e^{p} \bigr)\,dt \\ &\leq C(\delta ) \Vert y _{d}-\hat{y}_{d} \Vert _{L^{2}(0,T;L^{2}(\Omega ))}^{2}+\delta \bigl\Vert p(U_{N})-P _{N} \bigr\Vert _{L^{2}(0,T;L^{2}(\Omega ))}^{2} \\ &\leq C(\delta )\eta^{2}_{4}+ \delta \bigl\Vert p(U_{N})-P_{N} \bigr\Vert _{L^{2}(0,T;H^{1}(\Omega ))}^{2}. \end{aligned} \end{aligned}$$
(3.23)
$$\begin{aligned} \begin{aligned}[b] I_{5}&= \int_{0}^{T}\bigl(\tilde{\phi }''( \tilde{Y}_{N}) \bigl(\tilde{Y}_{N}-y(U _{N}) \bigr)\tilde{P}_{N},p(U_{N})-P_{N}\bigr)\,dt \\ &\leq C(\delta ) \int_{0}^{T} \int_{\Omega } \bigl\vert y(U_{N})- \tilde{Y}_{N} \bigr\vert ^{2}\,dx\,dt+\delta \bigl\Vert p(U_{N})-P _{N} \bigr\Vert _{L^{2}(0,T;L^{2}(\Omega ))}^{2} \\ &\leq C(\delta ) \bigl\Vert y(U_{N})-Y _{N} \bigr\Vert _{L^{2}(0,T;L^{2}(\Omega ))}^{2}+C(\delta ) \Vert Y_{N}- \tilde{Y} _{N} \Vert _{L^{2}(0,T;L^{2}(\Omega ))}^{2} \\ & \quad {} +\delta \bigl\Vert p(U_{N})-P_{N} \bigr\Vert _{L^{2}(0,T;H^{1}(\Omega ))}^{2} \\ &\leq C(\delta )\eta_{5}^{2}+C( \delta ) \bigl\Vert y(U_{N})-Y_{N} \bigr\Vert _{L^{2}(0,T;H^{1}(\Omega ))}^{2}+ \delta \bigl\Vert p(U _{N})-P_{N} \bigr\Vert _{L^{2}(0,T;H^{1}(\Omega ))}^{2}. \end{aligned} \end{aligned}$$
(3.24)
$$\begin{aligned} \begin{aligned}[b] I_{6}&= \int_{0}^{T}\bigl(y(U_{N})- \hat{Y}_{N},e^{p}\bigr)\,dt \\ &\leq C(\delta ) \int_{0}^{T} \int_{\Omega } \bigl\vert y(U_{N})- \hat{Y}_{N} \bigr\vert ^{2}\,dx\,dt+\delta \bigl\Vert p(U _{N})-P_{N} \bigr\Vert _{L^{2}(0,T;L^{2}(\Omega ))}^{2} \\ &\leq C(\delta ) \bigl\Vert y(U _{N})-Y_{N} \bigr\Vert _{L^{2}(0,T;L^{2}(\Omega ))}^{2}+C(\delta ) \Vert Y_{N}- \hat{Y}_{N} \Vert _{L^{2}(0,T;L^{2}(\Omega ))}^{2} \\ &\quad {} +\delta \bigl\Vert p(U_{N})-P _{N} \bigr\Vert _{L^{2}(0,T;H^{1}(\Omega ))}^{2} \\ &\leq C(\delta )\eta_{6}^{2}+C( \delta ) \bigl\Vert y(U_{N})-Y_{N} \bigr\Vert _{L^{2}(0,T;H^{1}(\Omega ))}^{2}+ \delta \bigl\Vert p(U _{N})-P_{N} \bigr\Vert _{L^{2}(0,T;H^{1}(\Omega ))}^{2} . \end{aligned} \end{aligned}$$
(3.25)
$$ \bigl\Vert p(U_{N})-P_{N} \bigr\Vert _{L^{2}(0,T;H^{1}(\Omega ))}^{2}\leq C(\delta ) \sum_{i=2}^{6} \eta_{i}^{2}+C(\delta ) \bigl\Vert y(U_{N})-Y_{N} \bigr\Vert _{L^{2}(0,T;H ^{1}(\Omega ))}^{2}. $$
(3.26)
Part II. Let \(e^{y}=y(U_{N})-Y_{N}\), \(e_{I}^{y}=P^{0}_{1,N} e ^{y}\in V^{N}\), where \(P^{0}_{1,N}\) is the orthogonal projection operator defined as in Lemma 2.2. Note that
then we have
Thus
From (2.30), it is easy to show that
By using (3.27), we can get
Combining (3.28) and (3.29), note that \(e^{y}-e_{I}^{y} \in H_{0}^{1}(\Omega )\), from (2.25) and (3.4), we can obtain
From Lemma (2.2), we have
For \(J_{2}\), by using the assumption of ϕ, we can get
For \(J_{3}\), it is clear that
For \(J_{4}\), we can obtain
By simplifying both sides of (3.30), we get
Then we derive (3.14) follows from (3.26) and (3.35). □
$$\begin{aligned} \begin{aligned}[b] \int_{0}^{T}\bigl(y_{t}(U_{N})-Y_{Nt},e^{y} \bigr)\,dt&= \int_{\Omega } \int_{0}^{T}e ^{y}\bigl(y_{t}(U_{N})-Y_{Nt} \bigr)\,dt\,dx \\ &= \int_{\Omega } \int_{0}^{T}e^{y}\frac{d(y(U_{N})-Y_{N})}{dt}\,dt\,dx \\ &= \int_{\Omega } \int_{0}^{T}e^{y}\,d\bigl(y(U_{N})-Y_{N} \bigr)\,dx \\ &= \int_{\Omega }\bigl(\bigl(y(U_{N})-Y_{N}\bigr) (x,T)\bigr)^{2}\,dx- \int_{\Omega }\bigl(\bigl(y(U _{N})-Y_{N} \bigr) (x,0)\bigr)^{2}\,dx \\ &\quad {} - \int_{0}^{T}\bigl(y_{t}(U_{N})-Y_{Nt},e^{y} \bigr)\,dt, \end{aligned} \end{aligned}$$
$$\begin{aligned} \begin{aligned} \int_{0}^{T}\bigl(y_{t}(U_{N})-Y_{Nt},e^{y} \bigr)\,dt&=\frac{1}{2} \int_{\Omega }\bigl(\bigl(y(U _{N})-Y_{N} \bigr) (x,T)\bigr)^{2}\,dx\\&\quad{} -\frac{1}{2} \int_{\Omega }\bigl(\bigl(y(U_{N})-Y_{N}\bigr) (x,0)\bigr)^{2}\,dx.\end{aligned} \end{aligned}$$
$$\begin{aligned} \int_{0}^{T}\bigl(y_{t}(U_{N})-Y_{Nt},e^{y} \bigr)\,dt+\frac{1}{2} \bigl\Vert y_{0}(x)-Y_{N}(x,0) \bigr\Vert _{L^{2}(\Omega )}^{2}\geq 0. \end{aligned}$$
(3.27)
$$\begin{aligned} \bigl(\phi \bigl(y(U_{N})\bigr)-\phi (Y_{N}),e^{y}\bigr)=\bigl(\tilde{\phi }' \bigl(y(U_{N})\bigr) \bigl(y(U _{N})-Y_{N} \bigr),e^{y}\bigr)\geq 0. \end{aligned}$$
(3.28)
$$\begin{aligned} \begin{aligned}[b] c \bigl\Vert e^{y} \bigr\Vert _{L^{2}(0,T;H^{1}(\Omega ))}^{2}&\leq \int_{0}^{T}a\bigl(y(U_{N})-Y _{N},e^{y}\bigr)\,dt+ \int_{0}^{T}\bigl(\phi \bigl(y(U_{N}) \bigr)-\phi (Y_{N}),e^{y}\bigr)\,dt \\ &\leq \int_{0}^{T}a\bigl(y(U_{N})-Y_{N},e^{y} \bigr)\,dt+ \int_{0}^{T}\bigl(\phi \bigl(y(U _{N}) \bigr)-\phi (\hat{Y}_{N}),e^{y}\bigr)\,dt \\ & \quad {} + \int_{0}^{T}\bigl(y_{t}(U_{N})-Y_{Nt},e ^{y}\bigr)\,dt+ \int_{0}^{T}\bigl(\phi (\hat{Y}_{N})-\phi (Y_{N}),e^{y}\bigr)\,dt \\ & \quad {} + \frac{1}{2} \bigl\Vert y_{0}(x)-Y_{N}(x,0) \bigr\Vert _{L^{2}(\Omega )}^{2} \\ &= \int_{0} ^{T}\bigl(A\nabla \bigl(y(U_{N})- \hat{Y}_{N}\bigr),\nabla e^{y}\bigr)\,dt+ \int_{0}^{T}\bigl( \phi \bigl(y(U_{N}) \bigr)-\phi (\hat{Y}_{N}),e^{y}\bigr)\,dt \\ & \quad {} + \int_{0}^{T}\bigl(A\nabla (\hat{Y}_{N}-Y_{N}), \nabla e^{y}\bigr)\,dt+ \int_{0}^{T}\bigl(y_{t}(U_{N})-Y_{Nt},e ^{y}\bigr)\,dt \\ &\quad {} + \int_{0}^{T}\bigl(\phi (\hat{Y}_{N})-\phi (Y_{N}),e^{y}\bigr)\,dt+ \frac{1}{2} \bigl\Vert y_{0}(x)-Y_{N}(x,0) \bigr\Vert _{L^{2}(\Omega )}^{2}. \end{aligned} \end{aligned}$$
(3.29)
$$\begin{aligned} c \bigl\Vert e^{y} \bigr\Vert _{L^{2}(0,T;H^{1}(\Omega ))}^{2}&\leq \int_{0}^{T}\bigl(\hat{f}+BU _{N}+ \operatorname{div}(A\nabla \hat{Y}_{N})-\phi (\tilde{Y}_{N})-Y_{Nt},e^{y}-e _{I}^{y}\bigr)\,dt \\ & \quad {} + \int_{0}^{T}\bigl(\tilde{\phi }'( \hat{Y}_{N}) (\hat{Y}_{N}-Y _{N}),e^{y} \bigr)\,dt+ \int_{0}^{T}\bigl(A\nabla (\hat{Y}_{N}-Y_{N}), \nabla e^{y}\bigr)\,dt \\ & \quad {} + \int_{0}^{T}\bigl(f-\hat{f},e^{y} \bigr)\,dt+\frac{1}{2} \bigl\Vert Y_{N}(x,0)-y_{0}(x) \bigr\Vert _{L^{2}(\Omega )}^{2} \\ &\equiv J_{1}+J_{2}+J_{3}+J_{4}+ \frac{1}{2} \eta_{10}^{2}. \end{aligned}$$
(3.30)
$$\begin{aligned} \begin{aligned}[b] J_{1}&= \int_{0}^{T}\bigl(\hat{f}+BU_{N}+ \operatorname{div}(A\nabla \hat{Y}_{N})- \phi (\tilde{Y}_{N})-Y_{Nt},e^{y}-e_{I}^{y} \bigr)\,dt \\ &\leq C(\delta ) \int _{0}^{T}N^{-2} \int_{\Omega }\bigl(\hat{f}+BU_{N}+\operatorname{div}(A\nabla \hat{Y} _{N})-\phi (\tilde{Y}_{N})-Y_{Nt} \bigr)^{2}\,dx\,dt \\ & \quad {} +\delta \int_{0}^{T} \bigl\Vert e ^{y} \bigr\Vert _{H^{1}(\Omega )}^{2}\,dt \\ &\leq C(\delta )\eta_{7}^{2}+\delta \bigl\Vert y(U_{N})-Y_{N} \bigr\Vert _{L^{2}(0,T;H^{1}(\Omega ))}^{2}. \end{aligned} \end{aligned}$$
(3.31)
$$\begin{aligned} \begin{aligned}[b] J_{2}&= \int_{0}^{T}\bigl(\tilde{\phi }'( \hat{Y}_{N}) (\hat{Y}_{N}-Y_{N}),e ^{y}\bigr)\,dt \\ &\leq C(\delta ) \Vert Y_{N}-\hat{Y}_{N} \Vert _{L^{2}(0,T;L^{2}( \Omega ))}^{2}+\delta \bigl\Vert e^{y} \bigr\Vert _{L^{2}(0,T;L^{2}(\Omega ))}^{2} \\ &\leq C(\delta )\eta_{6}^{2}+\delta \bigl\Vert e^{y} \bigr\Vert _{L^{2}(0,T;H^{1}(\Omega ))}^{2}. \end{aligned} \end{aligned}$$
(3.32)
$$\begin{aligned} \begin{aligned}[b] J_{3}&= \int_{0}^{T}\bigl(A\nabla (\hat{Y}_{N}-Y_{N}), \nabla e^{y}\bigr)\,dt \\ &\leq C(\delta ) \int_{0}^{T} \int_{\Omega } \bigl\vert A\nabla (\hat{Y}_{N}-Y_{N}) \bigr\vert ^{2}\,dx\,dt+ \delta \int_{0}^{T} \int_{\Omega } \bigl\vert A\nabla e^{y} \bigr\vert ^{2}\,dx\,dt \\ &\leq C( \delta )\eta_{8}^{2}+\delta \bigl\Vert y(U_{N})-Y_{N} \bigr\Vert _{L^{2}(0,T;H^{1}( \Omega ))}^{2}. \end{aligned} \end{aligned}$$
(3.33)
$$\begin{aligned} \begin{aligned}[b] J_{4}&= \int_{0}^{T}\bigl(f-\hat{f},e^{y}\bigr)\,dt \\ &\leq C(\delta ) \Vert f-\hat{f} \Vert _{L^{2}(0,T;L^{2}(\Omega ))}^{2}+\delta \bigl\Vert y(U_{N})-Y_{N} \bigr\Vert _{L^{2}(0,T;L ^{2}(\Omega ))}^{2} \\ &\leq C(\delta )\eta_{9}^{2}+\delta \bigl\Vert y(U_{N})-Y _{N} \bigr\Vert _{L^{2}(0,T;H^{1}(\Omega ))}^{2}. \end{aligned} \end{aligned}$$
(3.34)
$$ \bigl\Vert y(U_{N})-Y_{N} \bigr\Vert _{L^{2}(0,T;H^{1}(\Omega ))}^{2}\leq C(\delta ) \sum_{i=6}^{10} \eta_{i}^{2}. $$
(3.35)
Theorem 3.3
Let
\((y,p,u)\)
and
\((Y_{N},P_{N},U_{N})\)
be the solutions of (2.8)–(2.10) and (2.25)–(2.29), respectively. Assume that all the conditions in Theorem 3.1
and Theorem 3.2
are valid. Then
where
\(\eta_{i}^{2}\), \(i=1,\ldots,10\), are defined in Theorem 3.1
and Theorem 3.2.
$$ \Vert Y_{N}-y \Vert _{L^{2}(0,T;H^{1}(\Omega ))}^{2}+ \Vert P_{N}-p \Vert _{L^{2}(0,T;H ^{1}(\Omega ))}^{2}+ \Vert U_{N}-u \Vert _{L^{2}(0,T;L^{2}(\Omega ))}^{2}\leq C \sum _{i=1}^{10}\eta_{i}^{2}, $$
(3.36)
Proof
Combining Theorem 3.1 and Theorem 3.2, we have
Thanks to A being positive definite, according to the Poincaré inequality, we can obtain
Plugging (3.38) into (3.37) yields
Note that
and
From (3.39), (3.40), (3.42), and Theorem 3.2, we can derive
Similarly, from (3.39), (3.41), (3.43), and Theorem 3.2, we also can derive
Therefore, (3.36) follows from (3.39) and (3.44)–(3.45). □
$$\begin{aligned} \begin{aligned}[b] \Vert u-U_{N} \Vert _{L^{2}(0,T;L^{2}(\Omega ))}^{2}&\leq C\eta_{1}^{2}+C \bigl\Vert \tilde{P}_{N}-p(U_{N}) \bigr\Vert _{L^{2}(0,T;L^{2}(\Omega ))}^{2} \\ &\leq C \eta_{1}^{2}+C \Vert \tilde{P}_{N}-P_{N} \Vert _{L^{2}(0,T;L^{2}(\Omega ))}^{2}+C \bigl\Vert P_{N}-p(U_{N}) \bigr\Vert _{L^{2}(0,T;L^{2}(\Omega ))}^{2} \\ &\leq C\sum_{i=1}^{10} \eta_{i}^{2}+C \Vert \tilde{P}_{N}-P_{N} \Vert _{L^{2}(0,T;L ^{2}(\Omega ))}^{2}. \end{aligned} \end{aligned}$$
(3.37)
$$ \Vert \tilde{P}_{N}-P_{N} \Vert _{L^{2}(0,T;L^{2}(\Omega ))}^{2}\leq C \int_{0} ^{T} \int_{\Omega } \bigl\vert A^{*}\nabla ( \tilde{P}_{N}-P_{N}) \bigr\vert ^{2}\,dx\,dt=C\eta _{3}^{2}. $$
(3.38)
$$ \Vert u-U_{N} \Vert _{L^{2}(0,T;L^{2}(\Omega ))}^{2} \leq C\sum_{i=1}^{10}\eta _{i}^{2}. $$
(3.39)
$$\begin{aligned}& \Vert Y_{N}-y \Vert _{L^{2}(0,T;H^{1}(\Omega ))}^{2} \leq \bigl\Vert Y_{N}-y(U_{N}) \bigr\Vert _{L^{2}(0,T;H^{1}(\Omega ))}^{2} + \bigl\Vert y(U_{N})-y \bigr\Vert _{L^{2}(0,T;H^{1}( \Omega ))}^{2}, \end{aligned}$$
(3.40)
$$\begin{aligned}& \Vert P_{N}-p \Vert _{L^{2}(0,T;H^{1}(\Omega ))}^{2} \leq \bigl\Vert P_{N}-p(U_{N}) \bigr\Vert _{L^{2}(0,T;H^{1}(\Omega ))}^{2} + \bigl\Vert p(U_{N})-p \bigr\Vert _{L^{2}(0,T;H^{1}( \Omega ))}^{2}, \end{aligned}$$
(3.41)
$$\begin{aligned}& \bigl\Vert y(U_{N})-y \bigr\Vert _{L^{2}(0,T;H^{1}(\Omega ))}^{2}\leq C \Vert u-U_{N} \Vert _{L ^{2}(0,T;L^{2}(\Omega ))}^{2}, \end{aligned}$$
(3.42)
$$\begin{aligned}& \bigl\Vert p(U_{N})-p \bigr\Vert _{L^{2}(0,T;H^{1}(\Omega ))}^{2}\leq \bigl\Vert y(U_{N})-y \bigr\Vert _{L ^{2}(0,T;L^{2}(\Omega ))}^{2}\leq C \Vert u-U_{N} \Vert _{L^{2}(0,T;L^{2}( \Omega ))}^{2}. \end{aligned}$$
(3.43)
$$\begin{aligned} \begin{aligned}[b] \Vert Y_{N}-y \Vert _{L^{2}(0,T;H^{1}(\Omega ))}^{2}&\leq \bigl\Vert Y_{N}-y(U_{N}) \bigr\Vert _{L^{2}(0,T;H^{1}(\Omega ))}^{2} +C \Vert u-U_{N} \Vert _{L^{2}(0,T;L^{2}( \Omega ))}^{2} \\ &\leq C\sum_{i=1}^{10}\eta_{i}^{2}. \end{aligned} \end{aligned}$$
(3.44)
$$\begin{aligned} \Vert P_{N}-p \Vert _{L^{2}(0,T;H^{1}(\Omega ))}^{2} \leq C\sum_{i=1}^{10} \eta _{i}^{2}. \end{aligned}$$
(3.45)
4 \(L^{2}(L^{2})-L^{2}(L^{2})\) a posteriori error estimates
In this section, we shall carry out \(L^{2}(L ^{2})-L^{2}(L^{2})\) a posteriori error estimates for the spectral approximation of the optimal control problem governed by nonlinear parabolic equations. In order to estimate the error \(\Vert \tilde{P}_{N}-p(U _{N}) \Vert ^{2}_{L^{2}(0,T;L^{2}(\Omega ))}\), we shall use two auxiliary equations.
Now, we set the following dual auxiliary equations:
and
where
The following well-known stability results are presented in [10].
$$ \textstyle\begin{cases} -\frac{\partial \zeta }{\partial t}-\operatorname{div}(A^{*}\nabla {\zeta })+ \Phi \zeta =F, & x\in \Omega ,t\in (0,T]; \\ \zeta | _{\partial \Omega }=0, & t\in [0,T]; \\ \zeta (x,0)=0, & x\in \Omega , \end{cases} $$
(4.1)
$$ \textstyle\begin{cases} \frac{\partial \xi }{\partial t}-\operatorname{div}(A\nabla {\xi })+\phi '(y(U _{N}))\xi =F, & x\in \Omega , t\in (0,T]; \\ \xi | _{\partial \Omega }=0, & t\in [0,T]; \\ \xi (x,T)=0, & x\in \Omega , \end{cases} $$
(4.2)
$$ \Phi = \textstyle\begin{cases} \frac{\phi (y(U_{N}))-\phi (Y_{N})}{y(U_{N})-Y_{N}}, & y(U_{N})\neq Y _{N} , \\ \phi '(Y_{N}), & y(U_{N})= Y_{N} . \end{cases} $$
Lemma 4.1
Assume that Ω is a convex domain. Let
ζ
and
ξ
be the solutions of (4.1) and (4.2), respectively. Then, for
\(v=\zeta \)
or
\(v=\xi \),
where
\(D^{2} v=\partial^{2} v /{\partial x_{i}\partial x_{j}}\), \(1 \leq i,j\leq n\).
$$\begin{aligned}& \Vert v \Vert _{L^{\infty }(0,T;L^{2}(\Omega ))}\leq C \Vert F \Vert _{L^{2}(0,T;L^{2}( \Omega ))}, \\& \Vert \nabla v \Vert _{L^{2}(0,T;L^{2}(\Omega ))}\leq C \Vert F \Vert _{L^{2}(0,T;L ^{2}(\Omega ))}, \\& \bigl\Vert D^{2} v \bigr\Vert _{L^{2}(0,T;L^{2}(\Omega ))}\leq C \Vert F \Vert _{L^{2}(0,T;L ^{2}(\Omega ))}, \\& \biggl\Vert \frac{\partial v}{\partial t} \biggr\Vert _{L^{2}(0,T;L^{2}( \Omega ))}\leq C \Vert F \Vert _{L^{2}(0,T;L^{2}(\Omega ))}, \end{aligned}$$
Theorem 4.1
Let
\((Y_{N},P_{N},U_{N})\)
be the solutions of (2.25)–(2.29), let
\((y(U_{N}),p(U_{N}))\)
be defined by (3.4)–(3.7). Then
where
$$ \bigl\Vert Y_{N}-y(U_{N}) \bigr\Vert ^{2}_{L^{2}(0,T;L^{2}(\Omega ))}+ \bigl\Vert P_{N}-p(U_{N}) \bigr\Vert ^{2}_{L^{2}(0,T;L^{2}(\Omega ))}\leq C\sum_{i=2}^{10} \hat{\eta }_{i}^{2}, $$
(4.3)
$$\begin{aligned}& \hat{\eta }^{2}_{2} = \int_{0}^{T}N^{-4} \int_{\Omega }\bigl(\hat{Y}_{N}- \hat{y}_{d}+ \operatorname{div}\bigl(A^{*}\nabla \tilde{P}_{N}\bigr)-\phi '(\tilde{Y}_{N}) \tilde{P}_{N}+P_{Nt} \bigr)^{2}\,dx\,dt, \\& \hat{\eta }^{2}_{3} = \int_{0}^{T} \int_{\Omega } \bigl\vert A^{*}\nabla ( \tilde{P}_{N}-P_{N}) \bigr\vert ^{2}\,dx\,dt, \\& \hat{\eta }^{2}_{4} = \Vert y_{d}- \hat{y}_{d} \Vert _{L^{2}(0,T;L^{2}(\Omega ))}^{2}, \\& \hat{\eta }^{2}_{5} = \Vert Y_{N}- \tilde{Y}_{N} \Vert _{L^{2}(0,T;L^{2}( \Omega ))}^{2}, \\& \hat{\eta }^{2}_{6} = \Vert Y_{N}- \hat{Y}_{N} \Vert _{L^{2}(0,T;L^{2}(\Omega ))}^{2}, \\& \hat{\eta }^{2}_{7} = \int_{0}^{T}N^{-4} \int_{\Omega }\bigl(\hat{f}+BU _{N}+\operatorname{div}(A\nabla \hat{Y}_{N})-\phi (\hat{Y}_{N})-Y_{Nt} \bigr)^{2}\,dx\,dt, \\& \hat{\eta }^{2}_{8} = \int_{0}^{T} \int_{\Omega } \bigl\vert A\nabla (\hat{Y} _{N}-Y_{N}) \bigr\vert ^{2}\,dx\,dt, \\& \hat{\eta }^{2}_{9} = \Vert f-\hat{f} \Vert ^{2}_{L^{2}(0,T;L^{2}(\Omega ))}, \\& \hat{\eta }^{2}_{10} = \bigl\Vert Y_{N}(x,0)-y_{0}(x) \bigr\Vert ^{2}_{L^{2}(\Omega )}. \end{aligned}$$
Proof
Part I. Let ξ be the solution of (4.2) with \(F= p(U_{N})-P_{N}\). Let \(\xi_{I}=P^{0}_{1,N} \xi \in V^{N}\), where \(P^{0}_{1,N}\) is the orthogonal projection operator defined as in Lemma 2.2. Note that \(\xi -\xi_{I}\in H_{0}^{1}(\Omega )\), it follows from (2.27) and (3.6) that
From (4.4), we have
By using Lemma 2.2 and Lemma 4.1, we have
For \(K_{2}\), it is easy to see that
By using the assumption of ϕ, we have
For \(K_{4}\), we have
In the same way as (4.7), we have
Note that \(\phi (\cdot )\in W^{2,\infty }(-R,-R)\), we obtain
For the last term \(K_{7}\), we have
Putting (4.6)–(4.12) into (4.5), letting δ be small enough, we obtain
$$\begin{aligned}& \bigl\Vert p(U_{N})-P_{N} \bigr\Vert ^{2}_{L^{2}(0,T;L^{2}(\Omega ))} \\& \quad = \int_{0}^{T}\bigl(p(U _{N})-P_{N},F \bigr)\,dt \\& \quad = \int_{0}^{T} \bigl(-\bigl(p_{t}(U_{N})-P_{Nt}, \xi \bigr)+a\bigl( \xi , p(U_{N})-P_{N}\bigr)+\bigl(\phi '\bigl(y(U_{N})\bigr) \bigl(p(U_{N})-P_{N} \bigr),\xi \bigr) \bigr)\,dt \\& \quad = \int_{0}^{T} \bigl( -\bigl(p_{t}(U_{N})-P_{Nt}, \xi -\xi_{I}\bigr)+a\bigl(\xi - \xi_{I}, p(U_{N})-\tilde{P}_{N}\bigr) \bigr) \,dt \\& \quad \quad {} + \int_{0}^{T}\bigl(\phi '\bigl(y(U _{N})\bigr)p(U_{N})-\phi '( \tilde{Y}_{N})\tilde{P}_{N},\xi -\xi_{I}\bigr)\,dt \\& \quad \quad {} + \int_{0}^{T} \bigl( -\bigl(p_{t}(U_{N})-P_{Nt}, \xi_{I}\bigr)+a\bigl(\xi_{I}, p(U_{N})- \tilde{P}_{N}\bigr) \bigr) \,dt \\& \quad \quad {} + \int_{0}^{T} \bigl( a(\xi ,\tilde{P}_{N}-P _{N})+\bigl(\phi '(\tilde{Y}_{N}) \tilde{P}_{N}-\phi '\bigl(y(U_{N})\bigr) \tilde{P} _{N},\xi \bigr) \bigr) \,dt \\& \quad \quad {} + \int_{0}^{T}\bigl(\phi ' \bigl(y(U_{N})\bigr)p(U_{N})-\phi '( \tilde{Y}_{N})\tilde{P}_{N},\xi_{I}\bigr)\,dt \\& \quad \quad {} + \int_{0}^{T}\bigl(\phi ' \bigl(y(U_{N})\bigr) ( \tilde{P}_{N}-P_{N}),\xi \bigr)\,dt. \end{aligned}$$
(4.4)
$$\begin{aligned} \begin{aligned}[b] & \bigl\Vert p(U_{N})-P_{N} \bigr\Vert ^{2}_{L^{2}(0,T;L^{2}(\Omega ))} \\ &\quad = \int_{0}^{T}\bigl(P _{Nt}+\operatorname{div} \bigl(A^{*}\nabla \tilde{P}_{N}\bigr)+\hat{Y}_{N}- \hat{y}_{d}- \phi '(\tilde{Y}_{N}) \tilde{P}_{N}, \xi -\xi_{I}\bigr)\,dt \\ & \quad\quad {} + \int_{0}^{T}\bigl(y(U _{N})-Y_{N}, \xi \bigr)\,dt+ \int_{0}^{T} \bigl(\phi ' \bigl(y(U_{N})\bigr) (\tilde{P}_{N}-P _{N}),\xi \bigr)\,dt \\ &\quad\quad {} + \int_{0}^{T}a(\xi ,\tilde{P}_{N}-P_{N})\,dt+ \int_{0} ^{T}(\hat{y}_{d}-y_{d}, \xi )\,dt \\ &\quad\quad {} + \int_{0}^{T}\bigl(\tilde{\phi }''( \tilde{Y}_{N}) \bigl(\tilde{Y}_{N}-y(U_{N}) \bigr)\tilde{P}_{N},\xi \bigr)\,dt+ \int_{0} ^{T}(Y_{N}-\hat{Y}_{N}, \xi )\,dt \\ &\quad \equiv \sum_{i=1}^{7}K_{i}. \end{aligned} \end{aligned}$$
(4.5)
$$\begin{aligned} \begin{aligned}[b] K_{1}&= \int_{0}^{T}\bigl(P_{Nt}+\operatorname{div} \bigl(A^{*}\nabla \tilde{P}_{N}\bigr)+ \hat{Y}_{N}- \hat{y}_{d}-\phi '(\tilde{Y}_{N}) \tilde{P}_{N}, \xi -\xi _{I}\bigr)\,dt \\ &\leq C(\delta ) \int_{0}^{T}N^{-4} \int_{\Omega }\bigl(\hat{Y} _{N}-\hat{y}_{d}+ \operatorname{div}\bigl(A^{*}\nabla \tilde{P}_{N}\bigr)-\phi '( \tilde{Y}_{N})\tilde{P}_{N}+P_{Nt} \bigr)^{2}\,dx\,dt \\ & \quad {} +\delta \int_{0}^{T} \Vert \xi \Vert _{H^{2}(\Omega )}^{2}\,dt \\ &\leq C(\delta )\hat{\eta }_{2}^{2}+ \delta \bigl\Vert p(U_{N})-P_{N} \bigr\Vert _{L^{2}(0,T;L^{2}(\Omega ))}^{2}. \end{aligned} \end{aligned}$$
(4.6)
$$\begin{aligned} \begin{aligned}[b] K_{2}&= \int_{0}^{T}\bigl(y(U_{N})-Y_{N}, \xi \bigr)\,dt \\ &\leq C(\delta ) \bigl\Vert y(U _{N})-Y_{N} \bigr\Vert _{L^{2}(0,T;L^{2}(\Omega ))}^{2}+\delta \Vert \xi \Vert _{L^{2}(0,T;L ^{2}(\Omega ))}^{2} \\ &\leq C(\delta ) \bigl\Vert y(U_{N})-Y_{N} \bigr\Vert _{L^{2}(0,T;L ^{2}(\Omega ))}^{2}+\delta \bigl\Vert p(U_{N})-P_{N} \bigr\Vert _{L^{2}(0,T;L^{2}(\Omega ))}^{2}. \end{aligned} \end{aligned}$$
(4.7)
$$\begin{aligned} \begin{aligned} [b] K_{3}&= \int_{0}^{T}\bigl(\phi ' \bigl(y(U_{N})\bigr) (\tilde{P}_{N}-P_{N}),\xi \bigr)\,dt \\ &\leq C(\delta ) \Vert \tilde{P}_{N}-P_{N} \Vert _{L^{2}(0,T;L^{2}(\Omega ))} ^{2}+\delta \Vert \xi \Vert _{L^{2}(0,T;L^{2}(\Omega ))}^{2} \\ &\leq C( \delta ) \int_{0}^{T} \int_{\Omega } \bigl\vert A^{*}\nabla ( \tilde{P}_{N}-P_{N}) \bigr\vert ^{2}\,dx\,dt+ \delta \bigl\Vert p(U_{N})-P_{N} \bigr\Vert _{L^{2}(0,T;L^{2}(\Omega ))}^{2} \\ &\leq C( \delta )\hat{\eta }_{3}^{2}+\delta \bigl\Vert p(U_{N})-P_{N} \bigr\Vert _{L^{2}(0,T;L ^{2}(\Omega ))}^{2}. \end{aligned} \end{aligned}$$
(4.8)
$$\begin{aligned} \begin{aligned} [b] K_{4}&= \int_{0}^{T}a(\xi ,\tilde{P}_{N}-P_{N})\,dt \\ &\leq C(\delta ) \int_{0}^{T} \int_{\Omega } \bigl\vert A^{*}\nabla ( \tilde{P}_{N}-P_{N}) \bigr\vert ^{2}\,dx\,dt+ \delta \Vert \xi \Vert _{L^{2}(0,T;L^{2}(\Omega ))}^{2} \\ &\leq C(\delta ) \hat{\eta }_{3}^{2}+\delta \bigl\Vert p(U_{N})-P_{N} \bigr\Vert _{L^{2}(0,T;L^{2}(\Omega ))}^{2}. \end{aligned} \end{aligned}$$
(4.9)
$$\begin{aligned} \begin{aligned} [b] K_{5}&= \int_{0}^{T}(\hat{y}_{d}-y_{d}, \xi )\,dt \\ &\leq C(\delta ) \Vert y _{d}-\hat{y}_{d} \Vert _{L^{2}(0,T;L^{2}(\Omega ))}^{2}+\delta \Vert \xi \Vert _{L^{2}(0,T;L^{2}(\Omega ))}^{2} \\ &\leq C(\delta )\hat{\eta }_{4} ^{2}+\delta \bigl\Vert p(U_{N})-P_{N} \bigr\Vert _{L^{2}(0,T;L^{2}(\Omega ))}^{2}. \end{aligned} \end{aligned}$$
(4.10)
$$\begin{aligned} \begin{aligned}[b] K_{6}&= \int_{0}^{T}\bigl(\tilde{\phi }''( \tilde{Y}_{N}) \bigl(\tilde{Y}_{N}-y(U _{N}) \bigr)\tilde{P}_{N},\xi \bigr)\,dt \\ &\leq C(\delta ) \int_{0}^{T} \int_{ \Omega } \bigl\vert y(U_{N})- \tilde{Y}_{N} \bigr\vert ^{2}\,dx\,dt+\delta \Vert \xi \Vert _{L^{2}(0,T;L ^{2}(\Omega ))}^{2} \\ &\leq C(\delta ) \bigl\Vert y(U_{N})-Y_{N} \bigr\Vert _{L^{2}(0,T;L ^{2}(\Omega ))}^{2}+C(\delta ) \Vert Y_{N}- \tilde{Y}_{N} \Vert _{L^{2}(0,T;L ^{2}(\Omega ))}^{2} \\ &\quad {} +\delta \bigl\Vert p(U_{N})-P_{N} \bigr\Vert _{L^{2}(0,T;L^{2}( \Omega ))}^{2} \\ &\leq C(\delta )\hat{\eta }_{5}^{2}+C(\delta ) \bigl\Vert y(U _{N})-Y_{N} \bigr\Vert _{L^{2}(0,T;L^{2}(\Omega ))}^{2}+ \delta \bigl\Vert p(U_{N})-P_{N} \bigr\Vert _{L^{2}(0,T;L^{2}(\Omega ))}^{2}. \end{aligned} \end{aligned}$$
(4.11)
$$\begin{aligned} \begin{aligned}[b] K_{7}&= \int_{0}^{T}(Y_{N}-\hat{Y}_{N}, \xi )\,dt \\ &\leq C(\delta ) \int _{0}^{T} \int_{\Omega } \vert Y_{N}-\hat{Y}_{N} \vert ^{2}\,dx\,dt+\delta \Vert \xi \Vert _{L ^{2}(0,T;L^{2}(\Omega ))}^{2} \\ &\leq C(\delta )\hat{\eta }_{6}^{2}+ \delta \bigl\Vert p(U_{N})-P_{N} \bigr\Vert _{L^{2}(0,T;L^{2}(\Omega ))}^{2}. \end{aligned} \end{aligned}$$
(4.12)
$$ \bigl\Vert p(U_{N})-P_{N} \bigr\Vert _{L^{2}(0,T;L^{2}(\Omega ))}^{2}\leq C(\delta ) \sum_{i=2}^{6} \hat{\eta }_{i}^{2}+C(\delta ) \bigl\Vert y(U_{N})-Y_{N} \bigr\Vert _{L^{2}(0,T;L^{2}(\Omega ))}^{2}. $$
(4.13)
Part II. Let ζ be the solution of (4.1) with \(F=y(U_{N})-Y_{N}\). Let \(\zeta_{I}=P^{0}_{1,N} \zeta \in V^{N}\), where \(P^{0}_{1,N}\) is the orthogonal projection operator defined as in Lemma 2.2. Note that \(\zeta -\zeta_{I}\in H_{0}^{1}(\Omega )\), from (2.25) and (3.4), we have
Hence, let δ be small enough, by simplifying both sides, we have
Then (4.3) follows from (4.13) and (4.14). □
$$\begin{aligned} \begin{aligned} & \bigl\Vert y(U_{N})-Y_{N} \bigr\Vert ^{2}_{L^{2}(0,T;L^{2}(\Omega ))} \\ &\quad = \int_{0}^{T}\bigl(y(U _{N})-Y_{N},F \bigr)\,dt \\ &\quad = \int_{0}^{T} \bigl(\bigl(y_{t}(U_{N})-Y_{Nt}, \zeta \bigr)+a\bigl(y(U_{N})-Y_{N}, \zeta \bigr) \bigr)\,dt \\ &\quad \quad {} + \int_{0}^{T}\bigl(\phi \bigl(y(U_{N}) \bigr)-\phi (Y_{N}),\zeta \bigr)\,dt+\bigl(\bigl(y(U_{N})-Y _{N}\bigr) (x,0),\zeta (x,0)\bigr) \\ &\quad = \int_{0}^{T}\bigl(\hat{f}+BU_{N}+ \operatorname{div}(A\nabla \hat{Y}_{N})- \phi (\hat{Y}_{N})-Y_{Nt}, \zeta -\zeta_{I}\bigr)\,dt \\ &\quad \quad {} + \int_{0}^{T}(f-\hat{f},\zeta )\,dt+ \int_{0}^{T}a(\hat{Y}_{N}-Y_{N}, \zeta )\,dt \\ &\quad \quad {} + \int_{0}^{T}\bigl(\phi (\hat{Y}_{N})-\phi (Y_{N}),\zeta \bigr)\,dt+\bigl(\bigl(y(U_{N})-Y _{N}\bigr) (x,0),\zeta (x,0)\bigr) \\ &\quad \leq C(\delta ) \int_{0}^{T}N^{-4} \int_{\Omega }\bigl(\hat{f}+BU_{N}+ \operatorname{div}(A\nabla \hat{Y}_{N})-\phi (\hat{Y}_{N})-Y_{Nt} \bigr)^{2}\,dx\,dt \\ &\quad \quad {} +C(\delta ) \int_{0}^{T} \int_{\Omega } \bigl\vert A\nabla (\hat{Y}_{N}-Y_{N}) \bigr\vert ^{2}\,dx\,dt+C( \delta ) \Vert f-\hat{f} \Vert ^{2}_{L^{2}(0,T;L^{2}(\Omega ))} \\ &\quad \quad {} +C(\delta ) \Vert Y_{N}-\hat{Y}_{N} \Vert ^{2}_{L^{2}(0,T;L^{2}(\Omega ))}+C( \delta ) \bigl\Vert Y_{N}(x,0)-y_{0}(x) \bigr\Vert ^{2}_{L^{2}(\Omega )} \\ &\quad \quad {} +\delta \int_{0}^{T} \Vert \zeta \Vert _{H^{2}(\Omega )}^{2}\,dt+\delta \bigl\Vert \zeta (x,0) \bigr\Vert _{L^{2}(\Omega )}^{2} \\ &\quad \leq C(\delta )\sum_{6}^{10}\hat{\eta }_{i}^{2}+\delta \bigl\Vert Y _{N}-y(U_{N}) \bigr\Vert ^{2}_{L^{2}(0,T;L^{2}(\Omega ))}. \end{aligned} \end{aligned}$$
$$ \bigl\Vert y(U_{N})-Y_{N} \bigr\Vert ^{2}_{L^{2}(0,T;L^{2}(\Omega ))}\leq C\sum_{6}^{10} \hat{\eta }_{i}^{2}. $$
(4.14)
Theorem 4.2
Let
\((y,p,u)\)
and
\((Y_{N},P_{N},U_{N})\)
be the solutions of (2.8)–(2.10) and (2.25)–(2.29), respectively. Assume that all the conditions in Theorem 3.1
and Lemma 4.1
are valid. Then
where
\(\eta_{1}^{2}\), \(\hat{\eta }_{i}^{2}\), \(i=2,\ldots,10\)
are defined in Theorem 3.1
and Theorem 4.1.
$$ \begin{aligned}[b] & \Vert Y_{N}-y \Vert _{L^{2}(0,T;L^{2}(\Omega ))}^{2}+ \Vert P_{N}-p \Vert _{L^{2}(0,T;L ^{2}(\Omega ))}^{2}+ \Vert U_{N}-u \Vert _{L^{2}(0,T;L^{2}(\Omega ))}^{2}\\&\quad \leq C \eta_{1}^{2}+C\sum_{i=2}^{10} \hat{\eta }_{i}^{2},\end{aligned} $$
(4.15)
Proof
From Theorem 3.1 and Theorem 4.1, we have
By virtue of A being positive definite, by using the Poincaré inequality, we can get
Then it follows from (4.16) and (4.17) that
Note that
and
Therefore, (4.15) follows from (4.16) and (4.18)–(4.22). □
$$\begin{aligned} \begin{aligned}[b] \Vert u-U_{N} \Vert _{L^{2}(0,T;L^{2}(\Omega ))}^{2}&\leq C\eta_{1}^{2}+C \bigl\Vert \tilde{P}_{N}-p(U_{N}) \bigr\Vert _{L^{2}(0,T;L^{2}(\Omega ))}^{2} \\ &\leq C \eta_{1}^{2}+C \Vert \tilde{P}_{N}-P_{N} \Vert _{L^{2}(0,T;L^{2}(\Omega ))}^{2} \\ &\quad {} +C \bigl\Vert P_{N}-p(U_{N}) \bigr\Vert _{L^{2}(0,T;L^{2}(\Omega ))}^{2} \\ &\leq C \eta_{1}^{2}+C\sum_{i=2}^{10} \hat{\eta }_{i}^{2}+C \Vert \tilde{P} _{N}-P_{N} \Vert _{L^{2}(0,T;L^{2}(\Omega ))}^{2}. \end{aligned} \end{aligned}$$
(4.16)
$$ \Vert \tilde{P}_{N}-P_{N} \Vert _{L^{2}(0,T;L^{2}(\Omega ))}^{2}\leq C \int_{0} ^{T} \int_{\Omega } \bigl\vert A^{*}\nabla ( \tilde{P}_{N}-P_{N}) \bigr\vert ^{2}\,dx\,dt=C \hat{ \eta }_{3}^{2}. $$
(4.17)
$$ \Vert u-U_{N} \Vert _{L^{2}(0,T;L^{2}(\Omega ))}^{2} \leq C\eta_{1}^{2}+C\sum_{i=2}^{10} \hat{\eta }_{i}^{2}. $$
(4.18)
$$\begin{aligned}& \Vert Y_{N}-y \Vert _{L^{2}(0,T;L^{2}(\Omega ))}^{2} \leq \bigl\Vert Y_{N}-y(U_{N}) \bigr\Vert _{L^{2}(0,T;L^{2}(\Omega ))}^{2} + \bigl\Vert y(U_{N})-y \bigr\Vert _{L^{2}(0,T;L^{2}( \Omega ))}^{2}, \end{aligned}$$
(4.19)
$$\begin{aligned}& \Vert P_{N}-p \Vert _{L^{2}(0,T;L^{2}(\Omega ))}^{2} \leq \bigl\Vert P_{N}-p(U_{N}) \bigr\Vert _{L^{2}(0,T;L^{2}(\Omega ))}^{2} + \bigl\Vert p(U_{N})-p \bigr\Vert _{L^{2}(0,T;L^{2}( \Omega ))}^{2}, \end{aligned}$$
(4.20)
$$\begin{aligned}& \bigl\Vert y(U_{N})-y \bigr\Vert _{L^{2}(0,T;L^{2}(\Omega ))}^{2}\leq C \Vert u-U_{N} \Vert _{L ^{2}(0,T;L^{2}(\Omega ))}^{2}, \end{aligned}$$
(4.21)
$$\begin{aligned}& \bigl\Vert p(U_{N})-p \bigr\Vert _{L^{2}(0,T;L^{2}(\Omega ))}^{2}\leq \bigl\Vert y(U_{N})-y \bigr\Vert _{L ^{2}(0,T;L^{2}(\Omega ))}^{2}\leq C \Vert u-U_{N} \Vert _{L^{2}(0,T;L^{2}( \Omega ))}^{2}. \end{aligned}$$
(4.22)
5 Concluding remarks and future work
In this paper, we present a fully discrete scheme in which we use the backward Euler scheme in time and use the spectral approximation in space for the nonlinear parabolic optimal control problem. By using the orthogonal projection operator and some auxiliary equations, we obtain \(L^{2}(H^{1})-L^{2}(L^{2})\) a posteriori error estimates of the spectral approximation solutions for both the state and the control, and also obtained \(L^{2}(L^{2})-L^{2}(L^{2})\) a posteriori error estimates of the spectral approximation solutions for both the state and the control.
The results obtained and techniques used can be extended to such control problems with more general objective functions. Furthermore, we shall consider the spectral approximation for hyperbolic optimal control problems.
Acknowledgements
The authors express their thanks to the referees for their helpful suggestions, which led to improvements of the presentation.
Competing interests
The authors declare that they have no competing interests.
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.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.