Discretization of a Distributed Optimal Control Problem with a Stochastic Parabolic Equation Driven by Multiplicative Noise

Abstract

A discretization of an optimal control problem of a stochastic parabolic equation driven by multiplicative noise is analyzed. The state equation is discretized by the continuous piecewise linear element method in space and by the backward Euler scheme in time. The convergence rate $O(\tau ^{1/2} + h^2)$ is rigorously derived.

Proof of Lemma 4.3

Since (27) can be proved by the same argument as that used in the proof of [23, Theorem 5.1], we only prove (26), using the techniques in the proof of [26, Theorem 12.1]. Let

$$\begin{aligned} \theta _j := P_j - \eta (t_j), \quad 0 \le j \le J. \end{aligned}$$

By (25) we have

$$\begin{aligned} \begin{aligned}&\sum _{j=0}^{J-1} [P_j - P_{j+1}, (-\Delta _h)^{-1} \theta _j] + \sum _{j=0}^{J-1} \tau [P_j, \theta _j] \\&\quad ={} \sum _{j=0}^{J-1} \int _{t_j}^{t_{j+1}} [Q_hg(t), (-\Delta _h)^{-1} \theta _j] \, \mathrm {d}t. \end{aligned} \end{aligned}$$

(112)

Since (24) implies

$$\begin{aligned} {\left\{ \begin{array}{ll} -\eta '(t) - \Delta _h \eta (t) = Q_h g(t), \quad 0 \le t \le T, \\ \eta (T) = 0, \end{array}\right. } \end{aligned}$$

(113)

we obtain

$$\begin{aligned}&\sum _{j=0}^{J-1} \int _{t_j}^{t_{j+1}} [-\eta '(t), (-\Delta _h)^{-1}\theta _j] \, \mathrm {d}t + \sum _{j=0}^{J-1} \int _{t_j}^{t_{j+1}} [\eta (t), \theta _j] \, \mathrm {d}t \\&\quad ={} \sum _{j=0}^{J-1} \int _{t_j}^{t_{j+1}} [Q_hg(t), (-\Delta _h)^{-1} \theta _j] \, \mathrm {d}t. \end{aligned}$$

By integration by parts, we then obtain

$$\begin{aligned} \begin{aligned}&\sum _{j=0}^{J-1} [\eta (t_j) - \eta (t_{j+1}), (-\Delta _h)^{-1}\theta _j] + \sum _{j=0}^{J-1} \int _{t_j}^{t_{j+1}} [\eta (t), \theta _j] \, \mathrm {d}t \\&\quad ={} \sum _{j=0}^{J-1} \int _{t_j}^{t_{j+1}} [Q_hg(t), (-\Delta _h)^{-1} \theta _j] \, \mathrm {d}t. \end{aligned} \end{aligned}$$

(114)

Combining (112) and (114) yields

$$\begin{aligned} \sum _{j=0}^{J-1} [\theta _j - \theta _{j+1}, (-\Delta _h)^{-1} \theta _j] + \sum _{j=0}^{J-1} \int _{t_j}^{t_{j+1}} [P_j - \eta (t), \theta _j] \, \mathrm {d}t = 0. \end{aligned}$$

Since

$$\begin{aligned} \sum _{j=0}^{J-1} [\theta _j - \theta _{j+1}, (-\Delta _h)^{-1}\theta _j]&= \sum _{j=0}^{J-1} \Vert {\theta _j} \Vert _{\dot{H}_h^{-1}(\mathcal O)}^2 - [(-\Delta _h)^{-1/2}\theta _j, (-\Delta _h)^{-1/2} \theta _{j+1}] \\&\ge \sum _{j=0}^{J-1} \big ( \Vert {\theta _j} \Vert _{\dot{H}_h^{-1}(\mathcal O)}^2 - \Vert {\theta _j} \Vert _{\dot{H}_h^{-1}(\mathcal O)} \Vert {\theta _{j+1}} \Vert _{\dot{H}_h^{-1}(\mathcal O)} \big ) \\&\ge \frac{1}{2} \Vert {\theta _0} \Vert _{\dot{H}_h^{-1}(\mathcal O)}^2 - \frac{1}{2} \Vert {\theta _J} \Vert _{\dot{H}_h^{-1}(\mathcal O)}^2 \\&= \frac{1}{2} \Vert {\theta _0} \Vert _{\dot{H}_h^{-1}(\mathcal O)}^2 \quad \text {(by the fact } P_J = \eta (t_J) = 0 ), \end{aligned}$$

it follows that

$$\begin{aligned} \frac{1}{2} \Vert {\theta _0} \Vert _{\dot{H}_h^{-1}(\mathcal O)}^2 + \sum _{j=0}^{J-1} \int _{t_j}^{t_{j+1}} [P_j - \eta (t), \theta _j] \, \mathrm {d}t \le 0. \end{aligned}$$

Hence,

$$\begin{aligned}&\frac{1}{2} \Vert {\theta _0} \Vert _{\dot{H}_h^{-1}(\mathcal O)}^2 + \sum _{j=0}^{J-1} \tau \Vert {\theta _j} \Vert _{L^2(\mathcal O)}^2 \\&\quad \le {} \sum _{j=0}^{J-1} \int _{t_j}^{t_{j+1}} [\eta (t) - \eta (t_j), \theta _j] \, \mathrm {d}t \\&\quad \le {} \Big ( \sum _{j=0}^{J-1} \Vert {\eta - \eta (t_j)} \Vert _{ L^2(t_j,t_{j+1};L^2(\mathcal O)) }^2 \Big )^{1/2} \Big ( \sum _{j=0}^{J-1} \tau \Vert {\theta _j} \Vert _{L^2(\mathcal O)}^2 \Big )^{1/2} \\&\quad \le {} \frac{1}{2}\sum _{j=0}^{J-1} \Vert {\eta - \eta (t_j)} \Vert _{ L^2(t_j,t_{j+1};L^2(\mathcal O)) }^2 + \frac{1}{2} \sum _{j=0}^{J-1} \tau \Vert {\theta _j} \Vert _{L^2(\mathcal O)}^2, \end{aligned}$$

which implies

$$\begin{aligned} \Vert {\theta _0} \Vert _{\dot{H}_h^{-1}(\mathcal O)}&\le \Big ( \sum _{j=0}^{J-1} \Vert {\eta - \eta (t_j)} \Vert _{L^2(t_j,t_{j+1};L^2(\mathcal O))}^2 \Big )^{1/2}. \end{aligned}$$

Therefore, by the standard estimate

$$\begin{aligned} \sum _{j=0}^{J-1} \Vert {\eta - \eta (t_j)} \Vert _{L^2(t_j,t_{j+1};\mathcal O)}^2 \lesssim \tau ^2 \Vert {\eta '} \Vert _{L^2(0,T;L^2(\mathcal O))}^{2} \lesssim \tau ^2 \Vert {g} \Vert _{L^2(0,T;L^2(\mathcal O))}^{2}, \end{aligned}$$

we obtain

$$\begin{aligned} \Vert {\theta _0} \Vert _{\dot{H}_h^{-1}(\mathcal O)} \lesssim \tau \Vert {g} \Vert _{L^2(0,T;L^2(\mathcal O))}, \end{aligned}$$

namely,

$$\begin{aligned} \Vert {\eta (0) - P_0} \Vert _{\dot{H}_h^{-1}(\mathcal O)} \lesssim \tau \Vert {g} \Vert _{L^2(0,T;L^2(\mathcal O))}. \end{aligned}$$

Similarly, we can prove

$$\begin{aligned} \Vert {\eta (t_j) - P_j} \Vert _{\dot{H}_h^{-1}(\mathcal O)} \lesssim \tau \Vert {g} \Vert _{L^2(0,T;L^2(\mathcal O))}, \quad 1 \le j < J. \end{aligned}$$

Since $P_J = \eta (t_J) = 0$, this proves (26) and thus completes the proof.