Third order convergent time discretization for parabolic optimal control problems with control constraints

Abstract

We consider a priori error analysis for a discretization of a linear quadratic parabolic optimal control problem with box constraints on the time-dependent control variable. For such problems one can show that a time-discrete solution with second order convergence can be obtained by a first order discontinuous Galerkin time discretization for the state variable and either the variational discretization approach or a post-processing strategy for the control variable. Here, by combining the two approaches for the control variable, we demonstrate that almost third order convergence with respect to the size of the time steps can be achieved.

Appendix: Proof of Theorem 2

The proof is similar to Theorem 5 in [6, Chap. 7.1] where higher order stability estimates for a continuous parabolic problem are shown. Just as there, a Galerkin approximation with respect to the spatial variable is used. Let $\{v_{n}\}_{n\in\mathbb{N}}$ be an orthonormal basis of H consisting of eigenfunctions of −Δ defined on V. Then we define the spaces V _N and $X_{kN}^{r}$ by

Replacing the test and trial spaces in Eq. (20) by $X_{kN}^{r}$ leads to a sequence of Galerkin approximations y _kN of the semidiscrete solution y _k . In a first step we have to show that for those approximations the stated stability estimates hold.

Lemma 9

For the Galerkin approximations y _kN as defined above we have the stability estimate

$${ \bigl \Vert \Delta^2 y_{kN} \bigr \Vert _{I}}+ \Biggl( \sum_{m=1}^M{ \Vert \partial_t \Delta y_{kN} \Vert _{I_m}}^2 \Biggr)^{\frac{1}{2}} \leq C { \Vert \Delta w \Vert _{I}} $$

with a constant C independent of N.

Proof

To get the estimate for the first term we test with φ=Δ³ y _kN , which exists since y _kN is a linear combination of eigenvectors of Δ, resulting in

$$\begin{aligned} &-\sum_{m=1}^M { \bigl( \Delta^3 y_{kN},\partial_t y_{kN} \bigr)_{I_m}} - { \bigl( \Delta^3 y_{kN},\Delta y_{kN} \bigr)_{I}} \\ &\quad{}- \sum_{m=1}^{M-1} { \bigl( \Delta^3 y_{kN,m}^-,[y_{kN}]_m \bigr)} - { \bigl( \Delta^3 y_{kN,M}^-,y_{kN,M}^- \bigr)} = { \bigl( \Delta^3 y_{kN},w \bigr)_{I}}. \end{aligned}$$

We apply Green’s formula to each term and get

$$\begin{aligned} &\sum_{m=1}^M { ( \nabla\Delta y_{kN},\partial_t \nabla\Delta y_{kN} )_{I_m}} - { \bigl \Vert \Delta^2 y_{kN} \bigr \Vert _{I}}^2 \\ &\quad{}+ \sum_{m=1}^{M-1} { \bigl( \nabla\Delta y_{kN,m}^-,[\nabla\Delta y_{kN}]_m \bigr)} + { \bigl \Vert \nabla\Delta y_{kN,M}^- \bigr \Vert }^2 = { \bigl( \Delta^2 y_{kN},\Delta w \bigr)_{I}}. \end{aligned}$$

With the two identities

$$\begin{aligned} { ( \nabla\Delta y_{kN},\partial_t \nabla\Delta y_{kN} )_{I_m}} &= \frac{1}{2} { \bigl \Vert \nabla\Delta y_{kN,m}^- \bigr \Vert }^2-\frac{1}{2} { \bigl \Vert \nabla \Delta y_{kN,m-1}^+ \bigr \Vert }^2 \end{aligned}$$

and

$$\begin{aligned} { \bigl( \nabla\Delta y_{kN,m}^-,[\nabla\Delta y_{kN}]_m \bigr)} &= \frac{1}{2} { \bigl \Vert \nabla\Delta y_{kN,m}^+ \bigr \Vert }^2 - \frac{1}{2} { \bigl \Vert \nabla \Delta y_{kN,m}^- \bigr \Vert }^2 - \frac{1}{2} { \bigl \Vert [\nabla\Delta y_{kN}]_m \bigr \Vert }^2 \end{aligned}$$

we obtain

$$-{ \bigl \Vert \Delta^2 y_{kN} \bigr \Vert _{I}}^2 - \sum_{m=1}^{M-1} \frac{1}{2} { \bigl \Vert [\nabla \Delta y_{kN}]_m \bigr \Vert }^2 -\frac{1}{2} { \bigl \Vert \nabla\Delta y_{kN,M}^- \bigr \Vert }^2 = { \bigl( \Delta^2 y_{kN},\Delta w \bigr)_{I}}, $$

which immediately gives the estimate for the first term

$${ \bigl \Vert \Delta^2 y_{kN} \bigr \Vert _{I}} \leq{ \Vert \Delta w \Vert _{I}}. $$

In order to obtain the second estimate, we test with the interval-wise defined function φ where $\varphi |_{I_{m}} = (t-t_{m}) \partial_{t} \Delta^{2} y_{kN}$ for a fixed index m and φ=0 otherwise. Using the dual formulation (17) of the bilinear form, we note that the jump terms vanish and we get

$$\begin{aligned} &-{ \bigl( (t-t_m) \partial_t \Delta^2 y_{kN},\partial_t y_{kN} \bigr)_{I_m}} - { \bigl( (t-t_m) \partial_t \Delta^2 y_{kN},\Delta y_{kN} \bigr)_{I_m}} \\ &\quad= { \bigl( (t-t_m)\partial_t \Delta^2 y_{kN},w \bigr)_{I_m}}. \end{aligned}$$

We apply Green’s formula with respect to the spatial variable on each of the three terms and obtain after reordering

$$\begin{aligned} &\int_{I_m} (t_m-t) { \Vert { \partial_t}\Delta y_{kN} \Vert }^2 \,\mathrm{d}t= \int_{I_m}(t_m-t) { \bigl( { \partial_t}\Delta y_{kN},-\Delta w - \Delta^2 y_{kN} \bigr)}\,\mathrm{d}t \\ &\quad\leq \biggl( \int_{I_m}(t_m-t) { \Vert { \partial_t}\Delta y_{kN} \Vert }^2 \,\mathrm{d}t \biggr)^{\frac{1}{2}} \biggl( \int_{I_m}(t_m-t) { \bigl \Vert -\Delta w - \Delta^2 y_{kN} \bigr \Vert }^2\,\mathrm{d}t \biggr)^{\frac{1}{2}}. \end{aligned}$$

Together with the inverse estimate (4.5) from [17], which reads in our case

$${ \Vert y_{kN} \Vert _{I_m}}^2 \leq C k_m^{-1} \int_{I_m} (t_m-t) { \Vert y_{kN} \Vert }^2 \,\mathrm{d}t $$

with C independent of N, we obtain the estimate

$$\begin{aligned} { \Vert \partial_t \Delta y_{kN} \Vert _{I_m}}^2 &\leq C k_m^{-1} \int _{I_m} (t_m-t) { \Vert \partial_t \Delta y_{kN} \Vert }^2 \,\mathrm{d}t \\ &\leq C k_m^{-1} \int_{I_m}(t_m-t) { \bigl \Vert -\Delta w - \Delta^2 y_{kN} \bigr \Vert }^2\,\mathrm{d}t \\ &\leq C { \bigl \Vert -\Delta w - \Delta^2 y_{kN} \bigr \Vert _{I_m}}^2 \leq C \bigl( { \Vert \Delta w \Vert _{I_m}}^2 + { \bigl \Vert \Delta^2 y_{kN} \bigr \Vert _{I_m}}^2 \bigr). \end{aligned}$$

Summing over all time intervals yields

$$\sum_{m=1}^M { \Vert { \partial_t}\Delta y_{kN} \Vert _{I_m}}^2 \leq C \bigl( { \Vert \Delta w \Vert _{I}}^2 + { \bigl \Vert \Delta^2 y_{kN} \bigr \Vert _{I}}^2 \bigr) $$

which shows the second estimate. □

Proof of Theorem 2

From Lemma 9 we have

$${ \bigl \Vert \Delta^2 y_{kN} \bigr \Vert _{I}}+ \Biggl( \sum_{m=1}^M{ \Vert \partial_t \Delta y_{kN} \Vert _{I_m}}^2 \Biggr)^{\frac{1}{2}} \leq C { \Vert \Delta w \Vert _{I}} $$

with C independent of N. Therefore the sequence $\{y_{kN} \}_{n\in\mathbb{N}}$ is bounded with respect to the norm ∥⋅∥ _Y given by

$${ \Vert y_k \Vert _{Y}}^2 = { \Vert y_k \Vert _{I}}^2 + { \bigl \Vert \Delta^2 y_k \bigr \Vert _{I}}^2 + \sum_{m=1}^M { \Vert { \partial_t}\Delta y_k \Vert _{I_m}}^2 $$

and there exists a sub-sequence $(y_{kN_{j}} )_{j\in \mathbb{N}}$ that converges weakly with respect to the Y norm to a limit $\tilde{y}_{k}$ which satisfies the estimate

$${ \bigl \Vert \Delta^2 \tilde{y}_k \bigr \Vert _{I}}+{ \Vert {\partial _t}\Delta\tilde{y}_k \Vert _{I}} \leq C { \Vert \Delta w \Vert _{I}}. $$

To complete the proof, we need to show that $\tilde{y}_{k}$ is in fact the solution y _k of the semidiscrete problem (20). Therefore we note that the stability estimate in Corollary 1 also works for the Galerkin approximations y _kN with the constant C independent of N. We fix $\bar{N}$, then for any $\varphi\in X_{k\bar{N}}^{r}$ and for any $N_{j} \geq\bar{N}$ the identity

$$ -\sum_{m=1}^M { ( \varphi,\partial_t y_{kN_j} )_{I_m}} - { ( \varphi ,\Delta y_{kN_j} )_{I}} -\sum _{m=1}^{M} { \bigl( \varphi^-,[y_{kN_j}]_m \bigr)} = { ( \varphi,w )_{I}} $$

(70)

holds true. Since $\sum_{m=1}^{M}{ \Vert \partial_{t} y_{kN_{j}} \Vert _{I}}^{2}$, ${ \Vert \Delta y_{kN_{j}} \Vert _{I}}$ and $\sum_{m=1}^{M} { \Vert [y_{kN_{j}}]_{m} \Vert }^{2}$ are bounded by the stability estimate we can extract a subsequence such that (70) holds for the weak limit which has to be $\tilde{y}_{k}$ again. Passing to the limit $\bar{N} \to\infty$ shows that in fact $\tilde{y}_{k}=y_{k}$. □