A sharp error analysis for the DG method of optimal control problems

1.   Introduction

In the present work, we discuss discontinuous Galerkin (DG) approximations to a nonlinear optimal control problem (OCP) of ordinary differential equations (ODEs). More precisely, we consider the following optimal control problem:

subject to

Here $u(t) \in \mathbb{R}^m$ is the control, and $x(t) \in \mathbb{R}^d$ is the state of the system at time $t \in [0, T]$. Further, $g: [0, T] \times \mathbb{R}^d \times \mathbb{R}^m \to \mathbb{R}$ and $f: [0, T] \times \mathbb{R}^d \times \mathbb{R}^m \to \mathbb{R}^d$ are given, and the set of admissible controls $\mathcal{U}_{ad} \subset {\mathcal{U}: = L^{\infty}(0, T; \mathbb{R}^m)}$ is given by

for some $u_\ell, u_u \in \mathbb{R}^m$. Here the inequality is understood in the component-wise sense.

There have been a lot of study on the numerical computation for the above problem. The numerical schemes need a discretization of the ODEs, for example, the Euler discretization for the OCPs of ODEs are well studied for sufficiently smooth optimal controls based on strong second-order optimality conditions ^[2,13,14]. For optimal control problems with control appearing linearly, the optimal control may be discontinuous, for an instance, bang-bang controller, and such conditions may not be satisfied. In that respect, there have been many studies to develop new second-order optimality conditions for the optimal control problems with control appearing linearly ^[3,21,31,32]. The second-order Runge-Kutta approximations for the OPCs was studied in ^[15]. Recently, works ^[16,17] developed a novel stability technique to obtain new error estimates for the Euler discretization of OCPs.

The Pseudo-spectral method is also popularly used for the discretization due to its capability of high-order accuracy for smooth solutions to the OCPs ^[20,33]. However, the high-order accuracy of the Pseudo-spectral method is known to be often lost for bang-bang OCPs, where the solutions may not be smooth enough. To handle this issue, Henriques et al. ^[24] proposed a mesh refinement method based on a high-order DG method for the OCPs of ODEs. The DG method discretizes the time interval in small time subintervals, in which the weak formulation is employed. The test functions are usually taken as piecewise polynomials which can be discontinuous at boundaries of the time interval, see Section 2 for more detailed discussion. We refer to ^[7,19,34] and references therein for DG methods for ODEs. It is also worth to refer to papers for the analysis of the discretization of optimal control problems of PDEs, for example, the elliptic problems ^[1,23,35] and the parabolic problems ^{[9,12,25,26,27,28,29]}. In addition, the recent works ^[22,30] studied the discretization of the optimal control for fractional diffusion problems.

In this paper, we provide a rigorous analysis for the DG discretization applied to the nonlinear OCPs (1.1) and (1.2) with arbitrary order $r \in \mathbb{N} \cup \{0\}$ for general functions $f$ and $g$ with suitable smoothness. Motivated from a recent work by Neitzel and Vexler ^[29], we impose the non-degeneracy condition (2.4) on an optimal control $\bar{u}$ of the OCPs (1.1) and (1.2). We obtain the existence and convergence results for the semi-discretized case and the fully discretized case. The rates of the convergence results depend on the regularity of the optimal solution $\bar{u}$ and its adjoint state with the degree of piecewise polynomials mentioned above, see Section 2 for details.

It is worth noticing that the control is not required to be linear in the state Eq (1.2), and the control space $\mathcal{U}_{ad}$ allows to take into account discontinuous controls. The constraints for controls are defined by lower and upper bounds. Moreover, the cost functional is also given in a general form, not limited to be quadratic. We mention that the DG discretization of zeroth order was used in ^[29] for the optimal control problem for the semi-linear parabolic equation where the control is linearly applied to the system.

For notational simplicity, we denote by $I: = (0, T)$, $X: = L^2(I; \mathbb{R}^d)$, and $(v, w)_I = (v, w)_{L^2(I; \mathbb{R}^d)}$. We also use simplified notations:

for $1 \leq p \leq \infty$. Throughout this paper, for any compact set $K \subset \mathbb{R}^m$, we assume that $f, g \in C([0, T]; W^{3, \infty}(\mathbb{R}^d \times K))$ satisfy

for some $M > 0$.

We next introduce the control-to-state mapping $G: \mathcal{U} \to X \cap L^\infty(I; \mathbb{R}^d)$, $G(u) = x$, with $x$ solving (1.2). It induces the cost functional $j: \mathcal{U} \to \mathbb{R}_+$, $u \mapsto J(u, G(u))$. This makes the optimal control problems (1.1) and (1.2) equivalent to

Definition 1.1. A control $\bar u \in \mathcal{U}_{ad}$ is a local solution of (1.4) if there exists a constant $\varepsilon > 0$ such that $j(u) \geq j(\bar u)$ holds for all $u \in \mathcal{U}_{ad}$ with $\|\bar u - u\|_{L^2 (I)} \leq \varepsilon$.

In the proof of the existence and convergence results, the main task is to show that the strong convexity of $j$ induced by the second-order optimality condition (2.4) is preserved near the optimal control $\bar{u}$ and also for its DG discretized version $j_h$. It is achieved using the second-order analysis in Section 4. As a preliminary, we also justify that $j$ and $j_h$ are twice differentiable, by showing the differentiability of the control-to-state mapping $G$ and its discretized version $G_h$ in the appendix.

In Section 2, we explain the DG discretization of the ODEs and the OCP. Then we present the main results for the semi-discretized case and provide some preliminary results. In Section 3, the adjoint problems are studied. Section 4 is devoted to study the second order analysis of the cost functionals $j$ and $j_h$. In Section 5, we prove the existence of the local solution and obtain the convergence rate for the semi-discretized case. Section 6 is devoted to establish the existence and convergence results for the fully discretized case. Finally, in Section 7, we perform several numerical experiments for linear and nonlinear OCPs. In Appendix A, we obtain first and second order derivatives of the control-to-state mapping $G$. Appendix B is devoted to prove a Grönwall-type inequality for the discretization of the ODEs (1.2) involving the control variable. It is used in Appendix C to establish the differentiability of the discrete control-to-state mapping $G_h$ and obtain the derivatives. In Appendix D, we prove Lemmas 3.3 and 3.5, which reformulate the first derivatives of the cost functionals in terms of the adjoint states. In Appendix E, we derive the formulas on the second order derivatives of the cost functionals.

2.   DG formulation

In this section, we describe the approximation of the OCPs (1.1) and (1.2) with the DG method, and then we state the main results on the semi-discrete case. First, we illustrate the discretization of the ordinary differential equations

where $x: [0, T] \rightarrow \mathbb{R}^d$, $F:(0, T) \times \mathbb{R}^d \rightarrow \mathbb{R}^d$ is uniformly Lipschitz continuous with respect to $x$, i.e.,

with a constant $L > 0$. By the Cauchy Lipschitz theorem, we have the existence and uniqueness of classical solution $x$ of (2.1).

Given an integer $N \in \mathbb{N}$, we consider a partition of $I$ into $N$-intervals $\{I_n\}_{n = 1}^N$ given by $I_n = (t_{n-1}, t_n)$ with nodes $0 = :t_0 < t_1 < \cdots < t_{N-1} < t_N : = T$. Let $h_n$ be the length of $I_n$, i.e., $h_n = t_n - t_{n-1}$, and we set $h: = \max_{1 \leq n \leq N}h_n$. For a piecewise continuous function $\varphi :[0, T] \rightarrow \mathbb{R}^d$, we also define

The jumps across the nodes is denoted by $[\varphi]_n : = \varphi^+_n - \varphi^-_n$ for $1 \leq n \leq N-1$. For $r \in \mathbb{N} \cup \{0\}$, we define

where $P^r(I_n)$ represents the set of all polynomials of $t$ up to order $r$ defined on $I_n$ with coefficients in $\mathbb{R}^d$. Then the DG approximate solution $x_h$ of (2.1) is given as

for all $\varphi \in X^r_h$. Here $(\cdot, \cdot)$ denotes the inner product in $\mathbb{R}^d$, and

for integrable functions $A, B: I_n \rightarrow \mathbb{R}^d$.

We recall the error estimate for the DG approximation of (2.1) from [34,Corollary 3.15 & Theorem 2.6].

Theorem 2.1. Let $x(t)$ be the solution of $(2.1)$ such that $x \in W^{k, \infty}(I; \mathbb{R}^d)$ for some $k \geq 1$. Suppose that $h L < 1$. Then there exists a unique DG approximate solution $x_h \in X_h^r$ to (2.2) of order $r \in \mathbb{N} \cup \{0\}$. Furthermore, we have

where $C > 0$ is determined by $L$, $T$, and $r$.

Now, for given $u \in \mathcal{U}$, we consider the approximate solution $x \in X^r_h$ of the control problem (1.2) satisfying

for all $\varphi \in X^r_h$.

Throughout the paper, we will consider local solutions $\bar{u}$ to (1.4) satisfying the following non-degeneracy condition.

Assumption 1. Let $\bar{u} \in \mathcal{U}_{ad}$ be the local solution of (1.1). We assume that it satisfies

for some $\gamma > 0$.

The differentiability of the cost functional $j(u) = J(u, G(u))$ with respect to $u \in \mathcal{U}$ is induced by the differentiability of the solution mapping $G(u)$ justified in Appendix A (see also the proofs of Lemmas 3.3 and E.1). Note that the above second-order optimality condition holds under suitable regularity assumptions on the function $f$, $g$, and solutions, see Remark E.2 for a detailed discussion. We refer to ^[4,5] for further discussion on the second-order condition and also ^[8,10,11] for the optimal control problem of PDEs.

In addition, we assume that $\bar{u} \in \mathcal{U}_{ad}$ has bounded total variation, i.e., $V(\bar{u}) \leq R/2$ for a fixed value $R > 0$. Here the total variation $V (f)$ for $f \in L^{\infty}(0, T)$ is defined as

where $P$ is any partition $P = \{ 0 = x_0 < x_1 < x_2 < \cdots < x_n < x_{n+1} = T\}$.

Considering a discrete control-to-state mapping $G_h : \mathcal{U} \to X^r_h$, $u \mapsto G_h(u)$, where $G_h(u)$ is the solution of (2.3), we introduce the discrete cost functional $j_h : \mathcal{U} \to \mathbb{R}_+, u \mapsto J(u, G_h(u))$. Let us consider the following discretized version of (1.1):

where

We now define the local solution to (2.5) as follows.

Definition 2.2. A control $\bar u_h \in \mathcal{U}_{ad} \cap \mathcal{V}_R$ is called a local solution of (2.5) if there exists an $\delta > 0$ such that $j_h(u) \geq j_h(\bar u_h)$ holds for all $u \in \mathcal{U}_{ad}\cap \mathcal{V}_R$ with $\|u - \bar u_h\|_{L^2 (I)} \leq \delta$.

In the first main result, we prove the existence of the local solution to the approximate problem (2.5).

Theorem 2.3. Let $\bar{u} \in \mathcal{U}_{ad}\cap \mathcal{V}_{R/2}$ be a local solution of $(1.1)$ satisfying Assumption 1. Then, there are constants $\epsilon > 0$ and $h_0 > 0$ such that for $h \in (0, h_0)$ the approximate problem $(2.5)$ has a local solution $\bar{u}_h \in \mathcal{U}_{ad} \cap \mathcal{V}_R$ satisfying $\|\bar{u}_h - \bar{u}\|_{L^2 (I)} < \varepsilon$.

The second main result is the following convergence estimate of the approximate solutions.

Theorem 2.4. Let $\bar{u} \in \mathcal{U}_{ad}\cap \mathcal{V}_{R/2}$ be a local solution of $(1.4)$ satisfying Assumption 1, let $\bar{u}_h$ be the approximate solution found in Theorem 2.3, and let $\lambda (\bar{u})$ be the adjoint state defined in Definition 3.1 below. Assume that the state $\bar{x} = G(\bar{u})$ belongs to $W^{k_1, \infty}(I; \mathbb{R}^d)$ and the adjoint state $\lambda (\bar{u})$ belongs to $W^{k_2, \infty}(I; \mathbb{R}^d)$ for some $k_1, k_2 \geq 1$. Then we have

The required regularity of solutions $\bar x$ and $\lambda(\bar u)$ can be obtained under suitable smoothness assumptions on $f$, $g$, and $\bar u$, see Remark 3.2 below. The above result establishes the error estimate concerning the discretization of the ODEs in the OCPs. We will give the proofs of Theorems 2.3 and 2.4 in Section 5. On the other hand, to implement a numerical computation to the OCP (1.4), one needs also to consider an approximation of the control space with a finite dimensional space. In Section 6, we will see that the proof of Theorem 2.4 can be extended to the error analysis incorporating the discretization of the control space.

3.   Adjoint states

This section is devoted to study the adjoint states to the OCP (1.1) and its discretized version (2.5).

We introduce a bilinear form $b(\cdot, \cdot)$ for $x \in {W^{1, \infty} (0, T)}$ and $\varphi \in X$ by

Then, for a fixed control $u \in \mathcal{U}$ and initial data $x_0 \in \mathbb{R}^{d}$, a weak formulation of (1.2) can be written as

for all $\varphi \in X$ with $x(0) = x_0$.

Definition 3.1. For a control $u \in \mathcal{U}$, we define the adjoint state $\lambda = \lambda(u) \in W^{1, \infty}(0, T)$ as the solution to

with $\lambda(T) = 0$. It satisfies the weak formulation

for all $\varphi \in X$ with $\lambda (T) = 0$.

Remark 3.2. It follows from the Eqs (1.2) and (3.3) that if

we have

For $u, v \in \mathcal{U}$, the derivative of $j$ at $u$ in the direction $v$ is defined by

It is well-known that the derivative of the cost functional can be calculated with the adjoint state, as described below.

Lemma 3.3. We have

for all $v \in \mathcal{U}_{ad}$, where $x = G(u)$.

Proof. For the completeness of the paper, we give the proof in Appendix D.

Next we describe the adjoint problem for the approximate problem (2.5). For ${x}, \varphi \in X^r_h$, we define

For approximate solution $x_h = G_h (u) \in X^r_h$, the Eq (2.3) with control $u \in \mathcal{U}$ can be written as

Now we define the adjoint equation for the approximate problem (2.5).

Definition 3.4. The adjoint state $\lambda_h = \lambda_h(u) \in X^r_h$ is defined as the solution of the following discrete adjoint equation:

In Appendix D, we briefly explain how the adjoint Eq (3.8) can be derived from the Lagrangian related to (2.5). We also have an analogous result to Lemma 3.3.

Lemma 3.5. We have

where $x_h = G_h(u)$.

Proof. The proof is given in Appendix D.

In order to prove the main results in Section 2, we shall use the following lemma.

Lemma 3.6. Let $u \in \mathcal{U}$. Suppose that $x = G(u) \in W^{k_1, \infty} (I; \mathbb{R}^d)$ and $\lambda = \lambda (u) \in W^{k_2, \infty}(I; \mathbb{R}^d)$ for some $k_1, k_2 \geq 1$. Then we have

Proof. We recall from (3.4) and (3.8) that $\lambda = \lambda ({u})$ solves

and $\lambda_h = \lambda_h ({u})$ solves

Here $x \in G({u}) \in X$ and ${x}_h = G_h ({u}) \in X_h$. The estimate of ${x}-{x}_h$ is induced from Theorem 2.1 as follows:

As an auxiliary function, we consider $\zeta_h \in X_h$ solving

which is the DG discretization of (3.11) in a backward way (see Lemma 3.7 below). Then, by Theorem 2.1, we have

By (3.13), we obtain

and

Combining these estimates with (3.12) and (3.14) gives

where $R: I \rightarrow \mathbb{R}^d$ is given by

and it satisfies $\|R(t)\| = O(h^{\min\{k_1, r+1\}})$. This, together with Lemma B.4, yields

Combining this estimate with (3.15),

which completes the proof.

With abusing a notation for simplicity, let us define $J$ as the interval $I$ given a partition $0 = s_0 < s_1 < \cdots < s_{N-1} < s_N = T$ with $s_j = t_{N-j}$. Also we set $X_{h, J}^{r}$ as the DG space $X_h^r$ with the new partition. Then we have the following lemma.

Lemma 3.7. Assume that $\lambda \in X_h^r$ is a solution to

Then $W:I \rightarrow \mathbb{R}^d$ defined by $W(t) = \lambda (T-t)$ for $t \in I = [0, T]$ satisfies

Proof. By an integration by parts,

which leads to

We now observe that $W(t) = \lambda (T-t)$ satisfies $W' (t) = - \lambda' (T-t)$ and $[W]_{N-n} = - [\lambda]_{n}$. We also set $\psi (t) = \phi (T-t)$. Then $\psi \in X_{h, J}^{r}$ and we have $\phi_{n}^{-} = \psi_{N-n}^{+}$. Considering $J_n : = (s_{n-1}, s_n)$, it holds that $J_n = I_{N+1-n}$ for $1 \leq n \leq N$. Using these notations, we write (3.16) as

Rearranging this, we get

which is the desired equation $B(W, \psi) = (F(t, W), \psi)_{I}$. The proof is finished.

4.   Second order analysis

In this section, we analyze the second order condition of the functions $j$ and $j_h$, which are essential in the existence and convergence estimates in the next sections.

4.1.   Second order condition for $j$

We defined the solution mapping $G: \mathcal{U} \rightarrow X \cap L^{\infty}(I; \mathbb{R}^d)$ in the previous section. Here we present Lipschitz estimates for the solution mapping $G$, its derivative $G'$, and the solution to the adjoint Eq (3.4).

Lemma 4.1. There there exists $C > 0$ such that for all $u, \hat u \in \mathcal{U}_{ad}$ and $v \in \mathcal{U}$ we have

and

Proof. Let us denote by $x = G(u)$ and $\hat x = G(\hat u)$. Then it follows from (3.2) that

By (1.3), there exists a constant $C > 0$ such that

Using this estimate and applying the Grönwall inequality in (4.1), we get the inequality

This gives the first inequality. For the second one, if we set $y = G'(u)v$ and $\hat y = G'(\hat u)v$, then it follows from Lemma A.1 that

This together with the first assertion above yields

For notational simplicity, we denote by $\lambda = \lambda(u)$ and $\hat \lambda = \lambda(\hat u)$. Then we get

with $(\lambda - \hat{\lambda}) (T) = 0$. By applying the Grönwall inequality in a backward way, we obtain

where we used

due to (3.3) and $\hat\lambda(T) = 0$. This completes the proof.

We now show that the second order condition of $j$ holds near the optimal local solution $\bar{u} \in \mathcal{U}_{ad}$.

Lemma 4.2. Suppose that $\bar{u} \in \mathcal{U}_{ad}$ satisfies Assumption 1. Then there exists $\varepsilon > 0$ such that

holds for all $v \in \mathcal{U}$ and all $u \in \mathcal{U}_{ad}$ with $\|u - \bar{u}\|_{L^2 (I)} \leq 2 \varepsilon$. Here $\gamma > 0$ is given in (2.4).

Proof. Let $y (t) = G' (u)v$ and $y(\bar{u})(t) = G' (\bar{u}) v$. By using Lemma E.1, we find

where we denoted by $\lambda(t) : = \lambda(u)(t)$, $x(t) : = G(u)(t)$, $\bar\lambda(t) : = \lambda(\bar u)(t)$, and $\bar x(t) : = G(\bar u)(t)$. On the other hand, it follows from Lemma 4.1 that

This together with the following estimate

yields

Combining this with (2.4) we have

By choosing $\varepsilon = \frac{\gamma}{4C} > 0$ here, we obtain the desired result.

As a consequence of this lemma, we have the following result.

Theorem 4.3. Let $\bar{u} \in \mathcal{U}_{ad}$ satisfy the first optimality condition and Assumption 1. Then, there exist a constant $\varepsilon > 0$ such that

for any $u \in \mathcal{U}_{ad}$ with $\|u-\bar{u}\|_{L^2 (I)} \leq 2 \varepsilon$.

Proof. Choose $\varepsilon > 0$ as in Lemma 4.2. By Taylor's theorem, we get

where $\bar u_{s} = \bar{u} + s(u-\bar{u})$ for some $s \in [0, 1]$. On the other hand, the first optimality condition implies

Moreover, we also find

Using these observations and Lemma 4.2, we conclude

The proof is finished.

4.2.   Second order condition for $j_h$

In this part, we investigate the second order condition for the discrete cost functional $j_h$. Similarly as in the previous subsection, we first provide the Lipschitz estimates for $G_h$ and the discrete adjoint state.

Lemma 4.4. Let $u, \hat u \in \mathcal{U}_{ad}$ and $v \in \mathcal{U}$ be given. Then, there exists $C > 0$, independent of $h \in (0, 1)$, such that

and

Proof. The first and the third assertions are proved in Lemma B.5. The second estimate is proved in Lemma C.2.

Lemma 4.5. For $u \in \mathcal{U}_{ad}$, let $x = G(u)$ be given by the solution of the state Eq $(1.2)$, and let $y = G'(u)v$ for $v \in \mathcal{U}$. Let $x_h = G_h(u)$ be the solution of the discrete state Eq $(3.7)$, and let $y_h = G'_h(u)v$. Then we have

Proof. Define $\tilde y :[0, T] \rightarrow \mathbb{R}^d$ by the solution to

Recall from Lemma A.1 that $y$ satisfies

Combining these two equations, we get

Using the Grönwall inequality here with (4.2) and (3.13), we find that

On the other hand, $y_h$ satisfies

which is the DG discretization of (4.4) in a backward way in view of Lemma 3.7. Thus, we may use Theorem 2.1 to obtain the following error estimate:

This, together with (4.5) gives us the estimate

The proof is finished.

Lemma 4.6. For $\varepsilon > 0$ given in Lemma 4.2, there exists $h_0 > 0$ such that for $h \in (0, h_0)$ we have the following inequality

for any $u \in \mathcal{U}_{ad}$ satisfying $\|u -\bar{u}\|_{L^2(I)} \leq \varepsilon$.

Proof. We first claim that

for $h > 0$ small enough, where $C > 0$ is independent of $h$. Let $x(t) = G(u)(t)$, $\lambda (t) = \lambda(u) (t)$, $x_h (t) = G_h (u)(t)$, and $\lambda_h (t) = \lambda_h (u)(t)$. Also we let $y = G' (u)v$ and $y_h = {G_h}' (u)v$. It follows from Lemmas E.1 and E.3 that

In order to show (4.6), by using a similar argument as in the proof of Lemma 4.2, it suffices to show that there exists $C > 0$, independent of $h$, such that

and

The first and second inequalites in (4.7) hold due to Theorem 2.1 and Lemma 4.5. For the third one in (4.7) is proved in (C.2). By Lemma 3.6, the second inequality in (4.8) holds. We also find

which asserts the first inequality in (4.8). Finally, we obtain

due to (4.7). All of the above estimates enable us to prove the claim (4.6). This together with Lemma 4.2 yields

for $0 < h < h_0: = \gamma/(4C)$. The proof is finished.

5.   Existence and convergence results for the semi-discrete case

We first prove the existence of the local solution to the approximate problem (2.5).

Proof of Theorem 2.3. Choose $\varepsilon > 0$ as in Theorem 4.3. We consider the following set

and recall from Section 2 the space $\mathcal{V}_{R} = \{ u \in \mathcal{U}\; :\; V(u) \leq R\}.$ We will find a minimizer $\bar v$ of $j_h$ in the space $W_{ \varepsilon, R}: = \overline{B_{2 \varepsilon}(\bar{u})} \cap \mathcal{V}_R$, and then show that $\|\bar{v} - \bar{u} \|_{L^2 (I)} < \varepsilon$. It will imply that $\bar{v}$ is a local solution to (2.5).

Since $j_h$ is lower bounded on $W_{ \varepsilon, R}$, there exists a sequence $\{v_k\}_{k \in \mathbb{N}} \subset \overline{B_{ \varepsilon}(\bar{u})} \cap \mathcal{V}_R$ such that

Moreover, since $W_{ \varepsilon, R}$ is compactly embedded in $L^p(I)$ for any $p \in [1, \infty)$, up to a subsequence, there exists a function $\bar{v} \in W_{ \varepsilon, R}$ such that $\{v_k\}$ converges to $\bar v$ in $L^{2}(I)$ and converges a.e. to $\bar{v}$. By definition, the function $z_k : = G_h (v_k) \in X_h^r$ satisfies

for all $\varphi \in X^r_h$. Note that $\{z_k\}_{k \in \mathbb{N}}$ is a bounded set in the finite dimensional space $X_h^r$ by Theorem 2.4 (see also Lemma B.4). Therefore we can find a subsequence such that $z_k$ converges uniformly to a function $\bar{z} \in X_h^r$. We claim that $\bar{z} = G_h (\bar{v})$. Indeed, since $v_k (t)$ converges a.e. to $\bar{v}(t)$ for $t \in I$ and $f$ is Lipschitz continuous, we may take a limit $k$ to infinity in (5.2) to deduce

for all $\varphi \in X^r_h$. This yields that $\bar{z} = G_h (\bar{v})$, which enables us to derive

This together with (5.1) implies that $\bar{v} \in W_{ \varepsilon, R}$ satisfies

It remains to show that the minimizer $\bar{v} \in W_{ \varepsilon, R}$ is achieved in the interior of $B_ \varepsilon (\bar{u}) = \{u \in \mathcal{U}_{ad}\, :\, \|u-\bar{u}\|_{L^2 (I)} < \varepsilon\}$. To show this, we recall that

and

Since $\|G(u) \|_{W^{1, \infty}(I)} \leq C$ for all $u \in \mathcal{U}_{ad}$, we see from Theorem 2.1 that

where $C > 0$ is independent of $h$. Combining this with the Lipschitz continuity of $G$ yields that

Taking $h_0 = \gamma \varepsilon^2 /(8C)$. Using this and the estimate

from Theorem 4.3, it follows that for $h \in (0, h_0)$ we have

Thus, the minimizer $\bar{v}$ is achieved in $B_ \varepsilon (\bar{u})$. It gives that $j_h (u) \geq j_h (\bar{v})$ for all $u \in \mathcal{V}_R$ with $\|u-\bar{v}\|_{L^2} \leq \varepsilon$. We now provide the details of the convergence estimate of the approximate solutions.

Proof of Theorem 2.4. Analogous to (4.3), the discrete first order necessary optimality condition for $\bar u_h \in \mathcal{U}_{ad}$ reads

Inserting here $u = \bar{u}$ and summing it with (4.3), we get

Now, by applying the mean value theorem with a value $t \in (0, 1)$, we have

where we used Lemma 4.6 in the first inequality and (5.4) in the second inequality. For our aim, it only remains to estimate the right hand side. Let us express it using the adjoint states. From (3.5), we have

and it follows from (3.9) that

Here we remind that $\bar{x}_h \in X_h^r$ denotes the solution to (2.3) with control $\bar{u}$ and initial data $x_0$. Combining (5.6) and (5.7) we find

Applying Hölder's inequality here and using (1.3), we deduce

Now we apply (3.10) and (3.13) to get

Combining this with (5.5), we finally obtain

This completes the proof.

6.   Existence and convergence results for the fully discrete case

This section is devoted to the existence and convergence results for the fully discrete case. We consider a finite dimensional space $\mathcal{U}_h$ which discretizes the control space $\mathcal{U}_{ad}$, for example, the space of step functions

or the high-order DG space $\mathcal{U}_h = X_h^r \cap \mathcal{U}_{ad}$ with $r \in \mathbb{N}$.

We say that $\bar{u}_h \in \mathcal{U}_h$ is a local solution to

if there is a value $\varepsilon > 0$ such that $j_h (u) \geq j_h (\bar{u}_h)$ for all $u \in \mathcal{U}_h$ with $\|u- \bar{u}_h \|_{L^2} \leq \varepsilon$.

The existence result of local solution is provided in the following theorem.

Theorem 6.1. Choose $\varepsilon > 0$ as in Theorem 4.3. Let $\bar{u} \in \mathcal{U}_{ad}$ be a local solution of $(1.4)$ satisfying Assumption 1. Fix any $\varepsilon > 0$. Then there exists $h_0 > 0$ such that for $h \in (0, h_0)$ problem $(6.1)$ has a local solution $\bar{u}_h \in \mathcal{U}_h$ such that $\|\bar{u}- \bar{u}_h\|_{L^2} \leq \varepsilon$.

Proof. By compactness and continuity, $j_h$ has a minimizer $\bar u_h$ in

since $\mathcal{U}_h$ is finite dimensional. Next we aim to show that the minimizer $\bar{u}_h$ satisfies

To show this, we recall from (5.3) that there is a value $h_0 > 0$ such that for $h \in (0, h_0)$ we have

Combining this with the minimality of $\bar{u}_h$ for $j_h$ in $\overline{B_{2 \varepsilon}(\bar{u})}$, we find that $\|\bar{u}_h - \bar{u}\|_{L^2 (I)} \leq \varepsilon$. It then yields that

Thus $\bar{u}_h$ is a local solution of (6.1).

We establish the convergence result in the following theorem.

Theorem 6.2. Assume the same statements for $\bar{u}\in \mathcal{U}_{ad}$ and $\lambda (\bar{u})$ in Theorem 2.4. In addition, suppose that there exists a projection operator $P_h : \mathcal{U} \rightarrow \mathcal{U}_h$ and a value $a > 0$ such that

Let $\bar{u}_h \in \mathcal{U}_h$ be a local solution to $(6.1)$ constructed in Theorem 6.1. Then the following estimate holds:

If we further assume that $j' (\bar{u}) = 0$, then the above estimate can be improved to

Proof. In this case, by the first optimality conditions on $\bar{u}$ and $\bar{u}_{h}$, we have

The latter condition can be written as

where $R_h : = {j_h}' (\bar{u}_h) (P_h \bar{u} - \bar{u})$. Summing up the above two inequalities provides

i.e.,

By the assumption of the theorem,

On the other hand, by applying the mean value theorem and Lemma 4.6, we obtain

Combining this with (6.2) yields

Applying here the estimate (5.9) in the previous proof, we have

which together with (6.3) gives the desired estimate

When we further assume $j' (\bar{u}) = 0$, it follows that

Using this and the estimates in (5.8), we find

Inserting this into (6.4) yields

It gives the desired estimate

The proof is done.

7.   Numerical experiments

In this section, we present several numerical experiments which validate our theoretical results. The forward-backward DG methods ^[18] is employed to solve the examples of the OCPs.

7.1.   Linear problem

Let us consider the following simple one dimensional OCP, which has been used as an example ^[36], that consists of maximizing the functional

subject to the state equation

and $\mathcal{U} = L^2 ([0, 1])$. Using a similar idea as in Section 3 based on the maximum principle, we can derive the adjoint equation to the above optimal control problem:

Furthermore, we also find that the optimal solutions $\bar u = - \lambda$ and $\bar x$ satisfies (7.2). Thus we have the solution

and

For fixed $r \in \mathbb{N}$, we use $X_h^r$ for the approximate space of $\mathcal{U}$. In Table 1, we report the discrete $L^2$ error between optimal solutions and its approximations for the above optimal control problem. Here $r+1$ is the number of grid points on each time interval $I_n$, and we used the equidistant points for our numerical computations. The numerical result confirms that the error is of order $h^{r+1}$ as proved in Theorem 2.4.

7.2.   Nonlinear problem

In this part, we consider the following nonlinear optimal control problem:

subject to the state equation

In this case, the corresponding adjoint equation and optimal control are given as follows.

and thus the optimal solution $\bar x$ solves

In this case, since we have no explicit form of the actual solutions, we take the reference solutions $\bar{x}_h$ (resp., $\bar{u}_h$) with $h = (0.1) \times 2^{-9}$ instead of $\bar{x}$ (resp., $\bar{u}$). In Table 2, we arrange the discrete $L^2$ error between reference solutions and its approximations.

Next we consider a two dimensional problem given by

subject to the state equation

In this case, the corresponding adjoint equation and optimal control are given as follows.

This case also has no explicit form of the actual solutions and so we take the reference solutions $\bar{x}_h$ (resp., $\bar{u}_h$) with $h = (0.1) \times 2^{-9}$ instead of $\bar{x}$ (resp., $\bar{u}$). The discrete $L^2$ error between reference solutions and its approximations are arranged in Table 3.

8.   Conclusions

In this paper, we established the analysis for the DG discretization applied to the nonlinear OCP with arbitrary degree of piecewise polynomials $r$ for nonlinear functions $f$ and $g$ with suitable smoothness assumptions. Under the non-degeneracy condition on an optimal control of the OCP, we obtained the existence of the local solution to the approximate problem and the sharp $L^2$-error estimates of the approximated solutions. These results was extended to the fully discrete case, in which the control space is also discretized. Finally, we showed numerical experiments validating our theoretical results. Based on the results of this paper, it would be interesting to analyze the mesh refinement method for the discontinuous galerkin method of the optimal control problems. We would like to investigate this problem in the future.

Acknowledgments

The authors are grateful to the referees for valuable comments on the paper. The work of W. Choi is supported by NRF grants (No. 2016R1A5A1008055) and (No. 2021R1F1A1059671).

Conflict of interest

The authors declare no conflict of interest.

Appendix

A.   Differentiability of the control-to-state mapping

In this section, we show that the control-to-state mapping $G$ is twice differentiable, and obtain the derivatives.

Lemma A.1. Let $x^s = G(u+sv)$ and $y:[0, T] \rightarrow \mathbb{R}^d$ be the solution of

Then we have

Proof. Recall that $x^s$ and $x$ satisfy

respectively. Using this, we find that $r(t) : = x^s (t) - x (t) - s y(t)$ satisfies

where

and

Given that $|x^s (t) - x (t)|\leq Cs$ and (1.3), an elementary calculus shows that $|A_1| \leq Cs^2$ and $|A_2| \leq Cs^2$. With these bounds, we may apply the Grönwall's lemma for (C.3) to deduce $|r(t)| \leq Cs^2$ for $t \in [0, T]$. From this we find

which yields that

Next we show the twice differentiablity of the mapping $s \rightarrow G(u+sv)$ at $s = 0$.

Lemma A.2. Let $z:[0, T] \rightarrow \mathbb{R}^d$ be the solution of

Then we have

Proof. Let

Then we get

where

and

By Lemma 4.1, we have $|y^s (t) - y(t)|\leq Cs$. Given this estimate and that

from Lemma A.1, an elementary calculus shows that $|A_1 (t)| \leq Cs^2$ and $|A_2 (t)| \leq Cs^2$. Inserting this estimate into (C.5) and applying the Grönwall's lemma, we find

It proves that

This implies that

since

This completes the proof.

B.   Grönwall-type inequality for the DG discretization of ODEs

In this section, we provide a Grönwall-type inequality for the DG discretization of ODEs with inputs. It will be used in Section C to establish the differentiability of the discrete control-to-state mapping $G_h$.

We begin with recalling from [34,Lemma 2.4] the following lemma.

Lemma B.1. Let $I = (a, b)$ and $k = b-a > 0$. Then we have

for all $\phi (t) = (\phi_1 (t), \cdots, \phi_d (t)) \in P^r ((a, b); \mathbb{R}^d)$, $r \in \mathbb{N}_0$, where

The next result is from [34,Lemma 3.1].

Lemma B.2. For $I = (a, b)$ and $r \in \mathbb{N}_0$, we have

for all $\phi (t) = (\phi_1 (t), \cdots, \phi_d (t)) \in P^r ((a, b); \mathbb{R}^d)$. Here $C > 0$ is independent of $r$, $a$, $b$, and $d$.

We shall use the following Grönwall inequality.

Lemma B.3. Let $\{a_n\}_{n = 1}^{N}$ and $\{b_n\}_{n = 1}^{N}$ be sequences of non-negative numbers satisfying $b_1 \leq b_2 \leq\cdots \leq b_{N}$ and $b_1 = 0$. Assume that for a value $h \in (0, 1/2)$ we have

for $n \in \mathbb{N}$. Then there exists a constant $Q > 0$ independent of $h \in (0, 1/2)$ and $N \in \mathbb{N}$ such that

for any $n \in \mathbb{N}$ with $n \leq N/h$.

Proof. The proof can be obtained by induction.

Now we obtain the Grönwall-type inequality.

Lemma B.4. Suppose that

for all $\varphi \in X_h^r$. Then there exists a constant $C > 0$ independent of $h > 0$ such that

for all $u_1, u_2 \in \mathcal{U}_{ad}$ and $h > 0$ small enough.

Proof. From the condition (B.1) we have

for all $\varphi \in X_h^r$. To obtain the desired estimates, for each $n \in \{1, \cdots, N\}$ we shall take the following test functions $\varphi \in X_h^r$ supported on $I_n$ given as

where $1_{I_n}: I \to \{0, 1\}$ denotes the indicator function, that is, $1_{I_n}(t) = 1$ for $t \in I_n$ and $1_{I_n}(t) = 0$ for $t \in I\setminus I_n$. First we take $\varphi (t) = x(t)1_{I_n}(t)$ for $n = 1, 2, \cdots, N$. Then,

where for $n = 1$ we abuse a notation $[x]_{0}$ to mean $x_0^{+}$. Notice that

where for $n = 1$ the above is understood as $([x]_{0}, x_0^{+}) = (x_0^{+})^2.$ Using this in (B.2), we find

By applying Cauchy-Schwarz inequality, we obtain

Secondly, we take $\varphi (t) = (t-t_{n-1})x' (t)1_{I_n} (t)$ to have

By using Hölder's inequality, we get

Notice that

Thus, choosing $\varphi (t) = (t-t_{n-1})1_{I_n}(t)$ gives

and subsequently, this yields

where $h_n = t_n - t_{n-1}$. This together with Lemma A.1 asserts

for $h > 0$ small enough. Combining (B.3) and (B.4), we find

where we applied Lemma B.1 in the second inequality. This, together with (B.5), we obtain

for $h > 0$ small enough, where for $n = 1$ one has $|x_0^{-}| = 0$. This inequality trivially gives

for $n = 1, \cdots, N$. Now, by applying Lemma B.3 to find an estimate of $|x_n^{-1}|^2$ and inserting it into (B.6), we achieve

Finally, by applying Lemma B.2 to the above, we obtain the desired estimate.

As a corollary, we have the following Lipschitz estimates.

Lemma B.5. For $u, v \in \mathcal{U}_{ad}$ we have

and

Proof. Let us denote by $x = G_h(u)$ and $\hat{x} = G_h(v)$. Then it follows from (2.3) that

By (1.3), there exists a constant $C > 0$ such that

By applying Lemma B.4, we get the inequality

This gives the first inequality. For the second one, we denote by $\lambda = \lambda_h (u)$ and $\hat \lambda = \lambda_h (v)$. Then, we see from Lemma 3.8 that

By applying Lemma B.4 again in a backward way (see Lemma 3.7), we obtain

where we used

due to Lemma B.4. This completes the proof.

C.   Differentiability of discrete control-to-state mapping

This section is devoted to prove that the discrete control-to-state mapping $G_h$ is twice differentiable. We also obtain the first and second derivatives of $G_h$.

Theorem C.1. We denote $x_h^s = G_h(u+sv)$ and set $y_h \in X_h^r$ be the solution of the following discretized equation:

where $x_h = G_h (u)$. Then we have $\frac{d}{ds} x_h^s (t) = y_h(t)$.

Proof. By Theorem 2.1 there exists a solution $y_h \in X_h^r$ to

By Lemma B.4 we get

Recall that $x^s$ and $x$ satisfy

Using this, we find that $r(t) : = x_h^s (t) - x_h (t) - s y_h(t)$ satisfies

for all $\varphi \in X_h^r$, where

and

Given that $|x_h^s (t) - x_h (t)|\leq Cs$ and (1.3), an elementary calculus shows that $|A_1| \leq Cs^2$ and $|A_2| \leq Cs^2$. With these bounds, we may apply Lemma B.4 to deduce $|r(t)| \leq Cs^2$ for $t \in [0, T]$. From this we find that

which yields that

This completes the proof.

Lemma C.2. The following holds.

Proof. Let $y_h = {G_h}' (u_1) v \in X_h^r$ and $z_h = {G_h}' (u_2)v \in X_h^r$. Then we obtain

and

for all $\varphi \in X_h^r$. Combining these equalities, we have

for all $\varphi \in X_h^r$. On the other hand, the following two inequalities hold:

and

Given these estimates, by applying Lemma B.4 to (C.4), we obtain

where we used Lemma B.5 in the second inequality.

Lemma C.3. Let $z_h \in X^r_h$ be the solution of the following discretized equation:

for any $\varphi \in X_h^r$, where $y_h \in X^r_h$ is the solution of (C.1). Then we have

Proof. Let

It then follows that

where

and

We obtain from Lemma C.2 the estimate $|y_h^s (t) - y_h(t)|\leq Cs$. Upon this estimate and that $\frac{d}{ds} x_h^s (t)|_{s = 0} = y_h(t)$ from Lemma C.1, an elementary calculus reveals that $|A_1 (t)| \leq Cs^2$ and $|A_2 (t)| \leq Cs^2$. Putting this estimate into (C.5) and using Lemma B.4, we find

This yields that

and so we have

since

The proof is done.

D.   Derivations of the first order derivative of cost functionals

In this part, we give the proofs of Lemmas 3.3 and 3.5. Before presenting it, we shall explain how to derive the discrete adjoint Eq (3.8) from the Lagrangian associated to (2.5).

Let us first write the Lagrangian of the problems (1.1) and (3.7) as follows:

for $\lambda_h \in X_h^r$, where the bilinear operator $B(\cdot, \cdot)$ is given by (3.7). If we compute the functional derivatives of the above Lagrangian (D.1) with respect to the adjoint state $\lambda_h$, then $\delta \mathcal{L}_h/\delta \lambda_h = 0$ leads (3.7). We now derive the equation of discrete adjoint state. Using the integration by parts, we find

This enables us to rewrite the Lagrangian (D.1) as

and this further implies

for all $\psi_h \in X^r_h$, where we applied the integration by parts for $(\psi_h, {\lambda_h}')_{I_n}$ to derive the second equality. The above equality corresponds to the adjoint Eq (3.8).

Proof of Lemma 3.3. In order to compute the functional derivative of $j$ with respect to $u$, we consider $j(u+sv) = J(u + sv, G(u+sv))$ with $v \in \mathcal{U}$ and $s \in \mathbb{R}_+$. If we set $x^s(t) : = G(u(t)+sv(t))$ it follows from Lemma A.1 that $y = \frac{d}{ds} x^s (t)|_{s = 0}$ satisfies

with the initial condition $y(0) = 0$. Recall from (3.4) that the adjoint state $\lambda (t) = \lambda(u)(t)$ satisfies

Since $x^s (t)$ is differentiable with respect to $s$, the cost $j(u+sv)$ is differentiable with respect to $s$ and it is computed as

where we used

due to (D.3), (D.4), $y(0) = 0$, and $\lambda(T) = 0$.

Proof of Lemma 3.5. The proof is very similar to Lemma 3.3. We consider $j_h(u+sv) = J(u+sv, G_h(u + sv))$ with $v \in \mathcal{U}$ and $s \in \mathbb{R}_+$. We recall from Lemma C.1 that the function $x_{h}^s: = G_h(u + sv)$ is differentiable at $s = 0$ with

where $y_h \in X_h^r$ satisfies the following equation:

Using this, we obtain

We then take $\psi_h = y_h$ in (D.2) to get

On the other hand, by using the integration by parts, we find

where $B(\cdot, \cdot)$ is appeared in (3.6). This yields

due to (D.5). This together with (D.6) concludes

where $v \in \mathcal{U}$.

E.   Derivations of the second order derivative of cost functionals

In this appendix, we provide details of the derivation of the second order derivative of cost functional $j$ and its discrete version $j_h$.

Lemma E.1. Let $j$ be the cost functional for the optimal control problems $(1.1)$ and $(1.2)$. Then, for $u \in \mathcal{U}_{ad}$ and $v \in \mathcal{U}$, we have

Proof. Similarly as in Appendix D, we consider $j(u+sv) = J(u + sv, G(u+sv))$ with $v \in \mathcal{U}$ and $s \in \mathbb{R}_+$ and set $x^s(t) : = G(u(t)+sv(t))$. By Lemmas A.1 and A.2, it follows that

where $y \in X$ is given as in (D.3) and $z \in X$ is the solution to

with the initial condition $z(0) = 0$. Then we obtain

On the other hand, we use (D.4) to get

where we used $\lambda(T) = 0$ and $z(0) = 0$. By combining the above with (E.1), we have

This completes the proof.

Remark E.2. Solving the differential Eq (A.1) gives

and thus

where $C > 0$ depends only on $\|f\|_{L^\infty(0, T; W^{1, \infty})}$ and $T > 0$. This estimate for $y$ enables to bound the first four integrals on the right hand side of (E.2) by

where $C > 0$ depends only on $\|\lambda\|_{L^\infty(I)}$, $T$, $\|f\|_{L^\infty(0, T; W^{2, \infty})}$, and $\|g\|_{L^\infty(0, T; W^{2, \infty})}$. This implies that if $g$ is given by

then we have

which satisfies (2.4) if $\gamma > C/2$. It would be interesting to develop a numerical method to check (2.4) for general case.

Next we proceed the similar calculation for the approximate solution.

Lemma E.3. Let $j_h$ be the discrete cost functional for the optimal control problems $(1.1)$ and $(1.2)$. Then, for $u \in \mathcal{U}_{ad}$ and $v \in \mathcal{U}$, we have

Proof. Similarly as in the proof of Lemma 3.5, we consider $j_h(u+sv) = J(u+sv, G_h(u + sv))$ with $v \in \mathcal{U}$ and $s \in \mathbb{R}_+$ and set $x_{h}^s: = G_h(u + sv)$. We recall from Theorem C.1 and Theorem C.3 that

where $z_h \in X^r_h$ satisfies

Now a straightforward computation gives

Note that the discrete adjoint state $\lambda_h(t) = \lambda_h(u)(t)$ satisfies

for all $\psi \in X^r_h$. Thus by considering $\psi = z_h \in X^r_h$, we find

Combining the above equalities, we have

This completes the proof.