A Certified Reduced Basis Approach for Parametrized Optimal Control Problems with Two-Sided Control Constraints

Abstract

In this paper, we employ the reduced basis method for the efficient and reliable solution of parametrized optimal control problems governed by elliptic partial differential equations. We consider the standard linear-quadratic problem setting with distributed control and two-sided control constraints, which play an important role in many industrial and economical applications. For this problem class, we propose two different reduced basis approximations and associated error estimation procedures. In our first approach, we directly consider the resulting optimality system, introduce suitable reduced basis approximations for the state, adjoint, control, and Lagrange multipliers, and use a projection approach to bound the error in the reduced optimal control. For our second approach, we first reformulate the optimal control problem using two slack variables, we then develop a reduced basis approximation for both slack problems by suitably restricting the solution space, and derive error bounds for the slack based optimal control. We discuss benefits and drawbacks of both approaches and substantiate the comparison by presenting numerical results for a model problem.

3.1 Introduction

Optimal control problems governed by partial differential equations (PDEs) appear in a wide range of applications in science and engineering, such as heat phenomena, crystal growth, and fluid flow (see, e.g., [4, 8]). Their solution using classical discretization techniques such as finite elements (FE) or finite volumes can be computationally expensive and time-consuming. Often, additional parameters enter the problem, e.g., material or geometry parameters in a design exercise.

Previous work on reduced order methods for optimal control problems considered distributed but unconstrained controls or constrained but scalar controls. Elliptic optimal control problems with distributed control have been considered recently by Negri et al. [10]. The proposed error bound is based on the Banach-Nečas-Babuška (BNB) theory applied to the first order optimality system. The approach thus provides a combined bound for the error in the state, adjoint, and control variable, but it is only applicable to problems without control constraints. Since the bound requires the very costly computation of a lower bound to the inf-sup constant, Negri et al. [11] compute error estimates using a heuristic interpolant surrogate of that constant.

Based on the ideas in Tröltzsch and Volkwein [13], Kärcher and Grepl [6] proposed rigorous and online-efficient control error bounds for reduced basis (RB) approximations of scalar elliptic optimal control problems. These ideas are extended and improved in [7] to distributed control problems.

In a recent paper [1], we employed the RB method as a surrogate model for the solution of distributed and one-sided constrained optimal control problems governed by parametrized elliptic partial differential equations. In this paper we extend this work to two-sided control constraints. After stating the problem in Sect. 3.2 we present the following contributions:

• In Sect. 3.3 we extend previous work on reduced basis methods for variational inequalities in [1, 3] to the optimal control setting with two-sided control constraints. While we can derive an offline-online decomposable RB optimality system, we are only able to derive a partially offline-online decomposable control error bound that depends on the FE dimension of the control.

• In Sect. 3.4 we build on the recent work in [1, 14] and propose an RB slack approach for optimal control. We introduce two slack formulations for the optimal control problem, which we obtain by shifting the optimal control by each constraint. We are thus able to derive an offline-online decomposable RB optimality systems and control error bound. The evaluation of this bound is independent of the FE dimension of the problem, but requires the solution of three RB systems.

In Sect. 3.5 we propose a greedy sampling procedure to construct the RB spaces and in Sect. 3.6 we assess the properties of our methods by presenting numerical results for a Graetz flow problem.

3.2 General Problem Statement and Finite Element Discretization

In this section we introduce the parametrized linear-quadratic optimal control problem with elliptic PDE constraint and a constrained distributed control. We introduce a finite element (FE) discretization for the continuous problem and recall the first-order necessary (and in the convex setting sufficient) optimality conditions.

3.2.1 Preliminaries

Let Y _e with H ₀ ¹(Ω) ⊂ Y _e ⊂ H ¹(Ω) be a Hilbert space over the bounded Lipschitz domain $\varOmega \subset \mathbb{R}^{d},d \in \{ 1,2,3\}$, with boundary Γ.¹ The inner product and induced norm associated with Y _e are given by (⋅ , ⋅ ) _Y and $\Vert \cdot \Vert _{Y } = \sqrt{(\cdot, \cdot )_{Y}}$. We assume that the norm $\Vert \cdot \Vert _{Y }$ is equivalent to the H ¹(Ω)-norm and denote the dual space of Y _e by Y _e′. We also introduce the control Hilbert space U _e = L ²(Ω), together with its inner product (⋅ , ⋅ ) _U , induced norm $\Vert \cdot \Vert _{U} = \sqrt{(\cdot, \cdot )_{U}}$, and associated dual space U _e′.² Furthermore, let $\mathcal{D} \subset \mathbb{R}^{P}$ be a prescribed P-dimensional compact parameter set in which the P-tuple (input) parameter μ = (μ ₁, …, μ _P ) resides.

We directly consider a FE “truth” approximation for the exact infinite-dimensional optimal control problem. To this end, we define two conforming FE spaces Y ⊂ Y _e and U ⊂ U _e and denote their dimensions by $\mathcal{N}_{Y } =\dim (Y )$ and $\mathcal{N}_{U} =\dim (U)$. We assume that the truth spaces Y and U are sufficiently rich such that the FE solutions guarantee a desired accuracy over $\mathcal{D}$.

We next introduce the μ-dependent bilinear form $a(\cdot,\cdot;\mu ): Y \times Y \rightarrow \mathbb{R}$, and shall assume that a(⋅ , ⋅ ; μ) is (1) continuous for all $\mu \in \mathcal{ D}$ with continuity constant γ _a (μ) < ∞ and (2) coercive for all $\mu \in \mathcal{ D}$ with coercivity constant α _a (μ) > 0. Furthermore, we introduce the μ-dependent continuous linear functional $f(\cdot;\mu ): Y \rightarrow \mathbb{R}$ and the bilinear form $b(\cdot,\cdot;\mu ): U \times Y \rightarrow \mathbb{R}$ with continuity constant γ _b (μ) < ∞.

In anticipation of the optimal control problem, we introduce the parametrized control constraints u _a (μ), u _b (μ) ∈ U and a desired state y _d ∈ D. Here, $D \subset L^{2}(\varOmega _{\!_{D}}\!)$ is a suitable FE space for the observation subdomain $\varOmega _{\!_{ D}}\! \subset \varOmega$. Furthermore, we note that the semi-norm $\vert \cdot \vert _{D}$ for y ∈ L ²(Ω) is defined by $\vert \cdot \vert _{D} =\Vert \cdot \Vert _{L^{2}(\varOmega _{D})}$.

The involved bilinear and linear forms as well as the control constraint are assumed to depend affinely on the parameter. For example we require for all w, v ∈ Y and all parameters $\mu \in \mathcal{ D}$ that $a(w,v;\mu ) =\sum _{ q=1}^{Q_{a}}\varTheta _{a}^{q}(\mu )\,a^{q}(w,v)$ and $u_{a}(x;\mu ) =\sum _{ q=1}^{Q_{ua}}\varTheta _{ua}^{q}(\mu )\,u_{a}^{q}(x)$ for some (preferably) small integers Q _a and Q _ua . Here, the coefficient functions $\varTheta _{\bullet }^{q}(\cdot ):\mathcal{ D} \rightarrow \mathbb{R}$ are continuous and depend on μ, whereas the continuous bilinear and linear forms, e.g., a ^q(⋅ , ⋅ ) and u _a ^q ∈ U do not depend on μ. Although we choose y _d (x) to be parameter-independent, our approach directly extends to an affinely parameter-dependent y _d (x; μ) (see Kärcher et al. [7]).

For the development of a posteriori error bounds we also require additional ingredients. We assume that we are given a positive lower bound $\alpha _{a}^{\mathrm{LB}}(\mu ):\mathcal{ D} \rightarrow \mathbb{R}_{+}$ for the coercivity constant α _a (μ) of a(⋅ , ⋅ ; μ) such that $\alpha _{a}^{\mathrm{LB}}(\mu ) \leq \alpha _{a}(\mu )\ \forall \mu \in \mathcal{ D}$. Furthermore, we assume that we have upper bounds available for the constant $C_{D}^{\mathrm{UB}} \geq C_{D} =\sup _{w\in Y \setminus \{0\}}\frac{\vert w\vert _{D}} {\Vert w\Vert _{Y }} \geq 0\ \forall \mu \in \mathcal{ D},$ and the continuity constant of the bilinear form b(⋅ , ⋅ ; μ): $\gamma _{b}^{\mathrm{UB}}(\mu ) \geq \gamma _{b}(\mu )\ \forall \mu \in \mathcal{ D}.$ Here, the constant C _D depends on the parameter, since later we use $\vert \cdot \vert _{D} =\Vert \cdot \Vert _{L^{2}(\varOmega _{D}(\mu ))}$ (see Sect. 3.6). In our setting, it is possible to compute these constants (or their bounds) efficiently using an offline-online procedure (see [7, 12]).

3.2.2 Abstract Formulation of Linear-Quadratic Optimal Control Problems and the First-Order Optimality Conditions

We consider the following FE optimal control problem in weak form with u _a (μ) < u _b (μ)

$$\displaystyle\begin{array}{rcl} \min _{\hat{y},\hat{u}}J(\hat{y},\hat{u}) = \frac{1} {2}\vert \hat{y} - y_{d}\vert _{D}^{2} + \frac{\lambda } {2}\Vert \hat{u}\Vert _{U}^{2},\qquad \lambda> 0& &{}\end{array}$$

(P)

$$\displaystyle\begin{array}{rcl} & & \text{s.t.}\quad (\hat{y},\hat{u}) \in Y \times U\quad \text{solves}\quad a(\hat{y},\phi;\mu ) = b(\hat{u},\phi;\mu ) + f(\phi;\mu )\quad \forall \phi \in Y, {}\\ & & \quad \quad \,(u_{a}(\mu ),\rho )_{U} \leq (\hat{u},\rho )_{U} \leq (u_{b}(\mu ),\rho )_{U}\quad \forall \rho \in U^{+}, {}\\ \end{array}$$

where U ⁺: = { ρ ∈ U; ρ ≥ 0 almost everywhere} and we dropped the μ-dependence of the state and control $(\hat{y},\hat{u})$ for the sake of readability. We note that the last line of (P) is equivalent to $\hat{u}$ being in the convex admissible set

$$\displaystyle{ U_{\mathrm{ad}} =\{\psi \in U;(u_{a}(\mu ),\rho )_{U} \leq (\psi,\rho )_{U} \leq (u_{b}(\mu ),\rho )_{U}\ \forall \rho \in U^{+}\}. }$$

(3.1)

In the following we call problem (P) the “primal” problem, for which the existence and uniqueness of the solution is standard (see, e.g., [4]). The derivation of the necessary and sufficient first-order optimality system is straightforward: Given $\mu \in \mathcal{ D}$, the optimal solution (y, p, u, σ, σ _b ) ∈ Y × Y × U × U × U satisfies

$$\displaystyle\begin{array}{rcl} \begin{array}{rclrcl} a(y,\phi;\mu )& =&b(u,\phi;\mu ) + f(\phi;\mu )&&\forall \phi \in Y,\end{array} & &{}\end{array}$$

(3.2a)

$$\displaystyle\begin{array}{rcl} \begin{array}{rclrcl} a(\varphi,p;\mu )& =&(y_{d} - y,\varphi )_{D}&&\forall \varphi \in Y,\end{array} & &{}\end{array}$$

(3.2b)

$$\displaystyle\begin{array}{rcl} \begin{array}{rclrcll} (\lambda u,\psi )_{U} - b(\psi,p;\mu )& =&(\sigma,\psi )_{U} - (\sigma _{b},\psi )_{U}&&\forall \psi \in U,\end{array} & &{}\end{array}$$

(3.2c)

$$\displaystyle\begin{array}{rcl} \begin{array}{rclrcl} (u_{a}(\mu ) - u,\rho )_{U} \leq 0\quad \forall \rho \in U^{+},&\qquad (u_{a}(\mu ) - u,\sigma )_{U} = 0,&&\sigma \geq 0,\end{array} & &{}\end{array}$$

(3.2d)

$$\displaystyle\begin{array}{rcl} \begin{array}{rclrcl} (u_{b}(\mu ) - u,\rho )_{U} \geq 0\quad \forall \rho \in U^{+},&\qquad (u_{b}(\mu ) - u,\sigma _{b})_{U} = 0,&&\sigma _{b} \geq 0.\end{array} & &{}\end{array}$$

(3.2e)

Note that we follow a first-discretize-then-optimize approach here, for a more detailed discussion see [4, Sect. 3.2.4]).

In the following we comment on the FE-setting in this paper. We assume that the state variable is discretized by P ₁, i.e., continuous and piecewise linear, and the control variable by P ₀, i.e., piecewise constant finite elements. Next, we introduce two bases for the FE spaces Y and U, such that

$$\displaystyle{ Y =\mathrm{ span}\{\,\phi _{i}^{y},i = 1,\ldots,\mathcal{N}_{ Y }\}\quad \text{and}\quad U =\mathrm{ span}\{\,\phi _{i}^{u},i = 1,\ldots,\mathcal{N}_{ U}\}, }$$

where $\phi _{i}^{y} \geq 0,\;i = 1,\ldots,\mathcal{N}_{Y }$, and $\phi _{i}^{u} \geq 0,\;i = 1,\ldots,\mathcal{N}_{U}$, are the usual hat and bar basis functions. Using these basis functions we can express the functions y, p ∈ Y and u, σ, σ _b ∈ U as, e.g., $y =\sum _{ i=1}^{\mathcal{N}_{Y }}y_{i}\phi _{i}^{y}$. The corresponding FE coefficient vectors are given by, e.g., $\underline{y} = (y_{1},\ldots,y_{\mathcal{N}_{Y }})^{T} \in \mathbb{R}^{\mathcal{N}_{Y }}$. Note that by definition of U ⁺ and since ϕ _i ^u ≥ 0, the condition ρ ∈ U ⁺ in (3.2d) and (3.2e) translates into the condition ρ ≥ 0 for the corresponding coefficient vector. Further, we also introduce the control mass matrix M _U with entries (M _U ) _ij = (ϕ _i ^u, ϕ _j ^u) _U , which is for a P ₀ control discretization a positive diagonal matrix. Hence the point-wise and the ‘weak’ (averaged) constraint formulations are equivalent $u(x) \geq u_{a}(x;\mu )\, \Leftrightarrow (u,\rho )_{U} \geq (u_{a}(\mu ),\rho )_{U}\ \forall \rho \in U^{+}$. However, this is in general not true for other control discretizations, e.g., P ₁.

Based on the truth FE primal problem (P) we derive an RB primal problem (P _N ) and a rigorous a posteriori error bound for the error between the truth and RB control approximation in Theorem 1.

3.3 Reduced Basis Method for the Primal Problem

3.3.1 Reduced Basis Approximation

To begin, we define the RB spaces Y _N ⊂ Y, U _N , Σ _N , Σ _b, N ⊂ U as well as the convex cones Σ _N ⁺ ⊂ U ⁺, Σ _b, N ⁺ ⊂ U ⁺ as follows: given N parameter samples μ ¹, …, μ ^N, we set

$$\displaystyle\begin{array}{rcl} Y _{N} =\mathrm{ span}\{\,\zeta _{1}^{y},\ldots,\zeta _{ N_{Y }}^{y}\} =\mathrm{ span}\{\,y(\mu ^{1}),p(\mu ^{1}),\ldots,y(\mu ^{N}),p(\mu ^{N})\,\},& &{}\end{array}$$

(3.3a)

$$\displaystyle\begin{array}{rcl} U_{N} =\mathrm{ span}\{\zeta _{1}^{u},\ldots,\zeta _{ N_{U}}^{u}\} =\mathrm{ span}\{u(\mu ^{1}),\sigma (\mu ^{1}),\sigma _{ b}(\mu ^{1}),\!\ldots \!,u(\mu ^{N}),\sigma (\mu ^{N}),\sigma _{ b}(\mu ^{N})\!\},& &{}\end{array}$$

(3.3b)

$$\displaystyle\begin{array}{rcl} \varSigma _{N} =\mathrm{ span}\{\,\zeta _{1}^{\sigma },\ldots,\zeta _{ N_{\sigma }}^{\sigma }\} =\mathrm{ span}\{\,\sigma (\mu ^{1}),\ldots,\sigma (\mu ^{N})\,\},& &{}\end{array}$$

(3.3c)

$$\displaystyle\begin{array}{rcl} \varSigma _{b,N} =\mathrm{ span}\{\zeta _{1}^{\sigma _{b} },\ldots,\zeta _{N_{\sigma _{ b}}}^{\sigma _{b} }\} =\mathrm{ span}\{\sigma _{b}(\mu ^{1}),\ldots,\sigma _{ b}(\mu ^{N})\},& &{}\end{array}$$

(3.3d)

$$\displaystyle\begin{array}{rcl} \varSigma _{N}^{+} =\mathrm{ span}_{ +}\{\,\zeta _{1}^{\sigma },\ldots,\zeta _{ N_{\sigma }}^{\sigma }\}\quad \text{ and }\quad \varSigma _{ b,N}^{+} =\mathrm{ span}_{ +}\{\zeta _{1}^{\sigma _{b} },\ldots,\zeta _{N_{\sigma _{ b}}}^{\sigma _{b} }\},& &{}\end{array}$$

(3.3e)

where we assume that the basis functions, $\zeta _{1}^{\bullet },\ldots \zeta _{N_{\bullet }}^{\bullet }$, are linearly independent and span₊{⋅ } indicates the cone spanned by non-negative combinations of the elements, i.e.

$$\displaystyle{ \mathrm{span}_{+}\{\zeta _{1},\ldots,\zeta _{N}\} = \left \{\sum _{i=1}^{N}\alpha _{ i}\zeta _{i}\,\vert \alpha _{i} \geq 0\right \}. }$$

Note that we employ integrated spaces for the state and adjoint as well as for the control (see Remarks 1 and 2). For the spaces Y _N and U _N we additionally assume that the basis functions are orthogonal, i.e., (ζ _i ^y, ζ _j ^y) _Y = δ _ij and (ζ _i ^u, ζ _j ^u) _U = δ _ij , where δ _ij is the Kronecker delta. This orthogonality is favorable to keep the condition of the RB algebraic linear systems small [12]. In addition, we do not orthogonalize the basis $\zeta _{i}^{\sigma },\,\zeta _{i}^{\sigma _{b}}$ of the cones Σ _N ⁺, Σ _b, N ⁺ ⊂ U ⁺, because this non-negativity is used in the definition of the reduced problem (P _N ) of (P).³ Although the conditions ζ _i ^σ ∈ Σ _N ⁺ and $\zeta _{i}^{\sigma _{b}} \in \varSigma _{ b,N}^{+}$ appear to be much more restrictive than $\zeta _{i}^{\sigma },\,\zeta _{i}^{\sigma _{b}} \in U^{+}$, we observe in numerical tests (not shown) that the RB approximations converge to the FE solutions with a similar rate as the control approximations. In addition, the RB approximations are comparable to the best possible approximations derived by projecting σ to Σ _N or Σ _N ⁺, analogously for σ _b . We describe the greedy sampling approach to construct the RB spaces in Sect. 3.5. Next, given the RB spaces in (3.3) we derive the RB primal problem

The last line of (P _N ) defines the admissible set for u _N : $U_{\mathrm{ad},N} =\{\psi \in U_{N};(u_{a}(\mu ),\rho )_{U} \leq (\psi,\rho )_{U}\ \forall \rho \in \varSigma _{N}^{+},\ (u_{b}(\mu ),\rho )_{U} \geq (\psi,\rho )_{U}\ \forall \rho \in \varSigma _{b,N}^{+}\}$, which is in general not a subset of U _ad in (3.1). Analogously to the primal problem (P) we obtain the RB optimality system: Given $\mu \in \mathcal{ D}$, the optimal solution (y _N , p _N , u _N , σ _N , σ _b, N ) ∈ Y _N × Y _N × U _N ×Σ _N ×Σ _b, N satisfies

$$\displaystyle\begin{array}{rcl} \begin{array}{rclrcl} a(y_{N},\phi;\mu )& = b(u_{N},\phi;\mu ) + f(\phi;\mu )&&\forall \phi \in Y _{N},\end{array} & &{}\end{array}$$

(3.4a)

$$\displaystyle\begin{array}{rcl} \begin{array}{rclrcl} a(\varphi,p_{N};\mu )& = (y_{d} - y_{N},\varphi )_{D}&&\forall \varphi \in Y _{N},\end{array} & &{}\end{array}$$

(3.4b)

$$\displaystyle\begin{array}{rcl} \begin{array}{rclrcl} (\lambda u_{N},\psi )_{U} - b(\psi,p_{N};\mu )& = (\sigma _{N},\psi )_{U} - (\sigma _{b,N},\psi )_{U}&&\forall \psi \in U_{N},\end{array} & &{}\end{array}$$

(3.4c)

$$\displaystyle\begin{array}{rcl} \begin{array}{rclrcl} (u_{a}(\mu ) - u_{N},\rho )_{U} \leq 0\ \forall \rho \in \varSigma _{N}^{+},&\ (u_{a}(\mu ) - u_{N},\sigma _{N})_{U} = 0,&&\sigma _{N} \in \varSigma _{N}^{+},\end{array} & &{}\end{array}$$

(3.4d)

$$\displaystyle\begin{array}{rcl} \begin{array}{rclrcl} (u_{b}(\mu ) - u_{N},\rho )_{U} \geq 0\ \forall \rho \in \varSigma _{b,N}^{+},&\ (u_{b}(\mu ) - u_{N},\sigma _{b,N})_{U} = 0,&&\sigma _{b,N} \in \varSigma _{b,N}^{+}.\end{array} & &{}\end{array}$$

(3.4e)

Remark 1 (Existence, Uniqueness, Integrated Space Y _N )

Since (P _N ) is a linear-quadratic optimal control problem over the closed convex admissible set U _ad,N , the existence and uniqueness of the RB optimal control u _N follows from standard arguments (see, e.g., [4, Theorem 1.43]). Also note that we use a single “integrated” reduced basis trial and test space Y _N for the state and adjoint equations as one ingredient to ensure stability of the system (3.4), see e.g. Kärcher [5].

Remark 2 (Stability, Integrated Space U _N )

For the stability of the RB solutions we need to show that the RB inf-sup constants

$$\displaystyle{ \beta _{N}:=\inf _{\psi _{\sigma }\in \varSigma _{N}}\sup _{\psi _{u}\in U_{N}}\frac{(\psi _{\sigma },\psi _{u})_{U}} {\Vert \psi _{\sigma }\Vert _{U}\Vert \psi _{u}\Vert _{U}},\quad \beta _{b,N}:=\inf _{\psi _{\sigma _{ b}}\in \varSigma _{b,N}}\sup _{\psi _{u}\in U_{N}}\frac{(\psi _{\sigma _{b}},\psi _{u})_{U}} {\Vert \psi _{\sigma _{b}}\Vert _{U}\Vert \psi _{u}\Vert _{U}} }$$

are bounded away from zero. We guarantee that β _N , β _b, N ≥ β > 0 by enriching the RB control space with suitable supremizers [9]. Here, these supremizers are just the multiplier snapshots σ(μ ⁿ), σ _b (μ ⁿ),  1 ≤ n ≤ N; we thus have β _N = β _b, N = β = 1.

3.3.2 Primal Error Bound

We next propose an a posteriori error bound for the optimal control. The bound is based on [1], which uses an (1) RB approach for variational inequalities of the first kind [3], and (2) an RB approach for optimal control problems with a PDE constraint [7]. Before stating the main result, we define the following approximation errors (omitting μ-dependencies) of the RB primal system (3.4)

$$\displaystyle{ e_{y} = y - y_{N},\quad e_{p} = p - p_{N},\quad e_{u} = u - u_{N},\quad e_{\sigma } =\sigma -\sigma _{N},\quad e_{\sigma _{b}} =\sigma _{b} -\sigma _{b,N}, }$$

as well as the residuals in the next definition.

Definition 1 (Residuals)

The residuals of the state equation, the adjoint equation w.r.t. (3.2a)– (3.2c) are defined for all $\mu \in \mathcal{ D}$ by

$$\displaystyle\begin{array}{rcl} \begin{array}{rclrcl} r_{y}(\phi;\mu )& = b(u_{N},\phi;\mu ) + f(\phi;\mu ) - a(y_{N},\phi;\mu )&&\forall \phi \in Y,\end{array} & &{}\end{array}$$

(3.5a)

$$\displaystyle\begin{array}{rcl} \begin{array}{rclrcl} r_{p}(\varphi;\mu )& = (y_{d} - y_{N},\varphi )_{D} - a(\varphi,p_{N};\mu )&&\forall \varphi \in Y,\end{array} & &{}\end{array}$$

(3.5b)

$$\displaystyle\begin{array}{rcl} \begin{array}{rclrcl} r_{u}(\psi;\mu )& = -\lambda (u_{N},\psi )_{U} + b(\psi,p_{N};\mu ) + (\sigma _{N},\psi )_{U} - (\sigma _{b,N},\psi )_{U}&&\forall \psi \in U.\end{array} & &{}\end{array}$$

(3.5c)

Theorem 1 (Primal Error Bound)

Let u and u _N be the optimal controls of the FE primal problem  (P) and of the RB primal problem ( P _N ), respectively. Then the error in the optimal control satisfies for any given parameter $\mu \in \mathcal{ D}$

$$\displaystyle{ \Vert e_{u}\Vert _{U} \leq \varDelta _{N}^{\mathrm{pr}}(\mu ), }$$

where $\varDelta _{N}^{\mathrm{pr}}(\mu ):= c_{1}(\mu ) + \sqrt{c_{1 } (\mu )^{2 } + c_{2 } (\mu )}$ with nonnegative coefficients

$$\displaystyle\begin{array}{rcl} \begin{array}{rclrcl} c_{1}(\mu )& = \frac{1} {2\lambda }\left (\Vert r_{u}\Vert _{U'} + \frac{\gamma _{b}^{\mathrm{UB}}} {\alpha _{a}^{\mathrm{LB}}} \Vert r_{p}\Vert _{Y '} +\lambda (\delta _{1} +\delta _{1b})\right ),\end{array} & &{}\end{array}$$

(3.6a)

$$\displaystyle\begin{array}{rcl} \begin{array}{rclrcl} c_{2}(\mu )& = \frac{1} {\lambda } \Bigg[ \frac{2} {\alpha _{a}^{\mathrm{LB}}} \Vert r_{y}\Vert _{Y '}\Vert r_{p}\Vert _{Y '} + \frac{1} {4}\left (\frac{C_{D}^{\mathrm{UB}}} {\alpha _{a}^{\mathrm{LB}}} \left (\Vert r_{y}\Vert _{Y '} + \gamma _{b}^{\mathrm{UB}}(\delta _{1} +\delta _{1b})\right )\right )^{2} \\ & + \left (\Vert r_{u}\Vert _{U'} + \frac{\gamma _{b}^{\mathrm{UB}}} {\alpha _{a}^{\mathrm{LB}}} \Vert r_{p}\Vert _{Y '}\! +\! \sqrt{2(\sigma _{N },\sigma _{b,N } )_{U}}\right )\!(\delta _{1} +\delta _{1b}) +\delta _{2} +\delta _{2b}\Bigg],\end{array} & &{}\end{array}$$

(3.6b)

and $\delta _{1}\! =\!\Vert [u_{a} -u_{N}]_{+}\Vert _{U}\!,\ \delta _{2}\! =\! ([u_{a} -u_{N}]_{+},\sigma _{N})_{U}\!,\ \delta _{1b}\! =\!\Vert [u_{N} -u_{b}]_{+}\Vert _{U}\!,\ \delta _{2b}\! =\! ([u_{N} -u_{b}]_{+},\sigma _{b,N})_{U}\!$ .

Here [⋅ ]₊ = max(⋅ , 0) denotes the positive part (a.e.). Note that we sometimes use r _• instead of r _•(⋅ ; μ) and omit the μ-dependencies on the r.h.s. of (3.6) for a better readability.

Proof

This proof follows the proof of the primal error bound from [1]. Since the FE optimal solution (y, p, u, σ, σ _b ) satisfies the optimality conditions (3.2), we obtain the following error-residual equations:

$$\displaystyle\begin{array}{rcl} \begin{array}{rclrcl} a(e_{y},\phi;\mu ) - b(e_{u},\phi;\mu )& = r_{y}(\phi;\mu )&&\forall \phi \in Y,\end{array} & &{}\end{array}$$

(3.7a)

$$\displaystyle\begin{array}{rcl} \begin{array}{rclrcl} a(\varphi,e_{p};\mu ) + (e_{y},\varphi )_{D}& = r_{p}(\varphi;\mu )&&\forall \varphi \in Y,\end{array} & &{}\end{array}$$

(3.7b)

$$\displaystyle\begin{array}{rcl} \begin{array}{rclrcl} \lambda (e_{u},\psi )_{U} - b(\psi,e_{p};\mu ) - (e_{\sigma },\psi )_{U} + (e_{\sigma _{b}},\psi )_{U}& = r_{u}(\psi;\mu )&&\forall \psi \in U.\end{array} & &{}\end{array}$$

(3.7c)

From (3.7a) with ϕ = e _y , (3.7b) with φ = e _p , and α _a ^LB(μ) ≤ α _a (μ) we infer that

$$\displaystyle\begin{array}{rcl} \Vert e_{y}\Vert _{Y }& \leq \frac{1} {\alpha _{a}^{\mathrm{LB}}} \left (\Vert r_{y}\Vert _{Y '} + \gamma _{b}^{\mathrm{UB}}\Vert e_{u}\Vert _{U}\right ),\ \ \Vert e_{p}\Vert _{Y } \leq \frac{1} {\alpha _{a}^{\mathrm{LB}}} \left (\Vert r_{p}\Vert _{Y '} + C_{D}^{\mathrm{UB}}\vert e_{y}\vert _{D}\right ).&{}\end{array}$$

(3.8)

Choosing ϕ = e _p , φ = e _y , ψ = e _u in (3.7), adding (3.7b) and (3.7c), and subtracting (3.7a) results in

$$\displaystyle{ \lambda \Vert e_{u}\Vert _{U}^{2}\! +\!\vert e_{ y}\vert _{D}^{2}\! \leq \!\Vert r_{ y}\Vert _{Y '}\Vert e_{p}\Vert _{Y }\! +\!\Vert r_{p}\Vert _{Y '}\Vert e_{y}\Vert _{Y }\! +\!\Vert r_{u}\Vert _{U'}\Vert e_{u}\Vert _{U}\! +\! (e_{\sigma } - e_{\sigma _{b}},e_{u})_{U}. }$$

(3.9)

Next, we bound (e _σ , e _u ) _U and $-(e_{\sigma _{b}},e_{u})_{U}$. We first consider (e _σ , e _u ) _U and note that

$$\displaystyle\begin{array}{rcl} & & (e_{\sigma },e_{u})_{U} = (\sigma -\sigma _{N},u - u_{N})_{U} = (\sigma,u - u_{N})_{U} + (\sigma _{N},u_{N} - u)_{U} {}\\ & & = (\sigma,u - u_{a}(\mu ))_{U} + (\sigma,u_{a}(\mu ) - u_{N})_{U} + (\sigma _{N},u_{N} - u_{a}(\mu ))_{U} + (\sigma _{N},u_{a}(\mu ) - u)_{U}, {}\\ \end{array}$$

where, except for the second term, all terms are nonpositive, see (3.2d) and (3.4d). Hence

$$\displaystyle\begin{array}{rcl} (e_{\sigma },e_{u})_{U}& \leq & (\sigma,u_{a}(\mu ) - u_{N})_{U} \leq (\sigma,[u_{a}(\mu ) - u_{N}]_{+})_{U} \\ & =& (\sigma -\sigma _{N},[u_{a}(\mu ) - u_{N}]_{+})_{U} + (\sigma _{N},[u_{a}(\mu ) - u_{N}]_{+})_{U} \leq \Vert e_{\sigma }\Vert _{U}\;\delta _{1} +\delta _{2}.{}\end{array}$$

(3.10)

Analogously, we bound $-(e_{\sigma _{b}},e_{u})_{U} \leq \Vert e_{\sigma _{b}}\Vert _{U}\;\delta _{1b} +\delta _{2b}$. Most significantly, it remains to bound the terms $\Vert e_{\sigma }\Vert _{U}$ and $\Vert e_{\sigma _{b}}\Vert _{U}$, which we achieve in two steps: First, we relate $\Vert e_{\sigma }\Vert _{U}$ and $\Vert e_{\sigma _{b}}\Vert _{U}$ with $\Vert e_{\sigma } - e_{\sigma _{b}}\Vert _{U}$ by

$$\displaystyle{ \Vert e_{\sigma }\Vert _{U}^{2}\! +\!\Vert e_{\sigma _{ b}}\Vert _{U}^{2}\! =\!\Vert e_{\sigma } - e_{\sigma _{ b}}\Vert _{U}^{2}\! +\! 2\big((\sigma,\sigma _{ b})_{U}\! -\! (\sigma _{N},\sigma _{b})_{U}\! +\! (\sigma _{N},\sigma _{b,N})_{U}\! -\! (\sigma,\sigma _{b,N})_{U}\big) }$$

If we employ (σ, σ _b ) _U = 0, (σ _N , σ _b ) _U ≥ 0, and (σ, σ _b, N ) _U ≥ 0, we obtain

$$\displaystyle{ \Vert e_{\sigma }\Vert _{U},\Vert e_{\sigma _{b}}\Vert _{U} \leq \Vert e_{\sigma } - e_{\sigma _{b}}\Vert _{U} + \sqrt{2(\sigma _{N },\sigma _{b,N } )_{U}}. }$$

(3.11)

Second, we focus on the optimality residual (3.7c), use the inf-sup stability of (⋅ , ⋅ ) _U and (3.8) to derive $\Vert e_{\sigma } - e_{\sigma _{b}}\Vert _{U} \leq \Vert r_{u}\Vert _{U'} +\lambda \Vert e_{u}\Vert _{U} + \frac{\gamma _{b}^{\mathrm{UB}}} {\alpha _{a}^{\mathrm{LB}}} \left (\Vert r_{p}\Vert _{Y '} + C_{D}^{\mathrm{UB}}\vert e_{y}\vert _{D}\right ).$ Next, we employ the inequalities (3.8) and (3.11) in (3.9) to obtain

$$\displaystyle\begin{array}{rcl} \lambda \Vert e_{u}\Vert _{U}^{2}& +& \vert e_{ y}\vert _{D}^{2} \leq \,\Vert e_{ u}\Vert _{U}\left (\Vert r_{u}\Vert _{U'} + \frac{\gamma _{b}^{\mathrm{UB}}} {\alpha _{a}^{\mathrm{LB}}}\Vert r_{p}\Vert _{Y '} +\lambda (\delta _{1} +\delta _{1b})\right ) \\ & & + \frac{2} {\alpha _{a}^{\mathrm{LB}}}\Vert r_{y}\Vert _{Y '}\Vert r_{p}\Vert _{Y '} +\vert e_{y}\vert _{D}\frac{C_{D}^{\mathrm{UB}}} {\alpha _{a}^{\mathrm{LB}}} \left (\Vert r_{y}\Vert _{Y '} + \gamma _{b}^{\mathrm{UB}}(\delta _{ 1} +\delta _{1b})\right ) \\ & & +\left (\Vert r_{u}\Vert _{U'} + \frac{\gamma _{b}^{\mathrm{UB}}} {\alpha _{a}^{\mathrm{LB}}}\Vert r_{p}\Vert _{Y '} + \sqrt{2(\sigma _{N },\sigma _{b,N } )_{U}}\right )(\delta _{1} +\delta _{1b}) +\delta _{2} +\delta _{2b}.{}\end{array}$$

(3.12)

It thus follows from applying Young’s inequality to the $\vert e_{y}\vert _{D}$-terms in (3.12) that

$$\displaystyle{ \Vert e_{u}\Vert _{U}^{2} - 2c_{ 1}(\mu )\Vert e_{u}\Vert _{U} - c_{2}(\mu ) \leq 0, }$$

where c ₁(μ) and c ₂(μ) are given in (3.6). Solving the last inequality for the larger root yields $\Vert e_{u}\Vert _{U} \leq c_{1}(\mu ) + \sqrt{c_{1 } (\mu )^{2 } + c_{2 } (\mu )} =\varDelta _{ N}^{\mathrm{pr}}(\mu )$.

We note that most of the ingredients of the primal error bound Δ _N ^pr(μ) introduced in Theorem 1 are standard, i.e., the dual norms of state, adjoint, and control residuals, as well as coercivity and continuity constants or rather their lower and upper bounds [7, 12]. The only non-standard terms are δ ₁, δ ₂, δ _1b and δ _2b , which measure the constraint-violation of the RB optimal control u _N . As a result, the online computational cost to evaluate δ _•—and hence the error bound Δ _N ^pr(μ)—depends on the FE control dimension $\mathcal{N}_{U}$, requiring $\mathcal{O}((N_{U} + N_{\sigma } + N_{\sigma _{b}})\mathcal{N}_{U})$ operations.

3.4 Slack Problem and the Primal-Slack Error Bound

In this section we introduce a reformulation of the original primal problem by means of a slack variable. We extend the ideas presented for the one-sided control-constrained problem in [1] to the two-sided control-constrained problem. First, we reformulate the original optimization problem (P) by replacing the control variable with a slack variable that depends on one of the two constraints u _a (μ) or u _b (μ). Second, we use snapshots of the slack variable to construct an associated convex cone, leading to strictly feasible approximations w.r.t. either the lower or upper constraint. Third, we derive two RB slack problems by restricting the RB-slack coefficients to a convex cone. And finally, we propose an a posteriori $\mathcal{N}$-independent error bound for RB slack approximation w.r.t. either the lower or upper constraint in Theorem 2.

3.4.1 FE and RB Slack Problem

We consider the FE optimization problem (P) and introduce the slack variable s ∈ U ⁺ given by

$$\displaystyle{ s = u - u_{a}(\mu ) }$$

(3.13)

together with the corresponding FE coefficient vector s = u − u _a (μ), where we state the slack variable w.r.t. u _a (μ). Here, we again omit the explicit dependence of u and s on the parameter μ. We note that, by construction, the feasibility of u w.r.t. u _a (μ) is equivalent to M _U s ≥ 0, which in turn is equivalent to s ≥ 0, if we are using P ₀ elements.

If we substitute u by s + u _a (μ) in (P), we obtain the “slack” optimization problem

$$\displaystyle\begin{array}{rcl} & & \qquad \min _{\hat{y},\hat{s}}J_{s}(\hat{y},\hat{s}) = \frac{1} {2}\vert \hat{y} - y_{d}\vert _{D}^{2} + \frac{\lambda } {2}\Vert \hat{s} + u_{a}(\mu )\Vert _{U}^{2} \\ & & \text{s.t.}(\hat{y},\hat{s}) \in Y \times U^{+}\quad \text{solves}\quad a(\hat{y},\phi;\mu ) = b(\hat{s} + u_{ a}(\mu ),\phi;\mu ) + f(\phi;\mu )\ \forall \phi \in Y, \\ & & \qquad (u_{b}(\mu ),\rho )_{U} \geq (\hat{s} + u_{a}(\mu ),\rho )_{U}\quad \forall \rho \in U^{+}. {}\end{array}$$

(S)

Analogously, we define a slack variable s _b = u _b (μ) −u w.r.t. u _b (μ) and recast (P) w.r.t. s _b . We do not state this minimization problem explicitly since it is analogous to (S).

In the following we derive two RB slack problems w.r.t. u _a (μ) and u _b (μ). We start with the former and reuse the RB space Y _N , introduced in Sect. 3.3.1, for the state and adjoint variables. Furthermore, for the RB approximation of the slack variable s we simply introduce an RB slack space S _N and a convex cone S _N ⁺ by shifting the control snapshots of (P) with the control constraint u _a (μ)

$$\displaystyle\begin{array}{rcl} S_{N}& =& \mathrm{span}\{\,\zeta _{1}^{s},\ldots,\zeta _{ N_{S}}^{s}\,\} =\mathrm{ span}\{\,u(\mu ^{1}) - u_{ a}(\mu ^{1}),\ldots,u(\mu ^{N}) - u_{ a}(\mu ^{N})\,\},{}\end{array}$$

(3.14a)

$$\displaystyle\begin{array}{rcl} S_{N}^{+}& =& \mathrm{span}_{ +}\{\,\zeta _{1}^{s},\ldots,\zeta _{ N_{S}}^{s}\,\} \subset U^{+}.{}\end{array}$$

(3.14b)

We assume that the snapshots $\zeta _{1}^{s},\ldots,\zeta _{N_{s}}^{s}$ are linearly independent and not orthogonalized. Further, we need to consider a Lagrange multiplier σ _b ^s ∈ U ⁺ for the constraint u _b (μ) by incorporating the RB space Σ _b, N ⊂ U, as well as the convex cone Σ _b, N ⁺ from (3.3d) and (3.3e). Overall, for an RB approximation s _N ∈ S _N ⁺ ⊂ U ⁺ of s, we have s _N ≥ 0. From the definition of the slack variable s = u − u _a (μ), see (3.13), we derive the control approximation u ^s: = s _N + u _a (μ) that satisfies u ^s ≥ u _a (μ). However, we can not conclude u ^s ≤ u _b (μ) since the slack approximation s _N is constructed—as the slack variable s in (3.13)—using information from u _a (μ) but not u _b (μ).

Overall, employing the RB spaces in (S) results in the RB slack problem

As in the RB primal problem (P _N ), the existence and uniqueness of the RB optimal control follows from the same arguments as in the Remark 1. Next, we derive the optimality conditions for s _N ; however, we here follow the ‘first-discretize-then-optimize’ approach that will eventually lead to a feasible—w.r.t. u _a (μ)—approximation of the control. We perform two steps.

First, we use the RB-representations of y _N ^s, s _N with their RB-coefficient vectors y _N ^s, s _N to discretize (s _N ). Since the algebraic RB slack problem is simple to derive, we only state the main crucial condition that s _N ≥ 0. Next, we derive the first-order optimality conditions. We introduce a discrete Lagrange multiplier $\hat{\underline{\omega }}_{N} \in \mathbb{R}^{N_{S}},\ \hat{\underline{\omega }}_{N} \geq 0$, ensuring the non-negativeness of $\hat{\underline{s}}_{N}$ and derive the following necessary (and here sufficient) first-order optimality system: Given $\mu \in \mathcal{ D}$, the optimal RB slack solution coefficients $(\underline{y}_{N}^{s},\underline{s}_{N},\underline{p}_{N}^{s},\underline{\sigma }_{b,N}^{s},\underline{\omega }_{N}) \in \mathbb{R}^{N_{Y }} \times \mathbb{R}^{N_{Y }} \times \mathbb{R}^{N_{S}} \times \mathbb{R}^{N_{\sigma _{b}}} \times \mathbb{R}^{N_{S}}$ satisfy (omitting all μ-dependencies)

$$\displaystyle\begin{array}{rcl} \begin{array}{rclrcl} A_{N}\underline{y}_{N}^{s}& =&F_{N} + B_{N}^{s}\underline{s}_{N} + B_{a,N}^{s},&&\end{array} & &{}\end{array}$$

(3.15a)

$$\displaystyle\begin{array}{rcl} \begin{array}{rclrcl} A_{N}^{T}\underline{p}_{N}^{s}& = Y _{d,N} - D_{N}\underline{y}_{N}^{s},\end{array} & &{}\end{array}$$

(3.15b)

$$\displaystyle\begin{array}{rcl} \begin{array}{rclrcl} \lambda U_{N}^{s}\underline{s}_{N} +\lambda U_{a,N}^{s} - (B_{N}^{s})^{T}\underline{p}_{N}^{s} = \underline{\omega }_{N} - U_{N}^{\sigma _{b},s}\underline{\sigma }_{ b,N}^{s},\end{array} & &{}\end{array}$$

(3.15c)

$$\displaystyle\begin{array}{rcl} \begin{array}{rclrcl} \underline{s}_{N}^{T}\underline{\omega }_{N}& = 0,\quad \underline{s}_{N} \geq 0,\quad \underline{\omega }_{N} \geq 0,\end{array} & &{}\end{array}$$

(3.15d)

$$\displaystyle\begin{array}{rcl} \begin{array}{rclrcl} (U_{b,N}^{\sigma _{b}} - U_{ a,N}^{\sigma _{b}} - U_{ N}^{\sigma _{b},s}\underline{s}_{ N})^{T}\underline{\sigma }_{ b,N}^{s}& = 0,\quad U_{ b,N}^{\sigma _{b}} - U_{ a,N}^{\sigma _{b}} \geq U_{ N}^{\sigma _{b},s}\underline{s}_{ N},\quad \underline{\sigma }_{b,N}^{s} \geq 0.\end{array}& &{}\end{array}$$

(3.15e)

where the reduced basis matrices and vectors are given by

$$\displaystyle\begin{array}{rcl} \begin{array}{rclrclrclrclrcl} (A_{N})_{ij}& =&a(\zeta _{i}^{y},\zeta _{j}^{y}),\, & (F_{N})_{i}& =&f(\zeta _{i}^{y}),\, & (B_{N}^{s})_{ij}& =&b(\zeta _{j}^{s},\zeta _{i}^{y}),\, &(B_{a,N}^{s})_{i}& =&b(u_{a},\zeta _{i}^{y}), \\ (Y _{d,N})_{i}& =&(y_{d},\zeta _{i}^{y})_{D},\,& (D_{N})_{ij}& =&(\zeta _{i}^{y},\zeta _{j}^{y})_{D},\,& (U_{N}^{s})_{ij}& =&(\zeta _{i}^{s},\zeta _{j}^{s})_{U},\,&(U_{a,N}^{s})_{i}& =&(u_{a},\zeta _{i}^{s})_{U}, \\ (U_{N}^{\sigma _{b},s})_{ ij}& =&(\zeta _{i}^{\sigma _{b}},\zeta _{ j}^{s})_{ U},\, &(U_{b,N}^{\sigma _{b}})_{ i}& =&(u_{b},\zeta _{i}^{\sigma _{b}})_{ U},\, &(U_{a,N}^{\sigma _{b}})_{ i}& =&(u_{a},\zeta _{i}^{\sigma _{b}})_{ U} & & \end{array} & & {}\\ \end{array}$$

and 1 ≤ i, j ≤ N _• [see (3.3) and (3.14)].

Second, by solving (3.15) we have s _N ≥ 0 and through the definition of s we obtain a feasible—w.r.t. u _a (μ)—approximation for the control by u ^s = s _N + u _a (μ). In order to derive an error bound for $\Vert u - u^{s}\Vert _{U}$ we need, however, to analogously repeat the RB reduction for the second RB slack problem with s _b = u _b (μ) − u. There we likewise introduce the RB space S _b, N , as well as its convex cone S _b, N ⁺ and follow the previous steps to obtain s _b, N ≥ 0. Using this, we obtain a control approximation $u^{s_{b}} = u_{ b}(\mu ) - s_{b,N}$ that is feasible w.r.t. the constraint u _b (μ).

3.4.2 Primal-Slack Error Bound

In the following we will focus on the primal-slack error bound for $\Vert u - u^{s}\Vert _{U}$ w.r.t. u _a (μ). Similarly to the primal error bound in Theorem 1 we use residuals and properties of the bilinear and linear forms to derive a quadratic inequality in $\Vert u - u^{s}\Vert _{U}$. We consider the following RB primal-slack approximation (y _N ^s, p _N ^s, u ^s, σ _N , σ _b, N ) ∈ Y _N × Y _N × U _ad ×Σ _N ⁺ ×Σ _b, N ⁺, which depends on the solutions of the RB primal and slack problem. We define the corresponding errors e _y ^s = y − y _N ^s, e _p ^s = p − p _N ^s, e _u ^s = u − u ^s. Further, we revisit Definition 1 and insert on the r.h.s. of (3.5) the approximation (y _N ^s, p _N ^s, u ^s, σ _N , σ _b, N ) to obtain on the l.h.s. the residuals r _y ^s, r _p ^s, r _u ^s.

We state the main result in the following theorem.

Theorem 2 (Primal-Slack Error Bound)

Let u, s _N , and s _b, N be the optimal solutions of the FE primal problem  (P) and the RB slack problems ( s _N ) and its equivalent w.r.t. u _b (μ), respectively. Then the error in the optimal control satisfies for all parameters $\mu \in \mathcal{ D}$

$$\displaystyle{ \Vert e_{u}^{s}\Vert _{ U} \leq \varDelta _{N}^{\mathrm{pr-sl}}(\mu ), }$$

where $\varDelta _{N}^{\mathrm{pr-sl}}(\mu ):= c_{1}^{s}(\mu ) + \sqrt{c_{1 }^{s }(\mu )^{2 } + c_{2 }^{s }(\mu )}$ with nonnegative coefficients

$$\displaystyle\begin{array}{rcl} c_{1}^{s}(\mu )& =& \frac{1} {2\lambda }\left (\Vert r_{u}^{s}\Vert _{ U'} + \frac{\gamma _{b}^{\mathrm{UB}}} {\alpha _{a}^{\mathrm{LB}}}\Vert r_{p}^{s}\Vert _{ Y '} +\lambda \Vert u^{s} - u^{s_{b} }\Vert _{U}\right ),{}\end{array}$$

(3.16a)

$$\displaystyle\begin{array}{rcl} c_{2}^{s}(\mu )& =& \frac{1} {\lambda } \!\Bigg[\! \frac{2} {\alpha _{a}^{\mathrm{LB}}}\Vert r_{y}^{s}\Vert _{ Y '}\Vert r_{p}^{s}\Vert _{ Y '} + \frac{1} {4}\!\left (\frac{C_{D}^{\mathrm{UB}}} {\alpha _{a}^{\mathrm{LB}}} (\Vert r_{y}^{s}\Vert _{ Y '} + \gamma _{b}^{\mathrm{UB}}\Vert u^{s} - u^{s_{b} }\Vert _{U})\!\right )^{2}\!\!\! + (\sigma _{ N},s_{N})_{U} \\ & & +\Vert u^{s} - u^{s_{b} }\Vert _{U}\left (\Vert r_{u}^{s}\Vert _{ U'} + \frac{\gamma _{b}^{\mathrm{UB}}} {\alpha _{a}^{\mathrm{LB}}}\Vert r_{p}^{s}\Vert _{ Y '} + \sqrt{2(\sigma _{N },\sigma _{b,N } )_{U}}\right ) + (\sigma _{b,N},s_{b,N})_{U}\Bigg]{}\end{array}$$

(3.16b)

Proof

Let the FE primal solution (y, p, u, σ, σ _b ) satisfy the optimality conditions (3.2). We follow the proof of Theorem 1, and derive analogously to (3.9) the inequality

$$\displaystyle{ \lambda \Vert e_{u}^{s}\Vert _{ U}^{2}\! +\!\vert e_{ y}^{s}\vert _{ D}^{2}\! \leq \!\Vert r_{ y}^{s}\Vert _{ Y '}\Vert e_{p}^{s}\Vert _{ Y }\! +\!\Vert r_{p}^{s}\Vert _{ Y '}\Vert e_{y}^{s}\Vert _{ Y }\! +\!\Vert r_{u}^{s}\Vert _{ U'}\Vert e_{u}^{s}\Vert _{ U}\! +\! (e_{\sigma } - e_{\sigma _{b}},e_{u}^{s})_{ U}. }$$

(3.17)

We first focus on (e _σ , e _u ^s) _U and exploit the feasibility of u ^s w.r.t. u _a (μ). Again we have (e _σ , e _u ^s) _U = −(σ, u _a (μ) − u) _U − (σ, s _N ) _U + (σ _N , u _a (μ) − u) _U + (σ _N , s _N ) _U , where the first three terms are non-positive and hence (e _σ , e _u ^s) _U ≤ (σ _N , s _N ) _U . In order to bound $-(e_{\sigma _{b}},e_{u}^{s})_{U}$, we need to solve the second RB slack problem for $u^{s_{b}}$ and derive

$$\displaystyle{ -(e_{\sigma _{b}},e_{u}^{s})_{ U} = (e_{\sigma _{b}},u^{s} - u^{s_{b} } + u^{s_{b} } - u)_{U} \leq \Vert e_{\sigma _{b}}\Vert _{U}\Vert u^{s} - u^{s_{b} }\Vert _{U} + (\sigma _{b,N},s_{b,N})_{U}. }$$

We restate that $\Vert e_{\sigma _{b}}\Vert _{U} \leq \Vert e_{\sigma } - e_{\sigma _{b}}\Vert _{U} + \sqrt{2(\sigma _{N },\sigma _{b,N } )_{U}}$ and $\Vert e_{\sigma } - e_{\sigma _{b}}\Vert _{U}$ is bounded by $\Vert e_{\sigma } - e_{\sigma _{b}}\Vert _{U} \leq \Vert r_{u}^{s}\Vert _{U'} +\lambda \Vert e_{u}^{s}\Vert _{U} + \frac{\gamma _{b}^{\mathrm{UB}}} {\alpha _{a}^{\mathrm{LB}}} \left (\Vert r_{p}^{s}\Vert _{Y '} + C_{D}^{\mathrm{UB}}\vert e_{y}^{s}\vert _{D}\right ).$ Using the bounds for $(e_{\sigma },e_{u}^{s})_{U} - (e_{\sigma _{b}},e_{u}^{s})_{U}$ and the inequalities (3.8) in (3.17) we obtain

$$\displaystyle\begin{array}{rcl} \lambda \Vert e_{u}^{s}\Vert _{ U}^{2}& +& \vert e_{ y}^{s}\vert _{ D}^{2} \leq \,\Vert e_{ u}^{s}\Vert _{ U}\left (\Vert r_{u}^{s}\Vert _{ U'} + \frac{\gamma _{b}^{\mathrm{UB}}} {\alpha _{a}^{\mathrm{LB}}}\Vert r_{p}^{s}\Vert _{ Y '} +\lambda \Vert u^{s} - u^{s_{b} }\Vert _{U}\right ) + (\sigma _{N},s_{N})_{U} {}\\ & & + \frac{2} {\alpha _{a}^{\mathrm{LB}}}\Vert r_{y}^{s}\Vert _{ Y '}\Vert r_{p}^{s}\Vert _{ Y '} +\vert e_{y}^{s}\vert _{ D}\frac{C_{D}^{\mathrm{UB}}} {\alpha _{a}^{\mathrm{LB}}} \left (\Vert r_{y}^{s}\Vert _{ Y '} + \gamma _{b}^{\mathrm{UB}}\Vert u^{s} - u^{s_{b} }\Vert _{U}\right ) {}\\ & & +\left (\Vert r_{u}^{s}\Vert _{ U'} + \frac{\gamma _{b}^{\mathrm{UB}}} {\alpha _{a}^{\mathrm{LB}}}\Vert r_{p}^{s}\Vert _{ Y '} + \sqrt{2(\sigma _{N },\sigma _{b,N } )_{U}}\right )\Vert u^{s} - u^{s_{b} }\Vert _{U} + (\sigma _{b,N},s_{b,N})_{U}. {}\\ \end{array}$$

It thus follows from employing Young’s inequality to the $\vert e_{y}^{s}\vert _{D}$-term that $\Vert e_{u}^{s}\Vert _{U}^{2} - 2c_{1}^{s}(\mu )\Vert e_{u}^{s}\Vert _{U} - c_{2}^{s}(\mu ) \leq 0$, where c ₁ ^s(μ) and c ₂ ^s(μ) are given in (3.16). Solving the last inequality for the larger root yields $\Vert e_{u}^{s}\Vert _{U} \leq c_{1}^{s}(\mu ) + \sqrt{c_{1 }^{s }(\mu )^{2 } + c_{2 }^{s }(\mu )} =\varDelta _{ N}^{\mathrm{pr-sl}}(\mu )$.

3.5 Greedy Sampling Procedure

The reduced basis spaces for the two-sided control-constrained optimal control problem in Sects. 3.3.1, and 3.4.1 are constructed using the greedy sampling procedure outlined in Algorithm 1. Suppose $\varXi _{\mathrm{train}} \subset \mathcal{ D}$ is a finite but suitably large parameter train sample, μ ¹ ∈ Ξ _train is the initial parameter value, N _max the maximum number of greedy iterations, ε _tol,min > 0 is a prescribed desired error tolerance, and $\varDelta _{N}^{\bullet }(\mu )/\Vert u_{N}^{\bullet }(\mu )\Vert _{U}$, • ∈ {pr, pr−sl}, is the primal or primal-slack error bound from (3.6) or (3.16) with u _N ^• ∈ {u _N , u ^s}.

Algorithm 1 Greedy sampling procedure

We make two remarks: First, by using the bounds Δ _N ^•(μ), • ∈ {pr, pr−sl} we only refer to the bounds derived for the primal error $\Vert u - u_{N}\Vert _{U}$ and the slack error $\Vert u - u^{s}\Vert _{U}$ w.r.t. to u _a (μ). Therefore, using the primal-slack bound Δ _N ^pr−sl(μ) in the greedy sampling procedure, we expect not only to construct an accurate RB space S _N for s _N but also an accurate RB space S _b, N for s _b, N . Second, we comment on two special cases: (1) if one control constraint is fully active in each greedy step, i.e. we have, e.g., u(μ ⁿ) = u _a ,  n = 1, …, N, we set S _N = {} and s _N = 0 (analogously for u _b we set S _b, N = {} and s _b, N = 0); and (2) if the control constraint is never active, i.e., for all snapshots σ(μ ⁿ) = σ _b (μ ⁿ) = 0,  n = 1, …, N, we set Σ _N = Σ _N ⁺ = Σ _b, N = Σ _b, N ⁺ = {} and σ _N = σ _b, N = 0.

3.6 Numerical Results: Graetz Flow with Parametrized Geometry and Lower and Upper Control Constraints

We consider a Graetz flow problem, which describes a heat convection and conduction in a duct. The main goal of this example is to demonstrate the different properties of the approximations and their error bounds. The problem is parametrized by a varying Péclet number μ ₁ ∈ [5, 18] and a geometry parameter μ ₂ ∈ [0. 8, 1. 2]. Hence, the parameter domain is $\mathcal{D} = [5,18] \times [0.8,1.2]$. The parametrized geometry is given by Ω(μ) = [0, 1. 5 + μ ₂] × [0, 1] and is subdivided into three subdomains Ω ₁(μ) = [0. 2μ ₂, 0. 8μ ₂] × [0. 3, 0. 7], Ω ₂(μ) = [μ ₂ + 0. 2, μ ₂ + 1. 5] × [0. 3, 0. 7], and Ω ₃(μ) = Ω(μ)∖{Ω ₁(μ) ∪Ω ₂(μ)}. A sketch of the domain is shown in Fig. 3.1. We impose boundary condition of homogeneous Neumann and of non-homogeneous Dirichlet type: y _n = 0 on Γ _N (μ), and y = 1 on Γ _D (μ). Thus the trial space is given by $Y (\mu ) \subset Y _{\mathrm{e}}(\mu ) =\{ v \in H^{1}(\varOmega (\mu ));v\vert _{\varGamma _{D}(\mu )} = 1\}$. The amount of heat supply in the whole domain Ω(μ) is regulated by the distributed control u ∈ U(μ) ⊂ U _e(μ) = L ²(Ω(μ)) and bounded by the lower and upper constraints u _a = −0. 5 and u _b = 1. 25. The observation domain is $\varOmega _{\!_{D}}\!(\mu ) =\varOmega _{1}(\mu ) \cup \varOmega _{2}(\mu )$ and the desired state is given by y _d = 0. 5 on Ω ₁(μ) and y _d = 2 on Ω ₂(μ).

Fig. 3.1

Domain Ω(μ) for the Graetz flow problem with distributed control

Overall, the parametrized optimal control problem is given by

$$\displaystyle\begin{array}{rcl} & & \min _{\hat{y}\in Y (\mu ),\hat{u}\in U(\mu )}J(\hat{y},\hat{u};\mu ) = \frac{1} {2}\vert \hat{y} - y_{d}\vert _{D(\mu )}^{2} + \frac{\lambda } {2}\Vert \hat{u}\Vert _{L^{2}(\varOmega (\mu ))}^{2} {}\\ & & \text{s.t.}\quad \frac{1} {\mu _{1}} \int _{\varOmega (\mu )}\nabla \hat{y} \cdot \nabla \phi \;\mathrm{d}x +\int _{\varOmega (\mu )}\beta (x) \cdot \nabla \hat{y}\,\phi \;\mathrm{d}x =\int _{\varOmega (\mu )}\hat{u}\,\phi \;\mathrm{d}x\quad \forall \phi \in Y (\mu ), {}\\ & & \quad \quad \,(u_{a},\rho )_{U(\mu )} \leq (\hat{u},\rho )_{U(\mu )} \leq (u_{b},\rho )_{U(\mu )}\quad \forall \rho \in U(\mu )^{+}, {}\\ \end{array}$$

for the given parabolic velocity field β(x) = (x ₂(1 − x ₂), 0)^T. The regularization parameter λ is fixed to 0. 01.

After recasting the problem to a reference domain Ω = Ω(μ ^ref) = [0, 2. 5] × [0, 1] for μ ^ref = (5, 1), and introducing suitable lifting functions that take into account the non-homogeneous Dirichlet boundary conditions, we can reformulate the problem in terms of the parameter-independent FE space Y ⊂ Y _e = H ₀ ¹(Ω) and U ⊂ U _e = L ²(Ω) [12]. We then obtain the affine representation of all involved quantities with Q _a = Q _f = 4, Q _b = Q _d = Q _u = Q _yd = 2, and Q _ua = 1. The details of these calculations are very similar to the details presented by Rozza et al. [12] and Kärcher [5], and are thus omitted. The inner product for the state space is given by $(w,v)_{Y } = \frac{1} {\mu _{1}^{\mathrm{ref}}} \int _{\varOmega }\nabla w \cdot \nabla v\;\mathrm{d}x + \frac{1} {2}(\int _{\varOmega }\beta (x) \cdot \nabla w\,v\;\mathrm{d}x +\int _{\varOmega }\beta (x) \cdot \nabla v\,w\;\mathrm{d}x)$ and we obtain a lower bound α _a ^LB(μ) for the coercivity constant by the so-called min-theta approach [12]. Note that for the control space we obtain a parameter-dependent inner product (⋅ , ⋅ ) _U(μ) from the affine geometry parametrization. Hence the control error is measured in the parameter-dependent energy norm $\Vert \cdot \Vert _{U(\mu )}$. The derivations of the primal and primal-slack error bounds remain the same in this case and they bound the control error in the energy norm.

Although the introduction of a domain parametrization seems to add an entirely new μ-dependence to the primal and the slack problems (P) and (S), the reductions and the error bound derivations can be analogously derived w.r.t. (⋅ , ⋅ ) _U(μ) instead of $(\cdot,\cdot )_{U(\mu ^{\mathrm{ref}})}$. Also the definition of the integrated space U _N in (3.3) remains, while in the inf-sup condition of Remark 2 we use (⋅ , ⋅ ) _U(μ) instead of (⋅ , ⋅ ) _U .

We choose a P ₁ discretization for the state and adjoint, and a P ₀ discretization for the control to obtain $\dim (Y ) =\mathcal{ N}_{Y } \approx 11,000$ and $\dim (U) =\mathcal{ N}_{U} \approx 22,000$. The chosen discretization induces a discretization error of roughly 2%. In Fig. 3.2 we present control snapshots and associated active sets for two different parameters displayed on the reference domain Ω(μ ₂ ^ref = 1). We observe strongly varying control solutions and active sets.

Fig. 3.2

Snapshots of active sets (upper row) and optimal control (lower row) on the reference domain. The active (inactive) sets are displayed in light gray (gray). (a) μ = (5, 0. 8). (b) μ = (18, 1. 2)

We construct the RB spaces using the greedy procedure described in Algorithm 1 by employing an equidistant train sample $\varXi _{\mathrm{train}} \subset \mathcal{ D}$ of size 30 ⋅ 30 = 900 (log-scale in μ ₁ and lin-scale in μ ₂) and stop the greedy enrichment after 30 steps. We also introduce a test sample with 10 ⋅ 5 (log × lin) equidistant parameter points in $[5.2,17.5] \times [0.82,1.17] \subset \mathcal{ D}$.

In Fig. 3.3 we present, as a function of N, the resulting energy norm errors and bounds over Ξ _test. Here, the errors and bounds are defined as follows: the primal-slack bound is the maximum of $\varDelta _{N}^{\mathrm{pr-sl}}(\mu )/\Vert u(\mu )\Vert _{U(\mu )}$ over Ξ _test, the primal bound is the maximum of $\varDelta _{N}^{\mathrm{pr}}(\mu )/\Vert u(\mu )\Vert _{U(\mu )}$ over Ξ _test, and the u ^s and u _N errors are the maxima of $\Vert u(\mu ) - u^{s}(\mu )\Vert _{U(\mu )}/\Vert u(\mu )\Vert _{U(\mu )}$ and $\Vert u(\mu ) - u_{N}(\mu )\Vert _{U(\mu )}/\Vert u(\mu )\Vert _{U(\mu )}$ over Ξ _test, respectively. We observe that both errors and both bounds decay very similarly. Quantitatively, the error bounds are comparable throughout all N, since the dominating primal-slack terms $\Vert r_{u}^{s}\Vert _{U'(\mu )}$ and $\lambda \Vert u^{s} - u^{s_{b}}\Vert _{ U(\mu )}$ are comparable to the dominating primal terms $\Vert r_{u}\Vert _{U'(\mu )}$ and λ(δ ₁ + δ _1b ), resulting in Δ _N ^pr(μ) ≈ Δ _N ^pr−sl(μ).

Fig. 3.3

Maximal relative control errors and bounds over the number of greedy iterations. For each N the maximal value over Ξ _test is displayed for both errors and both bound

We briefly report the computational timings: the solution of the FE optimization problem takes ≈ 4 s (for a discretization error of 2%). The RB primal problem, for N = 25, is solved in ≈ 0. 066 s and the RB slack problem is solved faster in ≈ 0. 029 s, since dim(S _N ) = 25. We turn to the evaluation of error bounds: the primal bound takes 0. 01 s, whereas the primal-slack bound, given σ _N , σ _b, N , takes 0. 0065 s. From this we can conclude that for N = 25 the overall cost for one primal bound evaluation is roughly 0. 076 s = 0. 066 s + 0. 01 s and for the primal-slack bound evaluation is roughly 0. 13 ≈ 0. 029 + 0. 029 + 0. 066 + 0. 0065 s, since it relies on three RB solutions.

3.7 Conclusions

In this paper we extended the ideas from [1] to propose two certified reduced basis approaches for distributed elliptic optimal control problems with two-sided control constraints: a primal and a primal-slack approach. Albeit the reduction for the primal approach was straightforward, the primal-slack approach needed more consideration. We proposed for each constraint a corresponding RB slack problem with an additional Lagrange multiplier. The primal a posteriori error bound from [1] could be extended for the two-sided case by special properties of the Lagrange multipliers of the two-sided problem. The primal-slack error bound also relies on these properties and in addition uses three RB solutions to derive an $\mathcal{N}$-independent error bound. Both the primal and slack RB approximation can be evaluated efficiently using the standard offline-online decomposition. However, on the one hand the primal error bound depends on the FE control dimension and on the other hand the primal-slack error bound relies on three reduced order optimization problems.