Finite-Horizon Parameterizing Manifolds, and Applications to Suboptimal Control of Nonlinear Parabolic PDEs

Abstract

This article proposes a new approach for the design of low-dimensional suboptimal controllers to optimal control problems of nonlinear partial differential equations (PDEs) of parabolic type. The approach fits into the long tradition of seeking for slaving relationships between the small scales and the large ones (to be controlled) but differ by the introduction of a new type of manifolds to do so, namely the finite-horizon parameterizing manifolds (PMs). Given a finite horizon [0,T] and a low-mode truncation of the PDE, a PM provides an approximate parameterization of the high modes by the controlled low ones so that the unexplained high-mode energy is reduced—in a mean-square sense over [0,T]—when this parameterization is applied.

Analytic formulas of such PMs are derived by application of the method of pullback approximation of the high-modes introduced in Chekroun et al. (2014). These formulas allow for an effective derivation of reduced systems of ordinary differential equations (ODEs), aimed to model the evolution of the low-mode truncation of the controlled state variable, where the high-mode part is approximated by the PM function applied to the low modes. The design of low-dimensional suboptimal controllers is then obtained by (indirect) techniques from finite-dimensional optimal control theory, applied to the PM-based reduced ODEs.

A priori error estimates between the resulting PM-based low-dimensional suboptimal controller $u_{R}^{\ast}$ and the optimal controller u ^∗ are derived under a second-order sufficient optimality condition. These estimates demonstrate that the closeness of $u_{R}^{\ast}$ to u ^∗ is mainly conditioned on two factors: (i) the parameterization defect of a given PM, associated respectively with the suboptimal controller $u_{R}^{\ast}$ and the optimal controller u ^∗; and (ii) the energy kept in the high modes of the PDE solution either driven by $u_{R}^{\ast}$ or u ^∗ itself.

The practical performances of such PM-based suboptimal controllers are numerically assessed for optimal control problems associated with a Burgers-type equation; the locally as well as globally distributed cases being both considered. The numerical results show that a PM-based reduced system allows for the design of suboptimal controllers with good performances provided that the associated parameterization defects and energy kept in the high modes are small enough, in agreement with the rigorous results.

Appendices

Appendix A: Suboptimal Controller Synthesis Based on Galerkin Projections and Pontryagin Maximum Principle

To assess the performance of the PM-based reduced systems considered in Sects. 5 and 6 in synthesizing suboptimal controllers in the context of a Burgers-type equation, we derive in this appendix suboptimal control problems associated with the globally distributed optimal control problem (5.9) based on Galerkin approximations. Section A.1 concerns a two-mode Galerkin approximation; and Sect. A.2 deals with the more general m-dimensional case. The former serves as a basis of comparison to analyze the performance achieved by the PM-based approach, while the latter can in principle provide a good indication of the true optimal controller of the underlying optimal control problems by taking the dimension sufficiently large. Results for the general m-dimensional case will also be used in Sect. 7 to derive Galerkin-based reduced systems for the locally distributed problems (7.4) and (7.5).

1.1 A.1 Suboptimal Controller Based on a 2D Galerkin Reduced Optimal Problem

We first present the reduced optimal control problem based on a two-mode Galerkin approximation of the underlying PDE (5.1), which can be derived by simply setting $h^{(1)}_{\lambda}$ in (5.18)–(5.17) to zero. The corresponding operational forms for the cost functional and reduced system for the low modes can be obtained from (5.24)–(5.27) by setting α ₁(λ) and α ₂(λ) to be zero. The resulting cost functional reads:

$$ J_G(v, u_G) = \int_0^T \bigl[ \mathcal{G}^G\bigl(v(t)\bigr) + \mathcal{E} \bigl(u_G(t)\bigr) \bigr] \,\mathrm{d}t + C_T\bigl(v(T), P_{\mathfrak{c}}Y\bigr), $$

(A.1)

where $v = v_{1} e_{1} + v_{2} e_{2} \in L^{2}(0,T; \mathcal{H}^{\mathfrak {c}})$ is the state variable, $u_{G} = u_{G,1} e_{1} + u_{G,2} e_{2} \in L^{2}(0,T; \mathcal {H}^{\mathfrak{c}})$ is the control, C _T is the terminal payoff term defined by (5.26), and

$$ \mathcal{G}^G(v ) := \frac{1}{2} \|v \|^2 = \frac{1}{2} \bigl[(v_1 )^2 + (v_2 )^2\bigr], \qquad\mathcal{E}(u_G) := \frac{\mu_1}{2}\|u_G\|^2 = \frac {\mu_1}{2} \bigl[(u_{G,1} )^2 + (u_{G,2})^2 \bigr]. $$

(A.2)

The equations for v ₁ and v ₂ are given by:

$$ \begin{aligned} & \frac{\mathrm{d}v_1}{\mathrm{d}t} = \beta_1(\lambda) v_1 + \alpha v_1 v_2 + a_{11} u_{G,1}(t) + a_{21} u_{G,2}(t), \\ & \frac{\mathrm{d}v_2}{\mathrm{d}t} = \beta_2(\lambda) v_2 - \alpha(v_1)^2 + a_{12} u_{G,1}(t) + a_{22} u_{G,2}(t), \end{aligned} $$

(A.3)

which is subjected to the initial conditions:

$$ v_1(0) = \langle y_0, e_1 \rangle, \qquad v_2(0) = \langle y_0, e_2 \rangle, $$

(A.4)

where $\alpha= \frac{\gamma\pi}{\sqrt{2}l^{3/2}}$.

The corresponding Galerkin-based reduced optimal control problem for (5.9) reads:

$$ \min J_G(v, u_G) \quad\text{s.t.}\quad (v, u_G) \in L^2\bigl(0,T; \mathcal{H}^{\mathfrak{c}}\bigr) \times L^2\bigl(0,T; \mathcal {H}^{\mathfrak{c}}\bigr) \text{ solves (A.3)--(A.4)}. $$

(A.5)

It follows again from the Pontryagin maximum principle that for a given pair

$$\bigl(v_G^\ast, u_G^\ast\bigr) \in L^2\bigl(0,T; \mathcal{H}^{\mathfrak{c}}\bigr) \times L^2\bigl(0,T; \mathcal{H}^{\mathfrak{c}}\bigr) $$

to be optimal for the problem (A.5), it must satisfy the following conditions:

$$\begin{aligned} & \frac{\mathrm{d}v_{G,1}^\ast}{\mathrm{d}t} = \beta_1(\lambda) v_{G,1}^\ast+ \alpha v_{G,1}^\ast v_{G,2}^\ast+ a_{11} u_{G,1}^\ast(t) + a_{21}u_{G,2}^\ast (t), \end{aligned}$$

(A.6a)

$$\begin{aligned} & \frac{\mathrm{d}v_{G,2}^\ast}{\mathrm{d}t} = \beta_2(\lambda) v_{G,2}^\ast- \alpha \bigl(v_{G,1}^\ast\bigr)^2 + + a_{12} u_{G,1}^\ast(t) + a_{22}u_{G,2}^\ast(t), \end{aligned}$$

(A.6b)

$$\begin{aligned} & \frac{\mathrm{d}p_{G,1}^\ast}{\mathrm{d}t} = - v_{G,1}^\ast- \beta_1(\lambda) p_{G,1}^\ast- \alpha p_{G,1}^\ast v_{G,2}^\ast+ 2 \alpha p_{G,2}^\ast v_{G,1}^\ast, \end{aligned}$$

(A.6c)

$$\begin{aligned} & \frac{\mathrm{d}p_{G,2}^\ast}{\mathrm{d}t} = - v_{G,2}^\ast- \beta_2(\lambda) p_{G,2}^\ast- \alpha p_{G,1}^\ast v_{G,1}^\ast, \end{aligned}$$

(A.6d)

$$\begin{aligned} & \bigl(u_{G,1}^\ast, u_{G,2}^\ast \bigr)^{\mathrm{tr}} = - \biggl( \frac{a_{11} p_{G,1}^\ast(t) + a_{12} p_{G,2}^\ast(t)}{\mu_1}, \frac{a_{21} p_{G,1}^\ast(t) + a_{22} p_{G,2}^\ast(t)}{\mu_1} \biggr)^{\mathrm{tr}} = - \frac{1}{\mu_1} M^{\mathrm{tr}}p_G^\ast, \end{aligned}$$

(A.6e)

where $v_{G,1}^{\ast}= \langle v_{G}^{\ast}, e_{i} \rangle$, $u_{G,i}^{\ast}= \langle u_{G}^{\ast}, e_{i} \rangle$, i=1,2, and $p_{G}^{\ast}= p_{G,1}^{\ast}e_{1} + p_{G,2}^{\ast}e_{2}$ denotes the costate associated with $v_{G}^{\ast}$.

Thanks to (A.6e), we can express the controller $u_{G,i}^{\ast}$ in (A.6a)–(A.6b) in terms of the costate $p_{G,i}^{\ast}$, leading thus to the following BVP for $v_{G}^{\ast}$ and $p_{G}^{\ast}$:

$$ \begin{aligned} \frac{\mathrm{d}v_1}{\mathrm{d}t} &= \beta_1(\lambda) v_1 + \alpha v_1v_2 + f_3(p_1,p_2), \\ \frac{\mathrm{d}v_2}{\mathrm{d}t} & = \beta_2(\lambda) v_2 - \alpha(v_1)^2 + f_4(p_1,p_2), \\ \frac{\mathrm{d}p_1}{\mathrm{d}t} &= -2 v_1 - \beta_1(\lambda) p_1 - \alpha p_1 v_2 + 2 \alpha p_2 v_1, \\ \frac{\mathrm{d}p_2}{\mathrm{d}t} & = -2 v_2 - \beta_2(\lambda) p_2 - \alpha p_1 v_1, \end{aligned} $$

(A.7)

subject to the boundary condition

$$ \begin{aligned} &v_1(0) = \langle y_0, e_1 \rangle, \qquad v_2(0) = \langle y_0, e_2 \rangle, \\ &p_1(T) =\mu_2 \bigl(v_{1}(T) - Y_1\bigr), \qquad p_2(T) = \mu_2 \bigl(v_2(T) - Y_2\bigr), \end{aligned} $$

(A.8)

where f ₃ and f ₄ are defined by (5.33), and the boundary condition for the costate is derived in the same way as in (5.34) thanks to the Pontryagin maximum principle. Once this BVP is solved, the corresponding controller $u_{G}^{\ast}$ is determined by (A.6e) which provides the unique optimal controller for the Galerkin-based reduced optimal control problem (A.5), due again to the fact that the cost functional (A.1) is quadratic in u _G and the dependence on the controller is affine for the system of Eqs. (A.3); see e.g. [67, Sect. 5.3] and [99]. Note also that analogous results to those presented in Lemma 4 hold for the reduced optimal control problem (A.5) as well.

1.2 A.2 Suboptimal Controller Based on an m-Dimensional Galerkin Reduced Optimal Problem

We derive now a more general reduced optimal control problem based on higher-dimensional Galerkin approximation, where the subspace $\mathcal {H}^{\mathfrak{c}}$ is taken to be spanned by the first m eigenmodes:

$$ \mathcal{H}^{\mathfrak{c}} := \operatorname{span}\{e_1, \ldots, e_m\}. $$

(A.9)

The main interest is that by choosing m sufficiently large, such a reduced problem can serve in principle to provide a good estimate of the true optimal controllers of the globally distributed optimal control problem (5.9), which can be taken then as a benchmark for the numerical experiments reported in Sects. 5 and 6. Analogous reduced problems associated with the locally distributed cases (7.4) and (7.5) considered in Sect. 7 can be derived in the same way (and actually the corresponding results are the same as those presented in Sect. 7.2 by setting $h^{(1)}_{\lambda}$ therein to be zero).

The Galerkin-based reduced optimal control problem (A.5) when generalized to the case with m controlled modes reads:

$$ \min\widetilde{J}_G(v, \widetilde{u}_G) \quad \text{s.t.}\quad (v, \widetilde{u}_G) \in L^2\bigl(0,T; \mathcal{H}^{\mathfrak {c}}\bigr) \times L^2\bigl(0,T; \mathcal{H}^{\mathfrak{c}}\bigr) \text{ solves (A.11)--(A.12) below}, $$

(A.10)

where $\mathcal{H}^{\mathfrak{c}}$ is the m-dimensional reduced phase space defined in (A.9), and

$$\widetilde{J}_G(v, \widetilde{u}_G) = \int _0^T \Biggl[ \frac{1}{2} \sum _{i=1}^m(v_i)^2 + \frac{\mu_1}{2} \sum_{i=1}^m ( \widetilde{u}_{G,i} )^2 \Biggr] \,\mathrm{d}t + \frac{\mu_2}{2} \sum_{i=1}^m \bigl|v_i(T) - Y_i\bigr|^2. $$

The system of equations that $v(\cdot; \widetilde{u}_{G})$ satisfies is given by:

$$ \frac{\mathrm{d}v_i}{\mathrm{d}t} = \beta_i(\lambda) v_i + \Biggl\langle B \Biggl( \sum _{i=1}^m v_i e_i, \sum_{i=1}^m v_i e_i \Biggr), e_i \Biggr\rangle + \bigl[M^{\mathrm{tr}}\widetilde{u}_{G}(t)\bigr]_i, \quad i = 1,\ldots, m, $$

(A.11)

which is subjected to the initial conditions:

$$ v_i(0) = \langle y_0, e_i \rangle, \quad i = 1,\ldots, m, $$

(A.12)

where the matrix M _m×m is the representation of the linear operator $P_{\mathfrak{c}}\mathfrak{C}$ under the basis e ₁,…,e _m , i.e. the elements of M are given by $a_{ij} = \langle\mathfrak {C}e_{i}, e_{j} \rangle$ (see (5.16) for the case m=2) and $[M^{\mathrm{tr}}\widetilde{u}_{G}(t)]_{i}$ denotes the ith-component of the vector $M^{\mathrm{tr}}\widetilde{u}_{G}(t)$.

As before, by using the Pontryagin maximum principle, we can derive the following BVP to be satisfied by any optimal pair $(v_{G}^{\ast}, \widetilde {u}^{\ast}_{G})$ of (A.10):

$$\begin{aligned} & \frac{\mathrm{d}v_i}{\mathrm{d}t} = \beta_i(\lambda) v_i + i \alpha \Biggl( - \sum_{j = 1}^{\lfloor i/2 \rfloor} \omega_{i,j} v_j v_{i-j} + \sum _{j = i+1}^m v_j v_{j-i} \Biggr) - \frac{1}{\mu_1} \bigl[M^{\mathrm{tr}} M p\bigr]_i, \quad i = 1, \ldots, m, \end{aligned}$$

(A.13a)

$$\begin{aligned} & \frac{\mathrm{d}p_i}{\mathrm{d}t} = - v_i - \sum_{j = 1}^m p_j \frac{\partial f_j(v, p)}{\partial v_i},\quad i = 1, \ldots, m, \end{aligned}$$

(A.13b)

$$\begin{aligned} & v_{i}(0) = y_{0,i}, \qquad p_{i}(T) = \mu_2 \bigl(v_{i}(T) - Y_{i}\bigr), \quad i = 1, \ldots, m, \end{aligned}$$

(A.13c)

where the optimal controller $\widetilde{u}^{\ast}_{G}$ is related to the corresponding costate $p_{G}^{\ast}$ by

$$ \widetilde{u}^\ast_{G} = - \frac{1}{\mu_1} M p_G^\ast, $$

(A.14)

see (A.6e) for the case m=2. Here, f _i , i=1,…,m, denotes the RHS of (A.13a) and we have used the nonlinear interactions (5.20) to derive the quadratic parts of f _i . The formula for $\frac{\partial f_{j}(v, p)}{\partial v_{i}}$ is given by:

$$ \frac{\partial f_j(v,p)}{\partial v_i} = \beta_{j}(\lambda) \delta_{ij} + j \alpha I_{j,i}, $$

(A.15)

where δ _ij denotes the Kronecker delta, and

$$ I_{j,i} = \frac{\partial}{\partial v_i} \Biggl( - \sum _{k = 1}^{\lfloor j/2 \rfloor} \omega_{j,k} v_k v_{j-k} + \sum_{k = j+1}^m v_k v_{k-j} \Biggr) = \begin{cases} v_{i-j}, & \text{if }i > j, \\ v_{i+j}, & \text{if }i = j\mbox{ and }i+j\le m, \\ v_{i+j} - v_{j-i}, & \text{if }i < j\mbox{ and }i+j\le m, \\ -v_{j-i}, & \text{if }i < j\mbox{ and }i+j > m, \\ 0, & \text{otherwise}; \end{cases} $$

(A.16)

with ⌊x⌋ being the largest integer less than x and the coefficients ω _i,j given by

$$\omega_{i,j} := \begin{cases} 1, & \text{if $i$ is odd, or if $i$ is even and }j \neq i/2, \\ 1/2, & \text{if $i$ is even and }j = i/2. \end{cases} $$

Appendix B: Global Well-posedness for the Two-dimensional $h^{(1)}_{\lambda}$-based Reduced System (5.27)

In this appendix, we show that for any given initial datum and any fixed T>0, the $h^{(1)}_{\lambda}$-based reduced system (5.27) admits a unique mild solution in the space $C([0,T]; \mathbb{R}^{2})$.²¹ The result follows from classical ODE theory [2] once we can establish a priori bounds for the solution (z ₁(t),z ₂(t)). Similar (but more tedious) estimates can be used to deal with the Cauchy problem associated with the $h^{(2)}_{\lambda}$-based reduced system (6.17) derived in Sect. 6 and the more general m-dimensional $h^{(1)}_{\lambda}$-based reduced system (7.19) encountered in Sect. 7.

Let us first recall that the two-dimensional $h^{(1)}_{\lambda}$-based reduced system is given by:

$$\begin{aligned} & \frac{\mathrm{d}z_1}{\mathrm{d}t} = \beta _1(\lambda) z_1 + \alpha\bigl[ z_1z_2 + \alpha_1(\lambda) z_1z_2^2 + \alpha_1(\lambda) \alpha _2(\lambda ) z_1 z_2^3\bigr] + a_{11}u_{R,1}(t) + a_{21}u_{R,2}(t), \end{aligned}$$

(B.1a)

$$\begin{aligned} & \frac{\mathrm{d}z_2}{\mathrm{d}t} = \beta _2(\lambda) z_2 + \alpha \bigl[- z_1^2 + 2 \alpha_1(\lambda) z_1^2z_2 + 2 \alpha_2(\lambda) z_2^3\bigr] + a_{12}u_{R,1}(t) + a_{22}u_{R,2}(t), \end{aligned}$$

(B.1b)

where $u_{R}(\cdot):=u_{R,1}(\cdot)e_{1} + u_{R,2}(\cdot)e_{2} \in L^{2}(0,T; \mathcal{H}^{\mathfrak{c}})$ with T>0 being the fixed finite horizon, α ₁(λ) and α ₂(λ) are defined in (5.23), $\alpha= \frac{\gamma\pi}{\sqrt{2}l^{3/2}}$, and a _ij , 1≤i,j≤2, are elements of the coefficients matrix M associated with the operator $\mathfrak{C}$; see (5.15)–(5.16).

We check below by energy estimates that no finite time blow-up can occur for solutions to the system (B.1a), (B.1b) emanating from any initial datum $(z_{1,0}, z_{2,0}) \in\mathbb{R}^{2}$. For this purpose, let us define

$$R := \max \biggl\{ |z_{2,0}|, \ \frac{\alpha}{|2\alpha\alpha _2(\lambda )|}, \ \sqrt{ \frac{|\beta_2(\lambda)|}{|2\alpha\alpha_2(\lambda)|}} \biggr\} \quad\mbox{and}\quad C := \int _0^T \bigl|a_{12}u_{R,1}(t) + a_{22}u_{R,2}(t)\bigr| \,\mathrm{d}t. $$

We claim that

$$ \bigl|z_2(t)\bigr| \le e^{C/R}R \quad \forall t \in[0, T]. $$

(B.2)

It is clear that we only need to deal with those values of t such that |z ₂(t)|>R. Assume that there exists such time instances, otherwise we are done. Let us fix an arbitrary interval [t _∗,t ^∗]⊂[0,T] such that

$$ \bigl|z_2(t)\bigr| \ge R \quad\forall t \in \bigl[t_\ast, t^\ast\bigr]. $$

(B.3)

Since R≥|z _2,0| and z ₂ depends continuously on t, we can reduce t _∗ such that z ₂(t _∗)=R while the condition (B.3) remains true.

Now by multiplying z ₂(t) on both sides of (B.1b), we obtain

$$ \frac{1}{2} \frac{\mathrm{d}[(z_2)^2]}{\mathrm{d}t} = c(t) (z_2)^2, \quad \forall t \in\bigl[t_\ast, t^\ast\bigr], $$

(B.4)

where

$$c(t) := \biggl( \beta_2(\lambda) - \frac{\alpha(z_1)^2}{z_2} + 2\alpha \alpha_1(\lambda) (z_1)^2 + 2 \alpha \alpha_2(\lambda) (z_2)^2 + \frac {a_{12}u_{R,1}(t) + a_{22}u_{R,2}(t)}{z_2} \biggr). $$

It follows then that

$$ \bigl[z_2\bigl(t^\ast\bigr) \bigr]^2 = e^{2\int_{t_\ast}^{t^\ast} c(t) \mathrm{d}t}\bigl[z_2(t_\ast) \bigr]^2. $$

(B.5)

Since |z ₂(t)|≥R for all t∈[t _∗,t ^∗] by the choices of t _∗ and t ^∗, we get

$$\begin{aligned} \int_{t_\ast}^{t^\ast} c(t) \, \mathrm{d}t \le& \beta_2(\lambda) \bigl( t^\ast- t_\ast\bigr) + \int_{t_\ast}^{t^\ast} \biggl[ \frac{\alpha}{R} + 2 \alpha \alpha _1(\lambda)\biggr] (z_1)^2 \,\mathrm{d}t + 2 \alpha\alpha_2(\lambda) R^2 \bigl( t^\ast- t_\ast\bigr)\\ &{} + \frac{\int_{t_\ast}^{t^\ast} |a_{12}u_{R,1}(t) + a_{22}u_{R,2}(t)| \,\mathrm{d}t}{R}, \end{aligned}$$

where we have used $|- \frac{\alpha}{z_{2}}| \le\frac{\alpha}{R}$ and 2αα ₂(λ)(z ₂)²≤2αα ₂(λ)R ², which follow from the definition of R and the fact that α>0 and α ₂(λ)<0.

According again to the definition of R and the facts that α>0, α ₁(λ)<0 and α ₂(λ)<0, we get

$$\frac{\alpha}{R} + 2\alpha\alpha_1(\lambda) \le0 \quad\mbox{and} \quad\beta_2(\lambda) \bigl( t^\ast- t_\ast \bigr) + 2 \alpha\alpha _2(\lambda) R^2 \bigl( t^\ast- t_\ast\bigr) \le0. $$

We obtain then

$$\int_{t_\ast}^{t^\ast} c(t) \,\mathrm{d}t \le \frac{\int_{t_\ast }^{t^\ast} |a_{12}u_{R,1}(t) + a_{22}u_{R,2}(t)| \,\mathrm{d}t}{R} \le\frac{C}{R}. $$

By reporting the above estimate in (B.5) and using |z ₂(t _∗)|=R, we obtain

$$\bigl|z_2\bigl(t^\ast\bigr)\bigr| \le e^{C/R}\bigl|z_2(t_\ast)\bigr| = e^{C/R}R, $$

and (B.2) is thus proven.

Note also that by multiplying z ₁(t) on both sides of (B.1a), we obtain for any t∈[0,T] at which z ₁(t)≠0 that

$$\begin{aligned} &\frac{1}{2} \frac{\mathrm{d}[(z_1)^2]}{\mathrm{d}t} \\ &\quad = (z_1)^2 \biggl( \beta_1(\lambda) + \alpha z_2 + \alpha \alpha_1(\lambda) (z_2)^2 + \alpha\alpha _1(\lambda) \alpha_2(\lambda) (z_2)^3 + \frac{a_{11}u_{R,1}(t) + a_{21}u_{R,2}(t)}{z_1} \biggr). \end{aligned}$$

(B.6)

It follows then from the boundedness of z ₂ and (B.6) that z ₁ can grow at most exponentially. Consequently, no finite time blow-up can occur for the $h^{(1)}_{\lambda}$-based reduced system (B.1a), (B.1b).