Validated Numerical Approximation of Stable Manifolds for Parabolic Partial Differential Equations

Abstract

This paper develops validated computational methods for studying infinite dimensional stable manifolds at equilibrium solutions of parabolic PDEs, synthesizing disparate errors resulting from numerical approximation. To construct our approximation, we decompose the stable manifold into three components: a finite dimensional slow component, a fast-but-finite dimensional component, and a strongly contracting infinite dimensional “tail”. We employ the parameterization method in a finite dimensional projection to approximate the slow-stable manifold, as well as the attached finite dimensional invariant vector bundles. This approximation provides a change of coordinates which largely removes the nonlinear terms in the slow stable directions. In this adapted coordinate system we apply the Lyapunov-Perron method, resulting in mathematically rigorous bounds on the approximation errors. As a result, we obtain significantly sharper bounds than would be obtained using only the linear approximation given by the eigendirections. As a concrete example we illustrate the technique for a 1D Swift-Hohenberg equation.

Appendices

General Strategy for Bootstrapping Gronwall’s Inequality

We generalize the bootstrapping argument used in Sect. 3 so that it can be applied in Sects. 4 and 5. To unify the class of functions we wish to bound, and the set of assumptions we make on these functions, we define Condition A.1 below. In a slight abuse of notation, here we define ${\mathcal {B}}$ to be a tensor, distinct from its previous usage as a ball of functions in Definition 2.8.

Condition A.1

Fix $\lambda _1, \dots , \lambda _{N_{\lambda }} \in {\mathbb {R}}$, fix $H \in {\mathbb {R}}^{N_{\lambda }} \otimes {\mathbb {R}}^{N_{\lambda }}$ and define $\gamma _k := \lambda _k + H_k^k$ for $1 \le k \le N_\lambda$. For $N_\mu \in {\mathbb {N}}$, fix some $\mu _k \in {\mathbb {R}}$ for $1 \le k \le N_\mu$. Assume that $\{ \gamma _j \}_{j=1}^{N_\lambda } \subseteq \{ \mu _k \}_{k=1}^{N_\mu }$, and suppose that both $\gamma _{k} > \gamma _{k+1}$ and $\mu _{k} > \mu _{k+1}$. Assume further that $\mu _1 > \gamma _1$.

For $M \in {\mathbb {N}}$, and $N_i \in {\mathbb {N}}$ for $1 \le i \le M$ and basis elements $e_{n_i} \in {\mathbb {R}}^{N_i}$ where $1 \le n_i \le N_i$, we fix tensors

$$\begin{aligned} {\mathcal {A}}&\in \big ( \bigotimes _{i=1}^M {\mathbb {R}}^{N_i} \big ) \otimes {\mathbb {R}}^{N_\lambda } \otimes {\mathbb {R}}^{N_\mu } ,&{\mathcal {B}}&\in \big ( \bigotimes _{i=1}^M {\mathbb {R}}^{N_i} \big ) \otimes {\mathbb {R}}^{N_\lambda } \end{aligned}$$

component-wise by

$$\begin{aligned} {\mathcal {A}}_{j,k}&:= A_{j,k}^{n_1 \dots n_M} \cdot e_{n_1} \otimes \dots \otimes e_{n_M} ,&{\mathcal {B}}_j&:= B_j^{n_1 \dots n_M} \cdot e_{n_1} \otimes \dots \otimes e_{n_M} . \end{aligned}$$

For this arrangement of constants, we say that a pair $(u,\omega )$ satisfies Condition A.1 on a time interval [0, T] if the functions $u=(u_j)_{j=1}^{N_\lambda }$ and the positive tensor $\omega \in \bigotimes _{i=1}^M {\mathbb {R}}^{N_i}$ satisfy the inequalities

$$\begin{aligned} e^{ - \lambda _j t} u_j(t)&\le {\mathcal {B}}_j \omega + \int _0^t e^{-\lambda _j \tau } \sum _{0 \le k \le N_{\mu }} e^{\mu _k \tau } {\mathcal {A}}_{j,k} \omega \, d \tau + \int _0^t e^{-\lambda _j \tau } H_j^i u_i(\tau ) \, d\tau \qquad \text {for all } t \in [0,T]. \end{aligned}$$

(82)

In all cases where we consider constants satisfying Condition A.1, we take $N_\lambda = m_s$, and $\lambda _1 , \dots , \lambda _{N_\lambda }$ as in (6), and $H_j^i$ as in Definition 2.9. Hence, the definition of $\gamma _k$ here coincides with that given in Definition 3.3. For the other variables, we take them in the various sections according to the following table.

	Section 3	Section 4	Section 5
$u_j$	$| x_j(t,\xi ,\alpha ) - x_j(t,\zeta ,\alpha )|$	$\Vert \partial _{i} x_j(t,\eta ,\alpha ) - \partial _{i} x_j(t,\zeta ,\alpha )\Vert$	$| x_j(t , \xi , \alpha ) - x_j(t, \xi , \beta )|$
$\omega$	$| \xi _n - \zeta _n|$	$| \eta _l - \zeta _l|$	$|\xi _{n_1}| \otimes \Vert \alpha - \beta \Vert _{n_2', {\mathcal {E}}}^{n_3}$
${\mathcal {A}}_{j,k}$	0	$S_j^{nm} G_{m,k_1}^{l} G_{n,k_2}^i$	$C_j^{n_2'} G_{n_3, k}^{n_1}$
${\mathcal {B}}_j$	$\delta _j^n$	0	0
$\{ \mu _k \}$	$\{ \gamma _k\}_{k=0}^{m_s}$	$\{ \gamma _k \}_{k=0}^{m_s} \cup \{ \gamma _{k_1} + \gamma _{k_2} \}_{k_1,k_2=0}^{m_s}$	$\{ \gamma _k\}_{k=-1}^{m_s}$

We note that for ${\mathcal {A}}_{j,k}$ in Sect. 4 we use a double index $(k_1,k_2)$ to index over the elements of $\{\mu _k \}$. For a system given as in Condition A.1 we are interested in finding a tensor ${\mathcal {G}}$ satisfying Condition A.2 below.

Condition A.2

Given $\mu$ as in Assumption A.1 and a pair $(u,\omega )$ of functions $u=(u_j)_{j=1}^{N_\lambda }$ on [0, T] and a positive tensor $\omega \in \bigotimes _{i=1}^M {\mathbb {R}}^{N_i}$, we say that the tensor ${\mathcal {G}}\in \big ( \bigotimes _{i=1}^M {\mathbb {R}}^{N_i} \big ) \otimes {\mathbb {R}}^{N_\lambda } \otimes {\mathbb {R}}^{N_\mu }$ with components

$$\begin{aligned} {\mathcal {G}}_{j,k} := G_{j,k}^{n_1 \dots n_M} e_{n_1} \otimes \dots \otimes e_{n_M}, \end{aligned}$$

satisfies Condition A.2 if $u_j(t) \le \sum _{k=1}^{N_\mu } e^{\mu _k t} {\mathcal {G}}_{j,k} \omega$ for all $t\in [0,T]$.

From these two conditions, we can bootstrap our bounds on a tensor ${\mathcal {G}}$.

Proposition A.3

Assume the pair $(u,\omega )$ satisfies Condition A.1 on [0, T] and assume ${\mathcal {G}}$ satisfies Condition A.2. Fix $1 \le j \le N_\lambda$. If ${\mathcal {A}}_{j,k}=0$ and ${\mathcal {G}}_{i,k}=0$ whenever $\mu _k= \gamma _j$, then we have:

$$\begin{aligned} u_j(t)&\le e^{\gamma _j t} {\mathcal {B}}_j \omega + \sum _{\begin{array}{c} 1 \le k \le N_\mu \\ \mu _k \ne \gamma _j \end{array} } \frac{e^{\mu _k t} - e^{\gamma _j t}}{\mu _k - \gamma _j} \Big ({\mathcal {A}}_{j,k} + \sum _{\begin{array}{c} 1 \le i \le N_\lambda \\ i \ne j \end{array}} H_j^i {\mathcal {G}}_{i,k} \Big ) \omega \qquad \text {for all } t\in [0,T]. \end{aligned}$$

(83)

In other words, define a map ${\mathcal {T}}_{j,k} : \big ( \bigotimes _{i=1}^M {\mathbb {R}}^{N_i} \big ) \otimes {\mathbb {R}}^{N_\lambda } \otimes {\mathbb {R}}^{N_\mu } \rightarrow \bigotimes _{i=1}^M {\mathbb {R}}^{N_i}$ by:

$$\begin{aligned} {\mathcal {T}}_{j,k}({\mathcal {A}}, {\mathcal {B}},{\mathcal {G}}) := {\left\{ \begin{array}{ll} (\mu _k - \gamma _j)^{-1} \Big ( {\mathcal {A}}_{j,k} + \sum \limits _{\begin{array}{c} 1 \le i \le N_{\lambda } \\ i \ne j \end{array}} H_j^i {\mathcal {G}}_{i,k} \Big ) &{} \text{ if } \mu _k \ne \gamma _j \\ {\mathcal {B}}_j - \sum \limits _{\begin{array}{c} 0 \le m \le N_\mu \\ \mu _m \ne \gamma _j \end{array}} (\mu _m - \gamma _j)^{-1} \Big ( {\mathcal {A}}_{j,m}+ \sum \limits _{\begin{array}{c} 1 \le i \le N_{\lambda } \\ i \ne j \end{array}} H_j^i {\mathcal {G}}_{i,m} \Big )&\text{ if } \mu _k=\gamma _j. \end{array}\right. } \end{aligned}$$

(84)

Then ${\mathcal {G}}$ also satisfies Condition A.2 if we replace ${\mathcal {G}}_{j,k}$ by ${\mathcal {T}}_{j,k} ({\mathcal {A}}, {\mathcal {B}},{\mathcal {G}})$ for all k.

Proof of Proposition A.3

Splitting $H_j^i u_i = \sum _{i\ne j} H^i_j u_i + H_j^j u_j$, we write (82) as

$$\begin{aligned} e^{ - \lambda _j t} u_j(t)&\le {\mathcal {B}}_j \omega + \int _0^t e^{-\lambda _j \tau } v(\tau ,\omega ) d\tau + \int _0^t e^{-\lambda _j \tau } H_j^j u_j(\tau ) d\tau . \end{aligned}$$

where

$$\begin{aligned} v(\tau ,\omega )&= \sum _{\begin{array}{c} 1 \le k \le N_\mu \\ \mu _k \ne \gamma _j \end{array} }e^{\mu _k \tau } {\mathcal {A}}_{j,k} \omega + \sum _{\begin{array}{c} 1 \le i \le N_\lambda \\ i \ne j \end{array}} H_j^i u_i (\tau ) . \end{aligned}$$

By plugging in the bound assumed in Condition A.2, we obtain

$$\begin{aligned} v(\tau ,\omega )&\le \sum _{\begin{array}{c} 1 \le k \le N_\mu \\ \mu _k \ne \gamma _j \end{array} }e^{\mu _k \tau } \Big ( {\mathcal {A}}_{j,k} \omega + \sum _{\begin{array}{c} 1 \le i \le N_\lambda \\ i \ne j \end{array}} H_j^i {\mathcal {G}}_{i,k} \omega \Big ) . \end{aligned}$$

By applying Lemma 3.9 we obtain (83).

In order to obtain tensors satisfying the requirement that ${\mathcal {A}}_{j,k}, {\mathcal {G}}_{i,k}=0$ whenever $\mu _k= \gamma _j$, we define an operator ${\mathcal {Q}}_j$ as below.

Proposition A.4

Fix $1 \le j \le N_{\lambda }$ and define a map ${\mathcal {Q}}_j : \big ( \bigotimes _{i=1}^M {\mathbb {R}}_+^{N_i} \big ) \otimes {\mathbb {R}}^{N_\lambda } \otimes {\mathbb {R}}^{N_\mu } \rightarrow \big ( \bigotimes _{i=1}^M {\mathbb {R}}_+^{N_i} \big ) \otimes {\mathbb {R}}^{N_\lambda } \otimes {\mathbb {R}}^{N_\mu }$ by

$$\begin{aligned} {\mathcal {Q}}_j ({\mathcal {G}})_{i,k}^{n_{1} \dots n_{M}} := {\left\{ \begin{array}{ll} 0 &{} \text{ if } \mu _k= \gamma _j \\ G_{i,k}^{n_{1} \dots n_{M}} + G_{i,(k+1)}^{n_{1} \dots n_{M}} &{} \text{ if } \mu _{k+1}= \gamma _j, \text{ and } G_{i,(k+1)}^{n_{1} \dots n_{M}} >0 \\ G_{i,k}^{n_{1} \dots n_{M}} + G_{i,(k-1)}^{n_{1} \dots n_{M}} &{} \text{ if } \mu _{k-1}= \gamma _j, \text{ and } G_{i,(k-1)}^{n_{1} \dots n_{M}} <0 \\ G_{i,k}^{n_{1} \dots n_{M}} &{} \text{ otherwise. } \end{array}\right. } \end{aligned}$$

Then ${\mathcal {Q}}_j ({\mathcal {G}})_{i,k} =0$ whenever $\mu _k = \gamma _j$. Furthermore, if ${\mathcal {G}}$ satisfies Condition A.2 then ${\mathcal {Q}}_j ({\mathcal {G}})$ satisfies Condition A.2.

We are able to generalize Algorithm 3.11 as follows.

Algorithm A.5

Take as input all the constants in Condition A.1, an input tensor ${\widehat{{\mathcal {G}}}}$ satisfying Condition A.2, and a computational parameter $N_{bootstrap}$. The algorithm outputs a tensor ${\mathcal {G}}$.

Proposition A.6

If the input tensor ${\widehat{{\mathcal {G}}}}$ to Algorithm A.5 satisfies Condition A.2, then the output tensor ${\mathcal {G}}$ satisfies Condition A.2.

The proof of Proposition A.4 follows from the assumption that $\mu _{k} > \mu _{k+1}$. The proof of Proposition A.6 follows from an induction argument which uses Proposition A.3 for the inductive step. Both proofs are left to the reader.

Semigroup Estimates for Fast-Slow Systems

In Eq. (8) we require constants $C_s, \lambda _s$ satisfying

$$\begin{aligned} \begin{aligned} |e^{(\Lambda _s + L_s^s)t } \mathrm {x}_{s} |&\le C_s e^{\lambda _s t } |\mathrm {x}_{s}|,&\qquad&t \ge 0, \mathrm {x}_{s} \in X_s . \end{aligned} \end{aligned}$$

(85)

Our assumption that $\lambda _s <0$, and moreover that $\gamma _0 = \lambda _s + C_s {\hat{{\mathcal {H}}}} < 0$, is essential. In Proposition 3.13 this is used to prove that solutions $x(t, \xi ,\alpha )$ stay inside the ball $B_s(\rho )$ for all $t\ge 0$. While our method of bootstrapping Gronwall’s inequality greatly mitigates the effect of these constants $C_s, \lambda _s$ on our final estimates, for the Lyapunov-Perron operator to be well defined it is essential that we prove $\gamma _0 <0$.

There are two types of estimates which we will apply to obtain pairs $(C_s,\lambda _s)$ satisfying (85). First, for linear operators $A,B \in {\mathcal {L}}(X,X)$ with $| e^{At} \mathrm {x}| \le k e^{\lambda t} |\mathrm {x}|$ for all $\mathrm {x}\in X$ and $t \ge 0$, and $\Vert B\Vert <\infty$, we have (the proof is analogous to the one of Proposition 3.2)

$$\begin{aligned} | e^{(A+B)t} \mathrm {x}|&\le k e^{(\lambda + k \Vert B\Vert ) t} |\mathrm {x}|, \qquad \text {for all } t \ge 0, \mathrm {x}\in X. \end{aligned}$$

(86)

This estimate by itself is not enough, as the largest eigenvalue of $\Lambda _s$ is often small in comparison with $\Vert L_s^s \Vert$. For example, in Sect. 6 we showed that $|e^{\Lambda _i t} \mathrm {x}_i| \le e^{\lambda _i t} | \mathrm {x}_i|$ and $\Vert L_j^i \Vert \le D_j^i$ with values

$$\begin{aligned} \lambda _1&= -1.41,&\lambda _2&= -4.58 \times 10^{4} ,&D_s^s&= \left( \begin{matrix} 4 \times 10^{-10} &{} 1.6 \\ 1.6 &{} 5.7 \end{matrix} \right) . \end{aligned}$$

Since $\lambda _1 + \Vert L_s^s\Vert > 0$, just an estimate of the type in (86) with A the diagonal part of $D_s^s$ and B the off-diagonal part will not suffice. We further note that our estimates for $D_s^s$ do not improve with a larger Galerkin projection dimension. Hence we want to change basis to diagonalize $\Lambda _s + L_s^s$, at least approximately, and then take advantage of the identity $e^{PJP^{-1}t}=P e^{Jt}P^{-1}$ in our estimates. To motivate our construction, we first consider a $2 \times 2$ matrix

$$\begin{aligned} M&= \left( \begin{matrix} \lambda _1 &{} \delta _b \\ \delta _c &{} \lambda _\infty \end{matrix} \right) . \end{aligned}$$

If $\lambda _\infty$ is much larger in absolute value than the other matrix entries, then the eigenvalues of M are approximately given by $\lambda _1$ and $\lambda _\infty$. In particular, if $| \delta _b \delta _c| < | \lambda _1 \lambda _\infty |$ and $\lambda _1,\lambda _\infty <0$, then all of the eigenvalues of M have negative real part. Below in Theorem B.1 we prove an analogous theorem where we replace $\lambda _1$ by a finite dimensional matrix, and $\lambda _\infty$ by an infinite dimensional linear operator. This is the second type of estimate that we use to find pairs $(C_s,\lambda _s)$ satisfying (85).

Theorem B.1

Consider Banach spaces ${\mathbb {C}}^N$ and $X_\infty$ with arbitrary norms, and their product ${\mathbb {C}}^N \times X_\infty$ with norm $| (x_N,x_\infty )| = (|x_N|^p + |x_\infty |^p)^{1/p}$ for any $1 \le p \le \infty$.

Consider the linear operators $M,\Lambda ,L : {\mathbb {C}}^N \times X_\infty \rightarrow {\mathbb {C}}^N \times X_\infty$ given by

$$\begin{aligned} M&= \Lambda +L ,&\Lambda&= \left( \begin{matrix} \Lambda _1 &{} 0\\ 0 &{} \Lambda _{\infty } \end{matrix} \right) ,&L&= \left( \begin{matrix} L_1^1 &{} L_1^\infty \\ L_\infty ^1 &{} L_{\infty }^\infty \end{matrix} \right) . \end{aligned}$$

(87)

We require $\Lambda$ to be densely defined and L to be bounded. Suppose that $\Lambda _1$ is diagonal and that $\Lambda _\infty$ has a bounded inverse.

Fix constants $\mu _1,\mu _\infty ,C_1,C_\infty \in {\mathbb {R}}$ such that for all $t \ge 0$ we have

$$\begin{aligned} \Vert e^{\Lambda _1 t} \Vert&\le C_1 e^{\mu _1 t} ,&\Vert e^{\Lambda _\infty t} \Vert&\le C_\infty e^{\mu _\infty t} . \end{aligned}$$

Fix constants $\delta _1, \delta _b,\delta _c,\delta _d,\varepsilon >0$ such that

$$\begin{aligned} \Vert L_1^1 \Vert&\le \delta _a,&\Vert L_1^\infty \Vert&\le \delta _b ,&\Vert L_\infty ^1\Vert&\le \delta _c ,&\Vert L_\infty ^\infty \Vert&\le \delta _d , \end{aligned}$$

and set

$$\begin{aligned} \varepsilon&:= \sum _{ \lambda \in \sigma (\Lambda _1) } \frac{ \Vert \Lambda _\infty ^{-1} \Vert }{1 - \Vert \Lambda _\infty ^{-1} \Vert ( \delta _d + | \lambda | )} . \end{aligned}$$

Assume that the inequalities

$$\begin{aligned} \Vert \Lambda _\infty ^{-1} \Vert \left( \delta _d + \sup _{\lambda _k \in \sigma (\Lambda _1)} | \lambda _k| \right)&<1 ,&\mu _\infty + C_\infty \left( \delta _d+ \varepsilon \delta _b \delta _c (1 + \varepsilon ^2 \delta _b \delta _c)\right) < \mu _1 , \end{aligned}$$

(88)

are satisfied. Then we have

$$\begin{aligned} \Vert e^{M t} \Vert \le C_s e^{\lambda _s t}, \end{aligned}$$

where

$$\begin{aligned} C_s&:= (1+\varepsilon \delta _b)^2(1+\varepsilon \delta _c)^2 \max \{C_1,C_\infty \} \\ \lambda _s&:= \mu _1 + C_s \delta _a + \Delta \max \{C_1,C_\infty \}\\ \Delta&:= \varepsilon \delta _b \delta _c \left( 1 + \varepsilon (2 \delta _b + \delta _c) + \varepsilon ^2 \delta _b \delta _c (1 + \varepsilon \delta _b) \right) . \end{aligned}$$

First we prove a lemma for general Banach spaces which allows us to approximately diagonalize our matrix. When $|\cdot |$ denotes the norm on a Banach space, then by $|\cdot |_*$ we denote the norm on its dual.

Lemma B.2

For a Banach space $X_\infty$ consider the linear operator $M_1: {\mathbb {C}}^N \times X_\infty \rightarrow {\mathbb {C}}^N \times X_\infty$ defined as

$$\begin{aligned} M_1= \left( \begin{matrix} A &{} B \\ C &{}D \end{matrix} \right) . \end{aligned}$$

Suppose that $\sigma (A) \cap \sigma (D) = \emptyset$ and that A has distinct eigenvalues $\lambda _1 ,\dots , \lambda _N$ with eigenvectors $v_1 , \dots , v_N$, and dual eigenvectors $u_1 , \dots , u_N$ (the corresponding eigenvectors of $A^*$). Normalize the vectors so that $u^*_i v_j =\delta _{ij}$, the Kronecker delta.

We define $W_b : X_\infty \rightarrow {\mathbb {C}}^N$ and $W_c : {\mathbb {C}}^N \rightarrow X_\infty$ as a sum of products between vectors in their codomains, and dual vectors acting on their domains:

$$\begin{aligned} W_b&:= \sum _{k=1}^N v_k \left[ (D^* - \lambda ^*_k I_{\infty })^{-1} B^* u^*_k \right] ,&W_c&:= \sum _{k=1}^N - \left[ (D - \lambda _k I_{\infty })^{-1} C v_k \right] u^*_k , \end{aligned}$$

where $D^* : X_\infty ^* \rightarrow X_\infty ^*$ and $B^* :({\mathbb {C}}^N)^* \rightarrow X_{\infty }^*$ are the dual transformations. Define invertible operators $P_b,P_c: {\mathbb {C}}^N \times X_\infty \rightarrow {\mathbb {C}}^N \times X_\infty$ by

$$\begin{aligned} P_b&= \left( \begin{matrix} I_{N} &{} W_b \\ 0 &{} I_{\infty } \end{matrix} \right)&P_c&= \left( \begin{matrix} I_{N} &{} 0 \\ W_c &{} I_{\infty } \end{matrix} \right) . \end{aligned}$$

Then

$$\begin{aligned} (P_cP_b)^{-1} M_1 ( P_cP_b) = \left( \begin{matrix} A &{} 0 \\ 0 &{}D \end{matrix} \right) + E, \end{aligned}$$

where

$$\begin{aligned} E = \left( \begin{matrix} (I_N + W_b W_c) B W_c &{} B W_c W_b + W_bW_cB (I + W_c W_b) \\ -W_c B W_c &{} -W_c B ( I_{\infty } + W_c W_b ) \end{matrix} \right) . \end{aligned}$$

Proof

First we show that

$$\begin{aligned} P_b^{-1} \left( \begin{matrix} A &{} B \\ 0 &{} D \end{matrix} \right) P_b&= \left( \begin{matrix} A &{} 0 \\ 0 &{} D \end{matrix} \right) ,&P_c^{-1} \left( \begin{matrix} A &{} 0 \\ C &{} D \end{matrix} \right) P_c&= \left( \begin{matrix} A &{} 0 \\ 0 &{} D \end{matrix} \right) . \end{aligned}$$

(89)

We begin with the second equality in (89), and calculate

$$\begin{aligned} P_c^{-1} \left( \begin{matrix} A &{} 0 \\ C &{} D \end{matrix} \right) P_c&= \left( \begin{matrix} A &{} 0 \\ -W_cA + C +DW_c &{} D \end{matrix} \right) . \end{aligned}$$

We compute the action of $-W_cA + C +DW_c$ on an eigenvector $v_k$ of A as follows:

$$\begin{aligned} ( -W_cA + C +DW_c ) v_k&= C v_k +( D -\lambda _k I_{\infty } )W_c v_k . \end{aligned}$$

To see that the right hand side is equal to zero, we calculate, using $u^*_i v_j =\delta _{ij}$,

$$\begin{aligned} W_c v_k = - \left( D - \lambda _k I_{\infty } \right) ^{-1} C v_k . \end{aligned}$$

Since the eigenvectors $v_1 \dots v_N$ span ${\mathbb {C}}^N$, then $-W_c A + C +DW_c =0$, yielding the desired equality.

The argument is analogous for the first identity in (89). Again we begin by calculating

$$\begin{aligned} P_b^{-1} \left( \begin{matrix} A &{} B \\ 0 &{} D \end{matrix} \right) P_b&= \left( \begin{matrix} A &{} AW_b + B -W_bD \\ 0 &{} D \end{matrix} \right) . \end{aligned}$$

Hence, we would like to show the map $(AW_b + B -W_bD): X_\infty \rightarrow {\mathbb {C}}^N$ is the zero map, which we do by arguing that $u^*_k (AW_b + B -W_bD) =0$ for all k. The latter follows from a calculation similar to the one performed above.

Finally, we calculate $(P_cP_b)^{-1} M_1 P_cP_b$ as follows:

$$\begin{aligned} (P_cP_b)^{-1} M_1 ( P_cP_b)&= P_b^{-1} \left( \left( \begin{matrix} A &{} 0 \\ 0 &{}D \end{matrix} \right) + P_c^{-1} \left( \begin{matrix} 0 &{} B \\ 0 &{} 0 \end{matrix} \right) P_c \right) P_b \\&= P_b^{-1} \left( \left( \begin{matrix} A &{} B \\ 0 &{}D \end{matrix} \right) + \left( \begin{matrix} B W_c &{} 0 \\ -W_c B W_c &{} -W_c B \end{matrix} \right) \right) P_b \\&= \left( \begin{matrix} A &{} 0 \\ 0 &{}D \end{matrix} \right) + \left( \begin{matrix} (I_N + W_b W_c) B W_c &{} B W_c W_b + W_bW_cB (I + W_c W_b) \\ -W_c B W_c &{} -W_c B ( I_{\infty } + W_c W_b ) \end{matrix} \right) . \end{aligned}$$

$\square$

Proof of Theorem B.1

Let $M = M_1 + M_2$, where

$$\begin{aligned} M_1&:= \left( \begin{matrix} A&{}B\\ C&{}D \end{matrix} \right) := \left( \begin{matrix} \Lambda _1 &{} L_1^\infty \\ L_\infty ^1 &{} \Lambda _\infty + L_\infty ^\infty \end{matrix} \right) ,&M_2&:= \left( \begin{matrix} L_1^1 &{} 0 \\ 0 &{} 0 \end{matrix} \right) . \end{aligned}$$

We will apply Lemma B.2 to the matrix $M_1$. Since we have assumed that $\Lambda _1$ is diagonal we may take $u_k=v_k=e_k$, the standard basis vectors in ${\mathbb {C}}^{N}$. We begin by proving $\Vert W_b\Vert \le \varepsilon \delta _b$ and $\Vert W_c \Vert \le \varepsilon \delta _c$. We first calculate

$$\begin{aligned} (D-\lambda _k I_\infty )^{-1} = ( \Lambda _\infty + L_\infty ^\infty - \lambda _k I_{\infty } )^{-1}&= ( I_{\infty } + \Lambda _\infty ^{-1} ( L_\infty ^\infty -\lambda _k I_{\infty }))^{-1} \Lambda _\infty ^{-1} . \end{aligned}$$

By our hypothesis, we are allowed to apply the Neumann series and we obtain

$$\begin{aligned} \Vert (D-\lambda _k I_\infty )^{-1} \Vert&\le \frac{\Vert \Lambda _\infty ^{-1}\Vert }{1 - \Vert \Lambda _\infty ^{-1}\Vert ( \delta _d + | \lambda _k| )}. \end{aligned}$$

(90)

We note that the same estimate holds for the dual operator $(D^*-\lambda _k^* I_\infty )^{-1}$.

We now show that $\Vert W_b\Vert \le \varepsilon \delta _b$. Namely, by using that $\Vert u^*_k\Vert _{({\mathbb {C}}^N)^*}=\Vert v_k\Vert _{{\mathbb {C}}^N}=1$ we find that

$$\begin{aligned} \Vert W_b \Vert&= \sup _{ x \in X_\infty , \Vert x\Vert =1} \Big \Vert \sum _{ \lambda _k \in \sigma (\Lambda _1) } v_k \left[ (D^* - \lambda ^*_k I_{\infty } )^{-1} B^* u^T_k\right] x \Big \Vert _{{\mathbb {C}}^N} \\&\le \sup _{ x \in X_\infty , \Vert x\Vert =1} \sum _{ \lambda _k \in \sigma (\Lambda _1) } \Big | \left[ (D^* - \lambda ^*_k I_{\infty } )^{-1} B^* u^T_k\right] x \Big | \\&\le \sum _{ \lambda _k \in \sigma (\Lambda _1) } \Big \Vert (D^* - \lambda ^*_k I_{\infty } )^{-1} B^* \Big \Vert _{{\mathcal {L}}( ({\mathbb {C}}^{N})^* ,X_\infty ^*) } \\&\le \Vert B^* \Vert \sum _{ \lambda _k \in \sigma (\Lambda _1) } \frac{ \Vert \Lambda _\infty ^{-1} \Vert }{1 - \Vert \Lambda _\infty ^{-1} \Vert ( \delta _d + | \lambda _k| )} . \end{aligned}$$

Hence, by plugging in $\Vert B^*\Vert = \Vert L_1^\infty \Vert$ we obtain $\Vert W_b \Vert \le \varepsilon \delta _b$. The proof of the estimate $\Vert W_c\Vert \le \varepsilon \delta _c$ is analogous. Next, we note that

$$\begin{aligned} \Vert P_b\Vert ,\Vert P_b^{-1}\Vert&\le 1+\varepsilon \delta _b&\Vert P_c\Vert ,\Vert P_c^{-1}\Vert&\le 1+\varepsilon \delta _c . \end{aligned}$$

By Lemma B.2 we have

$$\begin{aligned} (P_cP_b)^{-1}( M_1 + M_2)( P_cP_b)&= M_3 +M_4 + (P_b P_b)^{-1} M_2 (P_c P_b), \end{aligned}$$

(91)

where

$$\begin{aligned} M_3&:= \left( \begin{matrix} \Lambda _1 &{} 0 \\ 0 &{} \Lambda _\infty + L_\infty ^\infty -W_c L_1^\infty ( I_{\infty } + W_c W_b ) \end{matrix} \right) , \\ M_4&:= \left( \begin{matrix} (I_N + W_b W_c) L_1^\infty W_c &{} L_1^\infty W_c W_b + W_bW_c L_1^\infty (Id + W_c W_b) \\ -W_c L_1^\infty W_c &{} 0 \end{matrix} \right) . \end{aligned}$$

For $( x_N , x_\infty ) \in {\mathbb {C}}^N \times X_\infty$ we see that

$$\begin{aligned} e^{M_3 t} (x_N , x_\infty )&= \left( e^{\Lambda _1 t}x_N, e^{(\Lambda _\infty + L_\infty ^\infty -W_c L_1^\infty ( I_{\infty } + W_c W_b )) t} x_\infty \right) . \end{aligned}$$

We also have $\Vert L_\infty ^\infty -W_c L_1^\infty ( I_{\infty } + W_c W_b ) \Vert \le \delta _d + \varepsilon \delta _b \delta _c (1 + \varepsilon _b \varepsilon _c)$. By applying the estimate (86) we obtain, for all $t \ge 0$,

$$\begin{aligned} \Vert e^{\Lambda _1 t} x_N \Vert&\le C_1 e^{\mu _1 t}\Vert x_N \Vert , \\ \Vert e^{(\Lambda _\infty + L_\infty ^\infty -W_c L_1^\infty ( I_{\infty } + W_c W_b )) t} x_\infty \Vert&\le C_\infty e^{(\mu _\infty + C_\infty [ \delta _d + \varepsilon \delta _b \delta _c (1 + \varepsilon _b \varepsilon _c)]) t} \Vert x_\infty \Vert . \end{aligned}$$

From our assumption in (88) that $\mu _1 > \mu _\infty + C_\infty [\delta _d + \varepsilon \delta _b \delta _c (1 + \varepsilon ^2 \delta _b \delta _c)]$, we obtain, for any p-norm, $1 \le p \le \infty$, on the product ${\mathbb {C}}^N \times X_\infty$,

$$\begin{aligned} \Vert e^{ M_3 t } (x_N,x_\infty ) \Vert \le \max \{ C_1, C_\infty \} e^{\mu _1 t} \Vert (x_N,x_\infty ) \Vert . \end{aligned}$$

We may estimate the norm of the components of $M_4$ as

$$\begin{aligned} \Vert (I_N + W_b W_c) L_1^\infty W_c \Vert&\le \varepsilon \delta _b \delta _c ( 1 + \varepsilon ^2 \delta _b \delta _c) , \\ \Vert -W_c L_1^\infty W_c \Vert&\le \varepsilon ^2 \delta _b \delta _c^2 , \\ \Vert L_1^\infty W_c W_b + W_bW_c L_1^\infty (Id + W_c W_b)\Vert&\le \varepsilon ^2 \delta _b^2 \delta _c ( 2 + \varepsilon ^2 \delta _b \delta _c) . \end{aligned}$$

We then obtain the bound

$$\begin{aligned} \Vert M_4\Vert \le \Delta := \varepsilon \delta _b \delta _c \left( 1 + \varepsilon (2 \delta _b + \delta _c) + \varepsilon ^2 \delta _b \delta _c (1 + \varepsilon \delta _b) \right) \end{aligned}$$

by summing the component bounds.

We now perform the final estimate. By using (91) we obtain

$$\begin{aligned} e^{Mt}&= (P_cP_b) \exp \left\{ \left[ M_3+M_4 + (P_cP_b)^{-1}M_2(P_cP_b)\right] t \right\} (P_cP_b)^{-1} . \end{aligned}$$

By then applying (86) to the sum of $M_3$ and the bounded operator $M_4 + (P_cP_b)^{-1}M_2(P_cP_b)$ we obtain, with $C_{1,\infty } := \max \{C_1,C_\infty \}$,

$$\begin{aligned} \Vert e^{Mt}\Vert&\le \Vert P_cP_b\Vert \cdot \Vert (P_cP_b)^{-1}\Vert C_{1,\infty } \exp \left\{ \mu _1 + C_{1,\infty } \left\| M_4 + (P_cP_b)^{-1}M_2 (P_cP_b) \right\| t \right\} . \end{aligned}$$

Defining $C_s = \max \{ C_1, C_\infty \} (1+\varepsilon \delta _b)^2 ( 1 + \varepsilon \delta _c)^2$ and plugging in our bounds, we finally infer

$$\begin{aligned} \Vert e^{Mt} \Vert \le C_s e^{( \mu _1 + C_s \delta _a+ \Delta \max \{ C_1, C_\infty \} )t} . \end{aligned}$$

$\square$

Remark B.3

If we use the $p=1$ norm for the product space ${\mathbb {C}}^N \times X_\infty$ then our bound for $\Delta$ can be sharpened to

$$\begin{aligned} \Vert M_4\Vert \le \varepsilon \delta _b \delta _c \max \left\{ 1 + \varepsilon \delta _c(1 + \varepsilon \delta _b ) , \varepsilon \delta _b ( 2 + \varepsilon ^2 \delta _b \delta _c) \right\} . \end{aligned}$$