Input/output reduced model of a damped nonlinear beam based on Volterra series and modal decomposition with convergence results

Abstract

This paper addresses the model reduction and the simulation of a damped Euler–Bernoulli–von Kármán pinned beam excited by a distributed force. This nonlinear problem is formulated as a PDE and reformulated as a well-posed state-space system. The model order reduction and simulation are derived by combining two approaches: a Volterra series expansion and truncation and a pseudo-modal truncation defined from the eigenbasis of the linearized problem. The interest of this approach lies in the large class of input waveshapes that can be considered and in the simplicity of the simulation structure. This structure only involves cascades of finite-dimensional decoupled linear systems and multilinear functions. Closed-form bounds depending on the model coefficients and the truncation orders are provided for the Volterra convergence domain and the approximation error. These theoretical results are generalized to a large class of nonlinear models, and refinement of bounds are also proposed for a large sub-class. Numerical experiments confirm that the beam model is well approximated by the very first Volterra terms inside the convergence domain.

Appendices

Well-posedness and definitions

1.1 General setting

The class of systems (1–2) is considered for the general following setting:

• ${\mathbb {T}}$ denotes the time set ${\mathbb {R}}_+$,

• ${\mathbb {U}}$ and ${\mathbb {X}}$ are Banach spaces on the field ${\mathbb {R}}$,

• ${\mathcal {L}}({\mathbb {U}},{\mathbb {X}})$ and ${\mathcal {L}}({\mathbb {X}})$ are the sets of bounded linear operators from ${\mathbb {U}}$ to ${\mathbb {X}}$, and from ${\mathbb {X}}$ to ${\mathbb {X}}$, respectively,

• $\mathcal {ML}_k({\mathbb {X}})$ ($k\ge 2$) is the set of bounded multilinear operators from $\underbrace{{\mathbb {X}}\times \dots \times {\mathbb {X}}}_{k}$ to ${\mathbb {X}}$, equipped with norm

  $$\begin{aligned} \displaystyle \Vert A_k\Vert = \sup _{ \underset{\Vert x_1\Vert =\dots =\Vert x_k\Vert =1}{(x_1,\dots ,x_k)\in {\mathbb {X}}^k} } \Vert A_k(x_1,\dots ,x_k)\Vert _{\mathbb {X}}, \end{aligned}$$

• ${\mathcal {U}}=L^\infty ({\mathbb {T}},{\mathbb {U}})$ and ${\mathcal {X}}=L^\infty ({\mathbb {T}},{\mathbb {X}})$ are standard Lebesgue spaces (in which trajectories u and x live).

1.2 Mild solutions

For the linearized version of systems (1–2), mild solutions are defined in the following way [7, 23].

Definition 1

(Mild solution of a linear system) Let $u\in {\mathcal {L}}^\infty _{loc}({\mathbb {T}},{\mathbb {U}})$. The mild solution of the linearized system is the function $x\in {\mathcal {C}}_0({\mathbb {T}},{\mathbb {X}})$ defined for all $t\in {\mathbb {T}}$ as

$$\begin{aligned} x(t) = S(t){x_{\mathrm {ini}}}+ \int _0^t S(t-\tau ) B u(\tau ) \, \mathrm {d}\tau . \end{aligned}$$

A similar definition is given for nonlinear systems.

Definition 2

(Mild solution of a nonlinear system) Let $u\in {\mathcal {L}}^\infty _{loc}({\mathbb {T}},{\mathbb {U}})$. Then, x is said to be a mild solution of (1–2) iff $x\in {\mathcal {C}}_0({\mathbb {T}},{\mathbb {X}})$ and satisfies, for all $t\in {\mathbb {T}},$

$$\begin{aligned} x(t)&= S(t){x_{\mathrm {ini}}}+\int _0^t S(t-\tau ) \times \nonumber \\&\left( B\,u(\tau ) + \sum _{k=2}^K A_k \big ( x(\tau ),\dots ,x(\tau )\big ) \right) \,\mathrm {d}\tau \end{aligned}$$

(89)

As a standard result from [23], for a fixed $u\in {\mathcal {C}}_0({\mathbb {T}},{\mathbb {U}})$, there exist $t_{\max }\in (0,T]$ and a unique function $x\in {\mathcal {C}}_0([0,t_{\max }),{\mathbb {X}})$ such that x is a mild solution in the sense of (89). For the class of systems (1–2), it can be easily shown that the local existence and uniqueness of mild solutions still holds when the input u is taken in ${\mathcal {L}}^\infty _{loc}({\mathbb {T}},{\mathbb {U}})$, as stated in Definition 2.

1.3 Functional setting for the linearized beam problem

In the linear problem⁸ (24), the bi-Laplacian ${\mathcal {B}}$ is defined as the unbounded operator on ${\mathbb {H}}=L^2(0,1)$ with domain⁹

$$\begin{aligned} D({\mathcal {B}})&= \big \{ w \in H^4(0,1) ~\text{ s.t. }~ w(0)=w(1)=0,\;\\&\quad w^{\prime \prime }(0)=w^{\prime \prime }(1)=0 \big \}, \end{aligned}$$

such that ${\mathcal {B}}(w)=w^{(4)}$ for all $w\in D({\mathcal {B}})$.

This operator is closed, densely defined, self-adjoint and positive on ${\mathbb {H}}$. Hence, we can introduce its uniquely defined positive square root ${\mathcal {K}}$, with domain

$$\begin{aligned} D({\mathcal {K}})=\big \{ w \in H^2(0,1) \text{ s.t. } w(0)=w(1)=0\big \}, \end{aligned}$$

such that ${\mathcal {K}}(w)=-\varDelta w$. The domain $D({\mathcal {K}})$ equipped with the ${\mathcal {K}}$-norm defines a Hilbert space¹⁰:

$$\begin{aligned} {\mathbb {H}}^{\frac{1}{2}}:= D({\mathcal {K}}) ~~\text {with}~ \Vert .\Vert _{{\mathbb {H}}^\frac{1}{2}}=\Vert {\mathcal {K}}\;.\Vert _{\mathbb {H}}. \end{aligned}$$

Then, according to [16], the linearized problem (24) admits the state-space representation (25–27), which is well-posed for the functional setting ${\mathbb {U}}$ and ${\mathbb {X}}$ introduced in (28–29) and defining operator A on domain D(A) as

$$\begin{aligned} A :\;&D(A) \longrightarrow {\mathbb {X}}~\text {with}~ D(A) = \{(u,v)\in {\mathbb {H}}^\frac{1}{2}\times {\mathbb {H}}^\frac{1}{2}~\text{ s.t. }~ \nonumber \\&\quad 2(a+b\,{\mathcal {B}})v+{\mathcal {B}}u \in {\mathbb {H}}\}. \end{aligned}$$

In this setting, A generates a $C_0$ contraction semigroup on ${\mathbb {X}}$ ([16, (A1–A2), p. 6]), so that there exists a negative growth bound $\alpha <0$. More precisely (see [16, corollary 5.2]), it is a Riesz spectral operator on ${\mathbb {X}}$ which generates an analytic semigroup S, provided that $-\frac{1}{b}$ is not in the point spectrum of A. In addition, operator B belongs ${\mathcal {L}}({\mathbb {U}},{\mathbb {X}})$ and $\Vert B\Vert =1$. Therefore, this linear problem is well-posed and is in the class of systems defined in Sect. 2.1.

Estimates for the nonlinear beam model

1.1 Proof of Eq. (32)

For all $X_1\in {\mathbb {H}}^{\frac{1}{2}}$ and $z\in [0,1]$, $X_1(z)=X_1^\prime (0) \, z + \int _0^z (z-\zeta ) X_1^{\prime \prime }(\zeta )\,\mathrm {d}\zeta$ which is zero at $z=1$. Hence,

$$\begin{aligned} X_1(z)=\int _0^1 K(z,\zeta ) X_1^{\prime \prime }(\zeta )\,\mathrm {d}\zeta , \end{aligned}$$

where $K(z,\zeta )=\zeta (z-1)$ if $0\le \zeta \le z$ and $K(z,\zeta )=z(\zeta -1)$ if $z\le \zeta \le 1$. Using the Cauchy-Schwartz inequality, it comes

$\big | X_1(z)\big | \le \mu (z) \Big (\int _0^1 X_1^{\prime \prime }(\zeta )^2\,\mathrm {d}\zeta \Big )^{\frac{1}{2}}$, where $\mu (z) = \Big (\int _0^1 K(z,\zeta )^2 \,\mathrm {d}\zeta \Big )^{\frac{1}{2}} = \frac{z(1-z)}{\sqrt{3}}$.

Then, $\Big ( \int _0^1 \big | X_1(z)\big |^2 \, \mathrm {d}z \Big )^{\frac{1}{2}} \le \nu \big \Vert X_1 \Vert _{{\mathbb {H}}^\frac{1}{2}}^2,$ where $\nu = \Big (\int _0^1 \mu (z)^2 \,\mathrm {d}z\Big )^{\frac{1}{2}} =\frac{1}{3\sqrt{10}}$. Finally, for all $(X,Y,Z)\in {\mathbb {X}}^3$,

$$\begin{aligned}&\big \Vert A_3(X,Y,Z)\Vert _{\mathbb {X}}\\&\quad \le \eta \Big | \int _0^1 X_1^\prime (z)\,Y_1^\prime (z)\, \mathrm {d}z \Big | \; \int _0^1 \Big | Z_1^{\prime \prime }(z) \Big |\, \mathrm {d}z\\&\quad \le \eta \Big |\int _0^1 X_1(z)\,Y_1^{\prime \prime }(z)\, \mathrm {d}z \Big | \; \Big (\int _0^1 \Big |Z_1^{\prime \prime }(z)\Big |^2\, \mathrm {d}z\Big )^{\frac{1}{2}}\\&\quad \le \eta \; \Big ( \int _0^1 \big | X_1(z)\big |^2 \, \mathrm {d}z \Big )^{\frac{1}{2}} \, \big \Vert Y_1 \Vert _{{\mathbb {H}}^\frac{1}{2}} \, \big \Vert Z_1 \Vert _{{\mathbb {H}}^\frac{1}{2}}\\&\quad \le \nu \; \eta \; \big \Vert X_1 \Vert _{{\mathbb {H}}^\frac{1}{2}} \, \big \Vert Y_1 \Vert _{{\mathbb {H}}^\frac{1}{2}} \, \big \Vert Z_1 \Vert _{{\mathbb {H}}^\frac{1}{2}}\\&\quad \le \nu \; \eta \; \big \Vert X \Vert _{{\mathbb {X}}} \, \big \Vert Y \Vert _{{\mathbb {X}}} \, \big \Vert Z \Vert _{{\mathbb {X}}}, \end{aligned}$$

proves the result.

1.2 Proof of Eqs. (33,43)

Functions $e^\pm _n$ form an orthogonal basis of ${\mathbb {X}}$, from which we define an orthonormal basis

$$\begin{aligned} E_n^+(z) =\frac{1}{k_n^2} \begin{bmatrix} 1 \\ 0 \end{bmatrix}s_n(z) ~~\text{ and }~~ E_n^-(z) = \begin{bmatrix} 0 \\ 1 \end{bmatrix}s_n(z) \end{aligned}$$

On each modal subspace, the linearized system behaves like second order system. Indeed, for all $x\in {\mathbb {H}}$,

$$\begin{aligned} S(t)Bx=\sum _{n=1}^\infty \kappa _n^-S(t)E_n^-, \end{aligned}$$

(90)

where $\kappa _n^-=<Bx,E_n^->_{\mathbb {X}}=<x,s_n>_{\mathbb {H}}$, and,

$$\begin{aligned} E_n^+(z)= & {} \frac{(e_n^+-e_n^-)\lambda _n^+\lambda _n^-}{k_n^2(\lambda _n^- - \lambda _n^+)}= \frac{1}{k_n^2}, \begin{bmatrix} 1 \\ 0 \end{bmatrix}s_n(z),\nonumber \\ E_n^-(z)= & {} \frac{(\lambda _n^+e_n^+-\lambda _n^-e_n^-)}{\lambda _n^+-\lambda _n^-}= \begin{bmatrix} 0 \\ 1 \end{bmatrix}s_n(z) \end{aligned}$$

(91)

From (91), we obtain

$$\begin{aligned}&S(t)Bx=\sum _{n=1}^\infty \kappa _n^-\left( \frac{\lambda _n^+e^{\lambda _n^+ t}}{\lambda _n^+ - \lambda _n^-}e_n^+ -\frac{\lambda _n^-e^{\lambda _n^-t}}{\lambda _n^+ - \lambda _n^-}e_n^-\right) =\nonumber \\&\quad \sum _{n=1}^\infty \kappa _n^-\left( k_n^2\frac{e^{\lambda _n^+t}-e^{\lambda _n^- t}}{\lambda _n^+ - \lambda _n^-}E_n^+ + \frac{\lambda _n^+e^{\lambda _n^+ t}-\lambda _n^-e^{\lambda _n^-t}}{\lambda _n^+ - \lambda _n^-}E_n^-\right) . \end{aligned}$$

(92)

Setting $h_n(t)=\frac{e^{\lambda _n^+ t}-e^{\lambda _n^- t}}{\lambda _n^+ - \lambda _n^-}$ and using the standard expressions

$$\begin{aligned} h_n(t)= \left\{ \begin{array}{ll} e^{-k_n^2\xi _n t}\frac{\sin (k_n^2\sqrt{1-\xi _n^2}t)}{k_n^2\sqrt{1-\xi _n^2}} &{}\text { if }\xi _n<1\\ te^{-k_n^2 t} &{}\text { if }\xi _n=1\\ e^{-k_n^2\xi _n t}\frac{\sinh (k_n^2\sqrt{\xi _n^2-1}t)}{k_n^2\sqrt{\xi _n^2-1}} &{}\text { if }\xi _n>1 \end{array}\right. \end{aligned}$$

we obtain that

$$\begin{aligned} \Vert S(t)Bx\Vert _{\mathbb {X}}&=\sqrt{\sum _{n=1}^\infty (\kappa _n^-)^2 (k_n^4h_n(t)^2+h_n'(t)^2)}\\&\le \sqrt{\sup _n|k_n^4h_n(t)^2+h_n'(t)^2|}\,\Vert x\Vert _{\mathbb {H}}, \end{aligned}$$

so that we can set

$$\begin{aligned} \gamma =\int _{\mathbb {T}}\sqrt{\sup _n|k_n^4h_n(t)^2+h_n'(t)^2|}\mathrm {d}t. \end{aligned}$$

(93)

Now we notice that for all (x, y, z) in ${\mathbb {X}}$, $A_3(x,y,z)=Bg(x,y,z)$ where g is a third order multilinear operator from ${\mathbb {X}}^3$ to ${\mathbb {H}}$, such that $\Vert g\Vert _{\mathcal {ML}({\mathbb {X}},{\mathbb {H}})}=a_3$. We therefore obtain that for all (a, b, c) in ${\mathcal {X}}$

$$\begin{aligned}&\Vert \int _0^t S(t-\tau )Bg(a,b,c)(\tau ) \mathrm {d}\tau \Vert _{{\mathbb {X}}} \nonumber \\&\quad \le \int _0^t \Vert S(t)Bx\Vert _{{\mathcal {L}}({\mathbb {H}},{\mathbb {X}})}a_3\mathrm {d}\tau \Vert a\Vert _{\mathcal {X}}\Vert b\Vert _{\mathcal {X}}\Vert c\Vert _{\mathcal {X}}\nonumber \\&\quad \le \gamma a_3 \Vert a\Vert _{\mathcal {X}}\Vert b\Vert _{\mathcal {X}}\Vert c\Vert _{\mathcal {X}}. \end{aligned}$$

(94)

This proves that $\zeta _3\le \gamma a_3$.

In the same way, for operator C, we obtain from (92) that

$$\begin{aligned} CS(t)Bx= \sum _{n=1}^\infty \kappa _n^- \frac{e^{\lambda _n^+ t}-e^{\lambda _n^- t}}{\lambda _n^+ - \lambda _n^-}s_n, \end{aligned}$$

and

$$\begin{aligned} \Vert CS(t)Bx\Vert _{{\mathcal {L}}({\mathbb {H}},{\mathbb {H}}^{\frac{1}{2}})}&=\sqrt{\sum _{n=1}^\infty (\kappa _n^-)^2 k_n^4h_n(t)^2}\\&\le \sup _n\left( k_n^2|h_n(t)|\right) \Vert x\Vert _{\mathbb {H}}. \end{aligned}$$

It follows that $\Vert CS(t)Bx\Vert _{{\mathcal {L}}({\mathbb {H}},{\mathbb {H}}^{\frac{1}{2}})}\le \sup _n k_n^2|h_n(t)|$, so that we can set

$$\begin{aligned}&\gamma ^\dagger =\int _{\mathbb {T}}\sup _n\left( k_n^2|h_n(t)|\right) \mathrm {d}t,\\&r=\frac{\int _{\mathbb {T}}\sqrt{\sup _n|k_n^2h_n(t)^2+h_n'(t)^2|}\mathrm {d}t}{\int _{\mathbb {T}}\sup _n\left( k_n^2|h_n(t)|\right) \mathrm {d}t}. \end{aligned}$$

Proof of Proposition 1 (truncation error estimate)

We assume (A1–A3) in Sect. 5.3 and set $\varPhi (z)=\sum _{m=1}^\infty \phi _m z^m$ where the sequence $(\phi _m)_{m\in {\mathbb {N}}^*}$ is defined by (see step 6 in page 4)

$$\begin{aligned} \phi _1\!=\!1 ~~\text{ and, } \text{ for } \text{ all }~ m\ge 2,\quad \phi _m\!=\!\gamma a_3\sum _{p\in {\mathbb {M}}_m^3}\phi _{p_1}\phi _{p_2}\phi _{p_3}, \end{aligned}$$

with ${\mathbb {M}}_m^{3}= \Big \{ {p}\in ({\mathbb {N}}^*)^{3} \,\, \big | \,\,{p}_1+{p}_2+{p}_{3} = m \Big \}$. Then, the trajectories ${\widetilde{x}}=\sum _{m=1}^\infty {\widetilde{x}}_m$ and $x=\sum _{m=1}^\infty x_m$ are normally convergent and such that, for all $m\ge 1$,

$$\begin{aligned} \Vert {\widetilde{x}}_m\Vert _{\mathcal {X}}&\le \phi _m \Vert {\widehat{x}}_1\Vert _{\mathcal {X}}^m,\\ \Vert x_m\Vert _{\mathcal {X}}&\le \phi _m \Vert x_1\Vert _{\mathcal {X}}^m\\&\le \phi _m \left( (1+\epsilon )\Vert {\widehat{x}}_1\Vert _{\mathcal {X}}\right) ^m \le \beta _m \Vert {\widehat{x}}_1\Vert _{\mathcal {X}}^m, \end{aligned}$$

where the sequence $(\beta _m)_{m\in {\mathbb {N}}^*}$ defined by $\beta _m=(1+\epsilon )^m\phi _m$ is such that

$$\begin{aligned} \beta _1\!=\!1\!+\!\epsilon ~~\text{ and } \text{ for } \text{ all }~ m\!\ge \! 2,\quad \beta _m\!=\!\gamma a_3\sum _{p\in {\mathbb {M}}_m^3}\beta _{p_1}\beta _{p_2}\beta _{p_3}. \end{aligned}$$

Now, for all $m\ge 1$, denote the error terms $e_m := x_m-{\widetilde{x}}_m$. The first terms are such that $\Vert e_1\Vert _{\mathcal {X}}=\epsilon \Vert {\widehat{x}}_1\Vert _{\mathcal {X}}$ (from (A2)), $\Vert e_2\Vert _{\mathcal {X}}=0$. Moreover, for $m\ge 3$, it follows from (9) that

$$\begin{aligned} e_m(t)&\!=\! \int _0^t S(t\!-\!\tau )\sum _{p\in {\mathbb {M}}_m^3} \left( A_3\big (x_{p_1}(\tau ),x_{p_2}(\tau ),x_{p_3}(\tau )\big )\right. \nonumber \\&\quad \left. -{\widehat{A_3}}\big ({\widetilde{x}}_{p_1}(\tau ),{\widetilde{x}}_{p_2}(\tau ),{\widetilde{x}}_{p_3}(\tau )\big )\right) \mathrm {d}\tau \nonumber \\&= \int _0^t S(t-\tau )\sum _{p\in {\mathbb {M}}_m^3}F_p(\tau ) \mathrm {d}\tau , \end{aligned}$$

(95)

where we set $F_p(\tau )=A_3\big (x_{p_1}(\tau ),x_{p_2}(\tau ),x_{p_3}(\tau )\big ) -A_3\big ({\widetilde{x}}_{p_1}(\tau ),{\widetilde{x}}_{p_2}(\tau ),{\widetilde{x}}_{p_3}(\tau )\big )$. It should be noted that (95) crucially depends on Eq. (53), from which in ${\widehat{{\mathbb {X}}}}$, ${\widehat{A_3}}$ is identical to $A_3$.

Now, replacing $x_{p_i}$ by ${\widetilde{x}}_{p_i}+e_{p_i}$ and exploiting the multi-linearity of $A_3$ yield the following expansion (omitting variable $\tau$ for sake of legibility)

$$\begin{aligned} F_p&= A_3\big ({\widetilde{x}}_{p_1},{\widetilde{x}}_{p_2},e_{p_3}\big ) +A_3\big ({\widetilde{x}}_{p_1},e_{p_2},{\widetilde{x}}_{p_3}\big )\\&\quad +A_3\big (e_{p_1},{\widetilde{x}}_{p_2},{\widetilde{x}}_{p_3}\big ) \\&\quad +A_3\big ({\widetilde{x}}_{p_1},e_{p_2},e_{p_3}\big ) +A_3\big (e_{p_1},{\widetilde{x}}_{p_2},e_{p_3}\big )\\&\quad +A_3\big (e_{p_1},e_{p_2},{\widetilde{x}}_{p_3}\big ) \\&\quad +A_3\big (e_{p_1},e_{p_2},e_{p_3}\big ). \end{aligned}$$

Then, introducing $\psi _m=\beta _m-\phi _m$ for all $m\ge 1$, we prove by induction that (claim $C_m$) $\Vert e_m\Vert _{\mathcal {X}}\le \psi _m\Vert {\widehat{x}}_1\Vert _{\mathcal {X}}^m$ :

• ($m=1$): the claim ($C_1$) is true for $m=1$ by construction;

• ($m\ge 2$):assume that $C_p$ holds for all $p\le m-1$, then using expressions above, it follows that

  $$\begin{aligned}&\Vert e_m\Vert _{\mathcal {X}}\\&\quad \le \gamma a_3 \sum _{p\in {\mathbb {M}}_m^3}\Vert {\widehat{x}}_1\Vert _{\mathcal {X}}^{(p_1+p_2+p_3)} \Big ( \phi _{p_1}\phi _{p_2}\psi _{p_3}\\&\qquad +\phi _{p_1}\psi _{p_2}\phi _{p_3}+\psi _{p_1}\phi _{p_2}\phi _{p_3} +\phi _{p_1}\psi _{p_2}\psi _{p_3}\\&\qquad +\psi _{p_1}\phi _{p_2}\psi _{p_3} +\psi _{p_1}\psi _{p_2}\phi _{p_3} +\psi _{p_1}\psi _{p_2}\psi _{p_3} \Big )\quad \\&\quad \le \gamma a_3 \Vert {\widehat{x}}_1\Vert _{\mathcal {X}}^m \sum _{p\in {\mathbb {M}}_m^3} \Big ( \beta _{p_1}\beta _{p_2}\beta _{p_3}-\phi _{p_1}\phi _{p_2}\phi _{p_3} \big )\\&\quad \le (\beta _m-\phi _m) \Vert {\widehat{x}}_1\Vert _{\mathcal {X}}^m = \psi _m \Vert {\widehat{x}}_1\Vert _{\mathcal {X}}^m, \end{aligned}$$

so that $C_m$ is satisfied. An immediate consequence is that

$$\begin{aligned} \Vert e\Vert _{\mathcal {X}}&\le \sum _{m=1}^\infty \psi _m\Vert {\widehat{x}}_1\Vert _{\mathcal {X}}^m =\sum _{m=1}^\infty \beta _m\Vert {\widehat{x}}_1\Vert _{\mathcal {X}}^m-\sum _{m=1}^\infty \phi _m\Vert {\widehat{x}}_1\Vert _{\mathcal {X}}^m\\&\le \varPhi \left( (1+\epsilon )\Vert {\widehat{x}}_1\Vert _{\mathcal {X}}\right) -\varPhi (\Vert {\widehat{x}}_1\Vert _{\mathcal {X}}), \end{aligned}$$

which concludes the proof.

Proof of generalized result

We assume (A1–A3) in Sect. 5.3 and consider a beam model with a third order nonlinearity $A_3$ for which ${\widehat{{\mathbb {X}}}}$ is not invariant. We assume that a bound $a_3$ of $\Vert A_3\Vert _{{\mathbb {X}}}$ is available, and define $\varPhi (z)=\sum _{m=1}^\infty \phi _m z^m$ as in “Appendix C”.

We denote ${\widehat{A}}_3=\varPi _{\mathbb {X}}A_3$. Since $\varPi _{\mathbb {X}}$ is an orthogonal projection, $\Vert {\widehat{A}}_3\Vert _{{\widehat{{\mathbb {X}}}}}\le \Vert A_3\Vert _{{\mathbb {X}}} \le a_3$, and therefore, as in “Appendix C”, the trajectories ${\widetilde{x}}=\sum _{m=1}^\infty {\widetilde{x}}_m$ and $x=\sum _{m=1}^\infty x_m$ are convergent and satisfy, for all $m\ge 1$,

$$\begin{aligned} \Vert {\widetilde{x}}_m\Vert _{\mathcal {X}}&\le \phi _m \Vert {\widehat{x}}_1\Vert _{\mathcal {X}}^m,\\ \Vert x_m\Vert _{\mathcal {X}}&\le \phi _m \Vert x_1\Vert _{\mathcal {X}}^m\le \phi _m \left( (1+\epsilon )\Vert {\widehat{x}}_1\Vert _{\mathcal {X}}\right) ^m \le \beta _m \Vert {\widehat{x}}_1\Vert _{\mathcal {X}}^m, \end{aligned}$$

where the sequence $(\beta _m)_{m\in {\mathbb {N}}^*}$ was defined in C. Now, for all $m\ge 1$, denote the error terms $e_m := x_m-{\widetilde{x}}_m$. The first terms are such that $\Vert e_1\Vert _{\mathcal {X}}=\epsilon \Vert {\widehat{x}}_1\Vert _{\mathcal {X}}$ (from (A2)), $\Vert e_2\Vert _{\mathcal {X}}=0$. Moreover, for $m\ge 3$, it follows from (9) that

$$\begin{aligned} e_m(t)&= \int _0^t S(t-\tau )\sum _{p\in {\mathbb {M}}_m^3} \left( A_3\big (x_{p_1}(\tau ),x_{p_2}(\tau ),x_{p_3}(\tau )\big )\right. \nonumber \\&\quad \left. -{\widehat{A_3}}\big ({\widetilde{x}}_{p_1}(\tau ),{\widetilde{x}}_{p_2}(\tau ),{\widetilde{x}}_{p_3}(\tau )\big )\right) \mathrm {d}\tau \nonumber \\ \nonumber \\&= \int _0^t S(t-\tau )\sum _{p\in {\mathbb {M}}_m^3}F_p(\tau ) \mathrm {d}\tau \nonumber \\&\quad + \int _0^t S(t-\tau )\sum _{p\in {\mathbb {M}}_m^3}G_p(\tau )\mathrm {d}\tau , \end{aligned}$$

(96)

where we set $F_p(\tau )=A_3\big (x_{p_1}(\tau ),x_{p_2}(\tau ),x_{p_3}(\tau )\big ) -A_3\big ({\widetilde{x}}_{p_1}(\tau ),{\widetilde{x}}_{p_2}(\tau ),{\widetilde{x}}_{p_3}(\tau )\big )$ and

$$\begin{aligned} G_p(\tau )=(A_3-{\widehat{A_3}})\big ({\widetilde{x}}_{p_1}(\tau ),{\widetilde{x}}_{p_2}(\tau ),{\widetilde{x}}_{p_3}(\tau )\big ) \end{aligned}$$

Setting

$$\begin{aligned} \Vert A_3-{\widehat{A}}_3\Vert _{{\widehat{{\mathbb {X}}}}}= & {} \sup _{\begin{array}{c} (a,b,c)\in ({\widehat{{\mathbb {X}}}})^3\\ \Vert a\Vert _{{\widehat{{\mathbb {X}}}}},\Vert b\Vert _{{\widehat{{\mathbb {X}}}}},\Vert c\Vert _{{\widehat{{\mathbb {X}}}}}=1 \end{array}}\!\!\!\!\!\!\Vert A_3(a,b,c)\\&-{\widehat{A}}_3(a,b,c)\Vert _{{\mathbb {X}}}, \end{aligned}$$

we have

$$\begin{aligned} \Vert G_p\Vert _{{\mathcal {X}}} \le \Vert A_3-{\widehat{A}}_3\Vert _{{\widehat{{\mathbb {X}}}}}\Vert {\widehat{x}}_1\Vert _{\mathcal {X}}^{m}\phi _{p_1}\phi _{p_2}\phi _{p_3}. \end{aligned}$$

Then, following the same steps as in “Appendix C”, we introduce $\psi _m=\beta _m-\phi _m$ and obtain by induction that for all $m\ge 1$, $\Vert e_m\Vert _{\mathcal {X}}\le (\psi _m + \frac{\Vert A_3-{\widehat{A}}_3\Vert _{{\widehat{{\mathbb {X}}}}}}{a_3}\phi _m)\Vert {\widehat{x}}_1\Vert _{\mathcal {X}}^m$. An immediate consequence is that

$$\begin{aligned} \Vert e\Vert _{\mathcal {X}}&\le \sum _{m=1}^\infty (\psi _m+ \frac{\Vert A_3-{\widehat{A}}_3\Vert _{{\widehat{{\mathbb {X}}}}}}{a_3}\phi _m)\Vert {\widehat{x}}_1\Vert _{\mathcal {X}}^m \\&\le \sum _{m=1}^\infty \beta _m\Vert {\widehat{x}}_1\Vert _{\mathcal {X}}^m\\&\quad -\left( 1-\frac{\Vert A_3-{\widehat{A}}_3\Vert _{{\widehat{{\mathbb {X}}}}}}{a_3}\right) \sum _{m=1}^\infty \phi _m\Vert {\widehat{x}}_1\Vert _{\mathcal {X}}^m\\&\le \varPhi \left( (1+\epsilon )\Vert {\widehat{x}}_1\Vert _{\mathcal {X}}\right) \\&\quad -\left( 1-\frac{\Vert A_3-{\widehat{A}}_3\Vert _{{\widehat{{\mathbb {X}}}}}}{a_3}\right) \varPhi (\Vert {\widehat{x}}_1\Vert _{\mathcal {X}}), \end{aligned}$$

which concludes the proof.