\right \|<{\varepsilon},\qquad k=1,\dots,r,$$ for every $t \in [0,T_\tau]$.

Notation
--------

For every $n$ in ${\mathbb{N}}$, define the orthogonal projection $$\pi_n: {\mathcal{H}}\ni \psi\mapsto \sum_{k=1}^{n} \langle \phi_k,\psi\rangle
\phi_k \in {\mathcal{H}}.$$ Given a linear operator $Q$ on ${\mathcal{H}}$ we identify the linear operator $\pi_{n} Q \pi_{n}$ preserving ${\mathrm{span}}\{\phi_{1},\ldots, \phi_{n}\}$ with its $n \times n$ complex matrix representation with respect to the basis $(\phi_{1},\ldots, \phi_{n})$. We define $$A^{(n)} = \pi_{n} A \pi_{n} \quad \mbox{ and } \quad B_j^{(n)} = \pi_{n} B_{j} \pi_{n},$$ for every $j=1,\ldots,p$.

Let us introduce the set $\Sigma_{n}$ of spectral gaps associated with the $n$-dimensional Galerkin approximation as $$\Sigma_n = \{|\lambda_l - \lambda_k| \mid l,k = 1, \ldots, n\}.$$

For every $\sigma \geq 0$, every $m\in{\mathbb{N}}$, and every $m\times m$ matrix $M$, let $${\cal E}_\sigma(M)=  (M_{l,k} \delta_{\sigma, |\lambda_l - \lambda_k|})_{l,k=1}^m.$$ The $n\times n$ matrix ${\cal E}_\sigma(B^{(n)}_j)$, $j=1,\dots,p$, corresponds to the “activation” of the spectral gap $\sigma$: it reflects the action of the convexification procedure detailed in the following sections, which annihilates all the matrix elements $(B^{(n)}_j)_{l,k}$ such that $|\lambda_l-\lambda_k|\ne \sigma$.

Define $$\label{sigmabar}
\Xi_n=\left\{ (\sigma,j) \in \Sigma_n \times \{1,\dots,p\}\mid \exists M\in \mathfrak{u}(n)  \mbox{ s.t. }\mathcal{E}_\sigma(B_j^{(N)}) = 
\left(
\begin{array}{c|c}
M&0\\ \hline 0 & *
\end{array}
\right)
\mbox{ for every } N>n
\right\}.$$ The 