Non-homogeneous state-affine systems: matrix polynomials, the system equation, the governing bounds

matPolyEval

For natural numbers $N$ and $r$, given a family of $N \times N$ real matrices $P = (P_j)_{j \in \mathrm{Fin}\,r}$ indexed by $j \in \{0,\dots,r-1\}$, and a real number $z$, this defines the $N \times N$ real matrix

$$p_P(z) \;=\; \sum_{j=0}^{r-1} z^{j}\, P_j ,$$

the scalar multiples being taken entrywise. The exponent is the numeral value of the index $j$. Both $N$ and $r$ are implicit and arbitrary natural numbers: if $r = 0$ the sum is empty and $p_P(z)$ is the zero matrix for every $z$; if $N = 0$ the value is the unique $0 \times 0$ matrix. The $j = 0$ term is $z^0 \cdot P_0$, which equals $P_0$ even at $z = 0$, since $0^0 = 1$ here. Nothing restricts $z$ to any interval.

vecPolyEval

Symmetrically, for a family $Q = (Q_j)_{j=0}^{s-1}$ of vectors of the Euclidean space $\mathbb{R}^N$ (carrying the $\ell^2$ norm) and a real $z$, this is the vector

$$q_Q(z) \;=\; \sum_{j=0}^{s-1} z^{j}\, Q_j .$$

Again $s = 0$ gives the zero vector identically, and $z$ ranges over all of $\mathbb{R}$.

IsSASSolution

For implicit naturals $N, r, s$, a matrix family $P = (P_j)_{j<r}$, a vector family $Q = (Q_j)_{j<s}$, a real-valued sequence $z : \mathbb{N} \to \mathbb{R}$ and a sequence $x : \mathbb{N} \to \mathbb{R}^N$, this is the proposition

$$\forall k \in \mathbb{N}, \qquad x_k \;=\; p_P(z_k)\, x_{k+1} \;+\; q_Q(z_k),$$

where $p_P(z_k)\,x_{k+1}$ is the matrix-vector product: the Euclidean vector $x_{k+1}$ is transported to the plain coordinate tuple, multiplied on the left by the matrix $p_P(z_k)$, and transported back to Euclidean space (this transport is the identity on coordinates and changes nothing but the norm carried by the type). Note the index direction: the value at step $k$ is determined by the value at step $k+1$, so the recurrence runs from larger indices toward $0$, and the condition is imposed at every $k \in \mathbb{N}$, $k = 0$ included. No initial or terminal condition is imposed, no boundedness of $x$ or $z$ is assumed, and in particular $z_k$ is unconstrained — it need not lie in $[-1,1]$. If $r = 0$ and $s = 0$ the condition reduces to $x_k = 0$ for all $k$; if $N = 0$ it holds vacuously for any $z, x$.

MatPolyOpBound

For a matrix family $P = (P_j)_{j<r}$ and a real number $K$, this is the proposition

$$\forall z \in [-1,1],\; \forall v \in \mathbb{R}^N: \quad \bigl\lVert p_P(z)\, v \bigr\rVert_2 \;\le\; K\,\lVert v \rVert_2 .$$

The quantification over $z$ is restricted to the closed interval $[-1,1]$, endpoints included, and $K$ is a single constant independent of both $z$ and $v$; equivalently, $K$ dominates the $\ell^2$ operator norm of $p_P(z)$ uniformly on $[-1,1]$. The bound is non-strict. $K$ is not assumed nonnegative, though for $N \ge 1$ nonnegativity follows by taking $v \ne 0$; for $N = 0$ the statement holds for every $K$, including negative ones. No claim is made that such a $K$ exists.

VecPolyBound

For a vector family $Q = (Q_j)_{j<s}$ and a real $K$:

$$\forall z \in [-1,1]: \quad \bigl\lVert q_Q(z) \bigr\rVert_2 \;\le\; K,$$

again non-strict, over the closed interval, with $K$ uniform in $z$ and not assumed nonnegative (nonnegativity is forced, since the left side is a norm). As above, this is a predicate on $(Q,K)$, not an existence assertion.

None of these five declarations refers to any of the reservoir, contraction, weighting, fading-memory or squashing-function notions from the imported definitions file; nothing here mentions echo state networks, and no theorem is stated — all five are definitions only.