R5 Memory compression, rational realization, and phase-type consequences
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Fix \(T>0\). For the next estimate let \(A\) be a bounded operator on a Banach space \(X\), let \({\mathcal K},\widetilde{\mathcal K}\in L^1((0,T),\mathcal B(X))\), and let \(f\) be continuous. Take continuously differentiable solutions of the displayed equations. This direct bounded-generator estimate retains a sharper exponential constant than the general mild-semigroup estimate in Part II.
Theorem R5.1: Stability of memory compression
Let \(x\) and \(\widetilde x\) solve
and
with the same initial value. Fix \(T>0\), and assume
Set
Then
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Proof of Theorem R5.1
Let \(e=x-\widetilde x\). Integrating the difference equation and changing the order of integration gives
Since \(t\leq T\), Gronwall's inequality gives
Take the supremum over \([0,T]\).
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For the exact closure criterion of Part II, allowing an arbitrary bounded local generator \(B\) does not enlarge the class: differentiating \(P e^{tA}P=e^{tB}P\) at \(t=0\) forces \(B=A_{PP}\) on the retained space. Thus its zero-memory criterion excludes every alternative bounded local generator.
A finite auxiliary realization of a matrix kernel means \({\mathcal K}(t)=B e^{tD}C\), with finite matrices of compatible sizes. Laplace transforms are taken in a right half-plane of absolute convergence. The rational characterization concerns continuous kernels of exponential order, so uniqueness of the ordinary Laplace transform applies.
Theorem R5.2: Rational memory realization
A matrix valued memory transform has a finite auxiliary realization if and only if it is a strictly proper rational matrix function. More precisely,
if and only if
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Proof of Theorem R5.2
The forward implication follows by taking the Laplace transform. The resulting matrix is rational and tends to zero as \(z\) tends to infinity.
Conversely, decompose every entry of a strictly proper rational matrix into partial fractions. A term
is an entry of the resolvent of an \(r\) by \(r\) Jordan block with eigenvalue \(\lambda\). Taking a direct sum of these blocks and choosing the input and output maps entry by entry gives matrices \(B,D,C\) with the required resolvent. Inverse Laplace transformation gives the time domain formula.
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Corollary R5.3: Exponential sum memory
If
then \({\mathcal K}\) has a finite auxiliary realization with one auxiliary block for each exponential. Repeated poles produce polynomial factors in \(t\) and are realized by Jordan blocks.
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Proof of Corollary R5.3
If \(B_j\) and \(C_j\) have an intermediate space of dimension \(d_j\), use a block diagonal matrix \(D\) with blocks \(\lambda_j I_{d_j}\), or the required Jordan blocks when poles repeat. Assemble the matrices \(B_j\) and \(C_j\) into the output and input maps.
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For the survival consequences, \(A\) is a finite Metzler matrix with nonpositive column sums and spectrum in the open left half-plane, \(p_0\) is a probability vector, \(S(t)={{\boldsymbol{1}}}^T e^{tA}p_0\), and \(f(t)=-S'(t)\). The one-phase model has \(\lambda>0\). The survival Laplace identity of Part II also holds for every complex \(s\) with \(\operatorname{Re}s\geq0\): stability gives absolute convergence of \(\int_0^\infty e^{-st}e^{tA}\,dt=(sI-A)^{-1}\), and multiplying by the exit row and \(p_0\) gives the same transform.
Corollary R5.4: Phase type rationality
The Laplace transform of every finite phase type distribution is rational. If \(A\) is diagonalizable, then \(S(t)\) and \(f(t)\) are finite linear combinations of exponential functions. Jordan blocks add polynomial factors in \(t\).
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Proof of Corollary R5.4
The resolvent \((sI-A)^{-1}\) is rational. The time domain statement follows from the Jordan form of \(A\).
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Corollary R5.5: One phase model
If \(A=(-\lambda)\) and \(p_0=1\), then
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Proof of Corollary R5.5
Insert the scalar generator into Theorem E16.2.
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