# Development of Operator Theory in the Capacity Adjustment of Job Shop Manufacturing Systems
## 2 Mathematical Preliminaries  
In this paper, general nonlinear input-output systems of the form:  
\[P:U\to Y, \tag{1}\]  
are considered, where the input and output spaces \(U\) and \(Y\) are two normed linear spaces over the field of complex numbers, endowed, respectively, with norms \(\|\cdot\|_{U}\) and \(\|\cdot\|_{Y}\). The set of all (nonlinear) operators is denoted by \(\mathcal{N}(U,Y)\) and call \(\mathcal{D}(P)\) and \(\mathcal{R}(P)\) the domain and range of \(P\). A (semi)-norm on (a subset of) \(\mathcal{N}(D_{s},Y)\) is defined via:  
\[\|P\|:=\sup_{x,x\in D_{s}\in\mathcal{K}\
eq\bar{x}}\frac{\|P(x)-P(\bar{x})\|_{ Y}}{\|x-\bar{x}\|_{U}}.\]  
Given such a system, our aim is to show the stability of the system, which is formally defined as follows:  
**Definition 1** (Finite-Gain Input-Output Stability). : _An operator \(P\in\mathcal{N}(U_{s},Y_{s})\) with \(U_{s}\subseteq U\) and \(Y_{s}\subseteq Y\) is called finite-gain input-output stable if:_  
1. _it is input-output stable, i.e.,_ \(P(U_{s})\subseteq Y_{s}\)_, and if:_
2. _the norm_ \(\|P\|\) _is well defined and finite, i.e.,_ \(\|P\|<\infty\)_._  
Here, \(U_{s}\) and \(Y_{s}\) are called the stable input subspace and the stable output subspace of the operator \(P\), respectively. Moreover, an operator \(P\) is called causal, stabilizable or unimodular if:  
1. for the projection: \[Q_{T}(x(t))= \begin{cases}x(t),&0\leq t\leq T\\ 0,&T\leq t\leq\infty\end{cases} ,\] it has \(Q_{T}\circ P\circ Q_{T}=Q_{T}\circ P\) for all \(x(t)\in U\) and all \(T\in[0,\infty)\);
2. there exists an operator \(Q:\mathcal{D}(Q)\to\mathcal{D}(Q)\) such that \(P\circ Q\) is input-output stable; (stabilizable)
3. \(P\) is stabilizable and \(P^{-1}\in\mathcal{N}(Y_{s},U_{s})\). (unimodular)  
These three properties allow us to introduce our main tool to show finite-gain input-output stability. **Definition 2** (Right Coprime Factorization (Rcf)). : _Let \(P:\mathcal{D}(P)\to\mathcal{R}(P)\) be a causal and stabilizable operator. We say that \(P\) has a right coprime factorization illustrated in Figure 1, if there exist finite-gain input-output stable and causal operators \(D:\mathcal{D}(P)\to\mathcal{D}(P)\), \(N:\mathcal{D}(P)\to\mathcal{R}(P)\) as well as \(A:\mathcal{R}(N)\to\mathcal{D}(P)\) and \(B:\mathcal{R}(D)\to\mathcal{D}(P)\) such that:_\(D\) is causal, invertible and \(P=N\circ D^{-1}\) holds on \(\mathcal{D}(P)\), and:
2. for the unimodular operator \(M:\mathcal{D}(P)\to\mathcal{D}(P)\), the Bezout identity is: \[A\circ N+B\circ D=M.\] (2)  
Here, \(y\), \(w\), and \(u\) represent the output, quasi-state, and input signal respectively, cf. [16] for details. \(v\) is the reference input, \(l\) is the feedback state, and \(e\) is the error between \(v\) and \(l\). Then, the latter definition allows us to convert the control system (1) to a dynamical system. Theorem 3.1: _If a causal and stabilizable operator \(P:\mathcal{D}(P)\to\mathcal{R}(P)\) has right coprime factorization, then the respective closed loop is finite-gain input-output stable. Moreover, for any reference \(v\), the closed loop simplifies to \(y=N\circ M^{-1}(v)\)._  
Proof: As \(P\) has a right coprime factorization, Figure 1 can be utilized to obtain \(l=A\circ N(w)\) and \(e=B\circ D(w)\). Therefore, we have \(v=l+e=(A\circ N+B\circ D)(w)\). Again, by the right coprime factorization property, the Bezout identity (2) can be applied to obtain \(v=M(w)\) and \(w=M^{-1}(v)\). Hence, by \(y=N(w)=N\circ M^{-1}(v)\), cf. Figure 2, the second assertion follows. Regarding finite-gain input-output stability, unimodularity of \(M\) and finite-gain input-output stability of \(N\) are first utilized to obtain:  
\[P(U_{s})=N\circ M^{-1}(U_{s})\subseteq N(U_{s})\subseteq Y_{s},\]  
which shows input-output stability.