**4. Example of SIR-Type Epidemic Models of Inter-Community Clusters**

The stability of the equilibrium points of the epidemic models is an interesting topic which is of grea<sup>t</sup> relevance to healthcare management. See, for instance, [15–19]. Now, we discuss an epidemic based-model related to stabilization under the given framework of the Cauchy's interlacing theorem.

**Example 1.** *Consider the subsequent continuous-time linearized epidemic model with Q community clusters:*

$$\begin{aligned} S\_{i+1} &= \nu\_i (1 - \mu\_i) I\_i - \beta\_{i+1} S\_{i+1} \\\\ \dot{I}\_{i+1} &= \beta\_{i+1} S\_{i+1} - \nu\_{i+1} I\_{i+1} \\\\ \dot{R}\_{i+1} &= \nu\_{i+1} \mu\_{i+1} I\_i \end{aligned} \tag{13}$$

*for i* = 0, 1, ... , *Q* − 1 *with Si*(0) = *Si*0 ≥ 0*, Ii*(0) = *Ii*0 ≥ 0*, Ri*(0) = *Ri*0 ≥ 0*;* ∀*i* ∈ *Q are the initial conditions of the susceptible, infectious and recovered subpopulations, respectively, and <sup>I</sup>*0(0) = 0*,* ν0 = 1*. In this model, the infectious subpopulation Ii of a community i* ∈ *Q* = {1, 2, ... , *Q*} *may infect the population of the neighboring community* (*i* + 1)*. The parameterization is as follows:* β(.) *are the disease transmission rates,* <sup>ν</sup>(.) *are the removal rates and* μ(.) *are the separation constants which bifurcate the disease rate between the local community and the total community. Note that the assumption* μ0 = 1 *implies that the first cluster is not a*ff*ected by contagions from any other cluster, [15]. A simple analysis of the trajectory solution of the first cluster shows that*

$$S\_1(t) = e^{-\beta\_1 t} S\_{10} \to 0 \text{ as } t \to \infty$$

*at an exponential rate, irrespective of the initial conditions, and is definitively bounded,*

$$\begin{aligned} I\_1(t) &= \varepsilon^{-\nu\_1 t} I\_{10} + \int\_0^t \varepsilon^{-\nu\_1 (t-\tau)} S\_1(\tau) d\tau = \varepsilon^{-\nu\_1 t} \Big( I\_{10} + S\_{10} \int\_0^t \varepsilon^{(\nu\_1 - \beta\_1)\tau} d\tau \Big) \\ &= \left( \varepsilon^{-\nu\_1 t} I\_{10} + \frac{\varepsilon^{-\beta\_1 t} - \varepsilon^{-\nu\_1 t}}{\nu\_1 - \beta\_1} S\_{10} \right) \to 0 \text{ as } t \to \infty \end{aligned}$$

*if* ν1 - β<sup>1</sup>*. If* ν1 = β1 *then <sup>I</sup>*1(*t*) = *<sup>e</sup>*<sup>−</sup>ν1*<sup>t</sup>*(*<sup>I</sup>*10 + *<sup>S</sup>*10*<sup>t</sup>*) → 0 *as t* → ∞*. In both cases, the convergence is of exponential order, irrespective of the initial conditions, and is definitively bounded, and*

*<sup>R</sup>*1(*t*) = *R*10 + <sup>ν</sup>1μ1! *t*0 *Ii*(τ)*d*<sup>τ</sup>= *R*10 + <sup>ν</sup>1μ1! *t*0 *e*<sup>−</sup>ν1<sup>τ</sup>*I*10 + *<sup>e</sup>*<sup>−</sup>β1<sup>τ</sup>−*e*<sup>−</sup><sup>ν</sup>1<sup>τ</sup> <sup>ν</sup>1−β<sup>1</sup> *<sup>S</sup>*10*<sup>d</sup>*<sup>τ</sup> = *R*10 + <sup>ν</sup>1μ1! *t*0 *e*<sup>−</sup>ν1<sup>τ</sup>*I*10 + *<sup>e</sup>*<sup>−</sup>β1<sup>τ</sup>−*e*<sup>−</sup><sup>ν</sup>1<sup>τ</sup> <sup>ν</sup>1−β<sup>1</sup> *<sup>S</sup>*10*<sup>d</sup>*<sup>τ</sup> = *R*10 + <sup>ν</sup>1μ1*I*10 1−*e*<sup>−</sup><sup>ν</sup>1*<sup>t</sup>* ν1 + <sup>ν</sup>1μ1*S*10 <sup>ν</sup>1−β<sup>1</sup> *e*<sup>−</sup>β1*<sup>t</sup>* − *e*<sup>−</sup>ν1*<sup>t</sup>* → *R*10 + μ1*I*10 + <sup>ν</sup>1μ1*S*10 <sup>ν</sup>1−β<sup>1</sup> *e*<sup>−</sup>β1*<sup>t</sup>* − *e*<sup>−</sup>ν1*<sup>t</sup>* as *t* → ∞

*if* ν1 - β1 *with solution which is definitively bounded, and, if* ν1 = β1 *then*

$$R\_1(t) = R\_{10} + \nu\_1 \mu\_1 \int\_0^t e^{-\nu\_1 \tau} (I\_{10} + S\_{10} \tau) d\tau$$

$$t = R\_{10} + \mu\_1 \left[ \left( 1 - e^{-\nu\_1 t} \right) I\_{10} + \left( \frac{1}{\nu\_1} \left( 1 - e^{-\nu\_1 t} \right) - te^{-\nu\_1 t} \right) S\_{10} \right] \to R\_{10} + \mu\_1 \left[ I\_{10} + \frac{S\_{10}}{\nu\_1} \right] \text{ as } t \to \infty$$

*with a solution which is definitively bounded. As a result, the total subpopulation at the first cluster is also definitively bounded, and it converges asymptotically to the limit value of the recovered subpopulation.*

*The solution trajectory is also definitively nonnegative since the matrix of dynamics* ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣ −β<sup>1</sup> 0 0 β1 −ν1 0 0 <sup>ν</sup>1μ1 0 ⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦ *is a*

*Metzler matrix and the initial conditions are non-negative. Interpretation shows that the total equilibrium subpopulation is that of the disease-free equilibrium which only has a recovered subpopulation. It can be surprising at a first glance to see that the usual nonlinear term* β1*S*1(*t*)*<sup>I</sup>*1(*t*) *is the susceptible and infectious subpopulations evolutions of the corresponding SIR Kermack-Mcendrick model counterpart is replaced by a linear term. However, for stability purposes, there is no substantial distinct qualitative behavior between both models, since in this case, <sup>S</sup>*1(*t*) = *e*<sup>−</sup>β1! *t*0 *<sup>I</sup>*1(τ)*d*<sup>τ</sup>*S*10 *is strictly decreasing for t* ≥ 0 *and <sup>I</sup>*1(*t*) = *e*! *t*0 (β1*S*1(τ)−ν1)*d*<sup>τ</sup>*I*10 *is also strictly decreasing for t* ≥ 0 *provided that S*10 < ν1β1 *. Now describe the whole model (14) of Nclusters in a more compact way through the individual states xi* = (*Si* , *Ii* ,*Ri*)*T; i* ∈ *N and associated matrices of self-dynamics for each i* ∈ *N and coupled dynamics with the respective preceding cluster* (*i* − 1) ∈ *N:*

$$A\_{i} = \begin{bmatrix} -\beta\_{i} & 0 & 0\\ \beta\_{i} & -\nu\_{i} & 0\\ 0 & \nu\_{i}\mu\_{i} & 0 \end{bmatrix}, \forall i \in \overline{\mathcal{N}};\ A\_{i,i-1} = \begin{bmatrix} 0 & \nu\_{i-1}(1-\mu\_{i-1}) & 0\\ 0 & 0 & 0\\ 0 & 0 & 0 \end{bmatrix} \tag{14}$$
 
$$\forall i(\ge 2) \in \overline{\mathcal{N}};\ A\_{0,-1} = 0$$

*so that (13) is equivalently described as*

$$\dot{\mathbf{x}}\_{i}(t) = A\_{i}\mathbf{x}\_{i}(t) + A\_{i,i-1}\mathbf{x}\_{i-1}(t), \; \mathbf{x}\_{i}(0) = \mathbf{x}\_{i0}(\geq 0); \; \forall i \in \overline{\mathbb{N}} \tag{15}$$

*with <sup>x</sup>*0(*t*) ≡ 0 *for t* ≥ 0*, and compactly, as follows:*

$$
\dot{\mathbf{x}}(t) = A\mathbf{x}(t), \; \mathbf{x}(0) = \mathbf{x}\_0 \tag{16}
$$

$$\begin{aligned} \text{where } \mathbf{x}(t) &= \begin{pmatrix} \mathbf{x}\_1^T(t), \mathbf{x}\_2^T(t), \dots, \mathbf{x}\_N^T(t) \end{pmatrix}^T \text{and} \\\\ A &= \begin{bmatrix} A\_1 & 0 & 0 & \cdots & 0 \\ A\_{21} & A\_2 & & 0 & \cdots & 0 \\\\ 0 & \vdots & A\_{32} & A\_3 0 & \cdots & \vdots & \vdots & \cdots & 0 \\\\ & \vdots & & & \vdots & \vdots & & \\\\ 0 & 0 & \cdots & \cdots & 0 & A\_{NN-1} & A\_N \end{bmatrix} \end{aligned} \tag{17}$$

*Thus, system (15), like (16), (17), can be interpreted as an aggregation model given by starting with the first cluster and successively incorporating the dynamics of the remaining clusters. Now, define the symmetric matrix M* = *M*(*N*) = *ATA. Then, define:*

> =

*M*(1) *A<sup>T</sup>*1 *A*1 *M*(2) = *A<sup>T</sup>* 1 *<sup>A</sup>T*21 0 *A<sup>T</sup>* 2 *A*1 0 *A*21 *A*2 = *A<sup>T</sup>* 1 *A*1 + *<sup>A</sup>T*21 *A*21 *<sup>A</sup>T*21 *A*2 *A<sup>T</sup>* 2 *A*21 *A<sup>T</sup>* 2 *A*2 = *M*(1) *<sup>A</sup>T*21 *A*2 *A<sup>T</sup>* 2 *A*21 *A<sup>T</sup>* 2 *A*2 + *M*(2) *M*(3) = ⎡ ⎢⎢⎢⎢⎢⎢⎢⎢⎣ *A<sup>T</sup>* 1 *<sup>A</sup>T*21 0 0 *A<sup>T</sup>* 2 *<sup>A</sup>T*32 0 0 *A<sup>T</sup>* 3 ⎤ ⎥⎥⎥⎥⎥⎥⎥⎥⎦ ⎡ ⎢⎢⎢⎢⎢⎢⎢⎢⎣ *A*1 0 0 *A*21 *A*2 0 0 *A*32 *A*3 ⎤ ⎥⎥⎥⎥⎥⎥⎥⎥⎦ = ⎡ ⎢⎢⎢⎢⎢⎢⎢⎢⎣ *M*(2) 0 *<sup>A</sup>T*32 *A*3 0 *A<sup>T</sup>* 3 *A*32 *A<sup>T</sup>* 3 *A*3 ⎤ ⎥⎥⎥⎥⎥⎥⎥⎥⎦ + *M*(3) *M*(*N*) = ⎡ ⎢⎢⎢⎢⎣ *M*(*<sup>N</sup>*−<sup>1</sup>) *m*(*<sup>N</sup>*−<sup>1</sup>) *m*(*<sup>N</sup>*−<sup>1</sup>) *T A<sup>T</sup>* 3 *A*3 ⎤ ⎥⎥⎥⎥⎦ + *M*(*N*) (18) *M*(2) = *<sup>A</sup>T*21 *A*21 0 0 0 , *M*(3) = ⎡ ⎢⎢⎢⎢⎢⎢⎢⎢⎣ *<sup>A</sup>T*21 *A*21 0 0 0 *<sup>A</sup>T*32 *A*32 0 0 00 ⎤ ⎥⎥⎥⎥⎥⎥⎥⎥⎦ *M*(*N*) = ⎡ ⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ *<sup>A</sup>T*21 *A*21 0 ··· 0 0 *<sup>A</sup>T*32 *A*32 0 ··· 0 0 ··· ... 0 ··· ··· *A<sup>T</sup> N*, *<sup>N</sup>*−1*<sup>A</sup> N*,*N*−1 0 ⎤ ⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

*if N* ≥ 2 *and M*(1) = 0*. By inspection of (18), one concludes that M*(*n*) = 0 \$*n n*=2 sup<sup>2</sup>*m*(*n*) *for i* = 2, 3, ... , *N which concludes that, if* - *An*−1,*<sup>n</sup> N n*=2 → 0 *as N* → ∞ *in such a way that* - *m*(*j*) *N j*=2 → 0 *as N* → <sup>∞</sup>*, for instance, if the convergence is at exponential rate, then* lim *N*→∞ \$ *N n*=2 sup<sup>2</sup>*m*(*n*) ≤ *Km* < + ∞*. Furthermore, if A*1 *is a stability matrix of absolute stability abscissa which are su*ffi*ciently larger than Km, then the dynamic system (16),(17) is globally asymptotically stable according to Lemma 5. In particular, note that if there is any pair of stable complex conjugate eigenvalues <sup>s</sup>*1,2 = μ ± *i*ν (μ < 0 , ν > 0) *for the first cluster, then there is a submatrix of <sup>M</sup>*(1)*,*

$$\mathcal{M}^{\varepsilon(1)} = A\_1^{sT} A\_1^s = \begin{bmatrix} \mu & -\nu \\ \nu & \mu \end{bmatrix} \begin{bmatrix} \mu & \nu \\ -\nu & \mu \end{bmatrix} = \begin{bmatrix} \mu^2 + \nu^2 & 0 \\ 0 & \mu^2 + \nu^2 \end{bmatrix} = \begin{bmatrix} \lambda\_1^{(1)} & 0 \\ 0 & \lambda\_2^{(1)} \end{bmatrix}$$

*in the real canonical form. Since* μ(1) = μ < 0 *then* λ(1) 2 = λ(1) 2 = μ2 + ν2 > 0*. The maximum and minimum corresponding eigenvalues of Ms*(2) *are no less than* λ(1) 2 *and no larger than* λ(1) 1 *, respectively, from Cauchy's* *interlacing theorem. Since the eigenvalues are continuous functions of the matrix entries, and since Ms*(*n*) *is positive definite any critical eigenvalue of a member Ms*(*j*) *of the sequence,* - *Ms*(*n*) ∞ *n*=2 *implies a lot of stability of the corresponding As j. This is avoided if* lim *N*→∞ \$ *N n*=2 sup<sup>2</sup>*m*(*n*) ≤ *Km* < + <sup>∞</sup>*, implying also that* - *m*(*n*) *N* 2 → 0 *as N* → <sup>∞</sup>*, and the sequence of separation constants* % μ*n* &*N n*=2 → 1 *as N* → ∞ *if Km is small enough related to* μ*. The physical interpretation relies on the fact that the contagion link from a cluster to the next one is weakened su*ffi*ciently quickly as the cluster index increases, due to the fact of the numbers of the infected subpopulations are rapidly decreasing as the cluster index increases at a su*ffi*ciently large rate.*

### **5. Dynamic Linear Discrete Aggregation Model with Output Delay and Linear Feedback Control**

In this section, the convergence results of Section 3 are applied to a dynamic discrete system which is built by the aggregation of discrete dynamic subsystems subject to linear output feedback control. Since we are dealing with a physical system, it turns out that the formalism of Section 2 can be developed by invoking conditions related to real symmetric systems, rather than to complex Hermitian ones, when necessary. It would su ffice to describe the state by expressing the matrix of dynamics in the real canonical form and to transform the control and output matrices by the appropriate similarity matrix. The necessary mathematical proof is given in Appendix A.

Consider the aggregation linear discrete dynamic system subject to *r* point delays under linear output-feedback:

$$\mathbf{x}^{0(n+1)} = A^{0(n)}\mathbf{x}^{(n)} + \hat{A}^{0(n)}\mathbf{x}^{(n)} + \sum\_{j=1}^{r} B\_j^{(n)} y^{(n-j)} + B\_0^{(n)} u^{(n)} \tag{19}$$

$$\mathbf{y}^{(n)} = \mathbf{C}^{(n)} \mathbf{x}^{(n)} \tag{20}$$

$$u^{(n)} = \sum\_{j=0}^{r\_n} K\_j^{(n)} y^{(n-j)} = \sum\_{j=0}^{r\_n} K\_j^{(n)} \mathbb{C} x^{(n-j)} \tag{21}$$

∀*n* ∈ **Z**0+, with initial conditions *x*<sup>0</sup>(0) = *x*0, where {*ni*} ∞ *i*=0 is a sequence of positive integer numbers, *x*(*n*) ∈ **R** \$*n i*=0 *ni* is the "a priori" vector state at the *n*-th iteration, *x*<sup>ˆ</sup>(*n*) ∈ **R***nn* is the aggregated "a priori" new substate at the *n*-th iteration (that is basically, the new information needed to update the state vector and its dimension) and *x*<sup>0</sup>(*n*+<sup>1</sup>) ∈ **R** \$*n i*=0 *ni* is the "a priori" whole state at the (*n* + 1)-th iteration. Also, *x*<sup>0</sup>(*n*) ∈ **R** \$*<sup>n</sup>*−<sup>1</sup> *i*=0 *ni* , *u*(*n*) ∈ **R** \$*n i*=0 *mi* and *y*(*n*) ∈ **R** \$*n i*=0 *pi* are, respectively, the "a priori" input and measurable output vectors at the *n*-th iteration and *r* ⊂ **Z**+ is a sequence of delays influencing the global dynamics. The sequences of matrices of dynamics - *A*<sup>0</sup>(*n*) ∞ *<sup>n</sup>*=0 and - *A*ˆ0(*n*) ∞ *<sup>n</sup>*=0 , control *B*(*n*) 0 ∞ *<sup>n</sup>*=0, output-state coupling *B*(*n*) *j* ∞ *<sup>n</sup>*=0 for *j* ∈ *r* are of members *A*<sup>0</sup>(*n*) ∈ **R**( \$*n i*=0 *ni*) ×( \$*n i*=0 *ni*) , *A*<sup>ˆ</sup>(*n*) ∈ **R**( \$*n i*=0 *ni*) ×*nn* , and *B*(*n*) 0 ∈ **R**( \$*n i*=0 *ni*) ×( \$*<sup>n</sup> i*=0 *mi*) and *B*(*n*) *j* ∈ **R**( \$*n i*=0 *ni*) ×( \$*<sup>n</sup>*−*<sup>j</sup> i*=0 *pi*) for *j* ∈ *r* and the output matrix *C*(*n*) ∈ **R**( \$*n i*=0 *pi*)×( \$*n i*=0 *ni*). The sequences of matrices *K*(*n*) *j* ∞ *<sup>n</sup>*=0 , with *K*(*n*) *j* ∈ **R**( \$*<sup>n</sup> i*=0 *mi*) ×( \$*n i*=0 *pi*) for *j* ∈ *r* ∪ {0}, are the output-feedback control gains which generate the control law sequence - *u*(*n*) ∞ .

*<sup>n</sup>*=0 The dynamics of the new dynamics at the (*n* + 1)-th iteration aggregated to the former global aggregation system of state *x*(*n*) obtained at the *n*-th iteration, are assumed to be described by:

$$\hat{\mathbf{x}}^{(n+1)} = \hat{A}^{(n+1)}\mathbf{x}^{(n)} + \left(\hat{D}^{(n)} + \overline{\hat{D}}^{(n)}\right)\hat{\mathbf{x}}^{(n)} + \sum\_{j=1}^{r} \left(\hat{B}\_{j}^{a(n)}y^{(n-j)} + \hat{B}\_{j}^{(n)}\hat{y}^{(n-j)}\right) + \hat{B}\_{0}^{(n)}\hat{u}^{(n)} \tag{22}$$

$$\mathfrak{F}^{(n)} = \mathbb{C}^{(n)} \mathfrak{X}^{(n)} \tag{23}$$

$$\mathfrak{A}^{(n)} = \sum\_{j=0}^{r} \left( \mathcal{K}\_j^{(n)} \mathfrak{G}^{(n-j)} + \mathcal{K}\_j^{a(n)} \mathfrak{z}^{(n-j)} \right) = \sum\_{j=0}^{r} \left( \mathcal{K}\_j^{(n)} \mathcal{C}^{(n)} \mathfrak{x}^{(n-j)} + \mathcal{K}\_j^{a(n)} \mathcal{C}^{(n)} \mathfrak{x}^{(n-j)} \right) \tag{24}$$

∀*n* ∈ **Z**0+, where *x*<sup>ˆ</sup>(*n*+<sup>1</sup>) ∈ **R***nn*+<sup>1</sup> is the "a posteriori" state of the aggregated subsystem at the (*n* + 1)-th iteration whose "a priori" value is *x*<sup>ˆ</sup>(*n*) ∈ **R***nn* , *y*<sup>ˆ</sup>(*n*) ∈ **R***pn* , *u*<sup>ˆ</sup>(*n*) ∈ **R***mn* , *A*<sup>ˆ</sup>(*n*+<sup>1</sup>) ∈ **<sup>R</sup>***nn*+1<sup>×</sup>( \$*n i*=0 *ni*) , *B*<sup>ˆ</sup>(*n*) 0 ∈ **R***nn*+1×*mn* , *B*<sup>ˆ</sup>(*n*) *j* ∈ **R***nn*+1×*pn*−*<sup>j</sup>* , *B*ˆ *a*(*n*) *j* ∈ **R***nn*+1<sup>×</sup>( \$*<sup>n</sup>*−*<sup>j</sup> i*=0 *pi*) for *j* ∈ *r*; *C*<sup>ˆ</sup>(*<sup>n</sup>*−*j*) ∈ **R***pn*−*j*×*nn*−*<sup>j</sup>* , *K*<sup>ˆ</sup>(*n*) *j* ∈ **R***mn*+1×*pn*−*<sup>j</sup>* , *K*ˆ *a*(*n*) *j* ∈ **R***mn*+1<sup>×</sup>( \$*<sup>n</sup>*−*<sup>j</sup> i*=0 *pi*) for *j* ∈ *r* ∪ {0}, and *D*ˆ (*n*) , *D* ˆ (*n*) ∈ **R***nn*+1×*nn* and *B*<sup>ˆ</sup>(*n*) 0 ∈ **R***nn*+1×*mn*+<sup>1</sup> , *B*<sup>ˆ</sup>(*n*) *j* ∈ **R***nn*+1×*pn*−*<sup>j</sup>* for *j* ∈ *r*; ∀*n* ∈ **Z**0+.

Note that the aggregated subsystem (22)–(24) is coupled to the former global state *x*(*n*) describing the total system's dynamics prior to the aggregation action. It can be seen that the coupling terms do not necessary demonstrate infinite memory requirements as *n* tends to infinity, since the matrices *<sup>A</sup>*(*n*+<sup>1</sup>), *B*ˆ *a*(*n*) *j* and *K*ˆ *a*(*n*) *j* can contain nonzero columns associated with the most recent state/output data related to the previous aggregation system; see, for instance, [12]. Note also that, due to the coupling between the a priori whole state at the *n*-th iterations with the a priori new aggregated substate, it can happen that the a posteriori vector after the new aggregated substate has a higher dimension than its a priori version. The various dynamics, control and output matrices have the appropriate orders.

After incorporating the control law, we can write this whole system of extended states *x*(*n*) = *x*<sup>0</sup>(*n*) *T* , *x*<sup>ˆ</sup>(*n*) *T T* ∈ **R** \$*n i*=0 *ni* ; ∀*n* ∈ **Z**0+ in a compact way:

$$\begin{aligned} \mathbf{x}^{(n+1)} &= \begin{bmatrix} \mathbf{x}^{0(n+1)} \\ \mathbf{x}^{(n+1)} \end{bmatrix} = \begin{bmatrix} A^{0(n)} + \mathcal{B}\_{0}^{(n)} K\_{0}^{(n)} \mathbf{C}^{(n)} & A^{0(n)} \\ \vdots & \mathcal{A}^{(n+1)} + \mathcal{B}\_{0}^{(n)} \mathcal{K}\_{0}^{(n)} \mathbf{C}^{(n)} & \mathcal{D}^{(n)} + \widetilde{\mathcal{D}}^{(n)} + \mathcal{B}\_{0}^{(n)} \mathcal{K}\_{0}^{(n)} \mathbf{C}^{(n)} \end{bmatrix} \begin{bmatrix} \mathbf{x}^{(n)} \\ \mathbf{x}^{(n)} \end{bmatrix} \\ &+ \boldsymbol{\Sigma}\_{j-1}' \begin{bmatrix} \begin{Bmatrix} \mathcal{B}\_{j}^{(n)} + \mathcal{B}\_{0}^{(n)} K\_{j}^{(n)} \end{Bmatrix} \mathbf{C}^{(n)} & \mathbf{0} \\ \begin{Bmatrix} \mathcal{B}\_{j}^{a(n)} + \mathcal{B}\_{0}^{(n)} \mathcal{K}\_{j}^{a(n)} \end{bmatrix} \mathbf{C}^{(n-j)} & \begin{Bmatrix} \mathcal{B}\_{j}^{(n)} + \mathcal{B}\_{0}^{(n)} K\_{j}^{(n)} \end{Bmatrix} \mathbf{C}^{(n-j)} \end{bmatrix} \mathbf{x}^{(n-j)}, \end{aligned} \tag{25}$$

so that *<sup>x</sup>*(*n*), *x*<sup>0</sup>(*n*+<sup>1</sup>) ∈ **R** \$*n i*=0 *ni* and *x*<sup>ˆ</sup>(*n*) ∈ **R***nn* and *x*<sup>ˆ</sup>(*n*+<sup>1</sup>) ∈ **R***nn*+<sup>1</sup> imply that *x*(*n*+<sup>1</sup>) ∈ **R** \$*n*+<sup>1</sup> *i*=0 *ni* , *x*(*n*+<sup>1</sup>) = *x*<sup>0</sup>(*n*+<sup>1</sup>) *T* , *x*<sup>ˆ</sup>(*n*+<sup>1</sup>) *T T* ∈ **R** \$*n*+<sup>1</sup> *i*=0 *ni* .

In order to construct a state vector which includes delayed dynamics, we now define the modified extended state *x*(*n*) defined by *x*(*n*) = *x*(*n*) *T* , *x*(*<sup>n</sup>*−<sup>1</sup>) *T* , ... , *x*(*<sup>n</sup>*−*<sup>r</sup>*) *T T* ∈ **R** \$*n j*=*n*−*<sup>r</sup>* \$*j i*=0 *ni* ; ∀*n* ∈ **Z**0+. Thus, one determines from (25) that:

$$
\overline{\mathfrak{X}}^{(n+1)} = \overline{A}^{(n)} \overline{\mathfrak{X}}^{(n)}; \; \forall n \in \mathbb{Z}\_{0+} \tag{26}
$$

where

$$
\mathbf{A}^{(n)} = \begin{bmatrix}
\mathbf{A}^{(n)} + \mathbf{I}\_{\mathbf{b}}^{(n)} \mathbf{A}\_{\mathbf{b}}^{(n)} \circ \mathbf{c}^{(n)} & \mathbf{A}^{(n)} & \left(\mathbf{I}\_{\mathbf{b}}^{(n)} + \mathbf{I}\_{\mathbf{b}}^{(n)} \mathbf{A}\_{\mathbf{b}}^{(n)}\right) \circ \mathbf{c}^{(n-1)} & \mathbf{0} & \dots & \left(\mathbf{I}\_{\mathbf{b}}^{(n)} + \mathbf{I}\_{\mathbf{b}}^{(n)} \mathbf{A}\_{\mathbf{b}}^{(n)}\right) \circ \mathbf{c}^{(n-1)} \\\vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\\mathbf{A}^{(n)} + \mathbf{I}\_{\mathbf{b}}^{(n)} \circ \mathbf{c}^{(n)} & \mathbf{0} & \dots & \mathbf{0} & \dots & \mathbf{0} \\\vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\\mathbf{C}^{(n)} & \mathbf{0} & \dots & \mathbf{0} & \dots & \mathbf{0} \\\vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\\mathbf{C}^{(n)} & \mathbf{0} & \dots & \mathbf{0} & \dots & \mathbf{0} \\\end{bmatrix}
$$

∀*n* ∈ **Z**0+, with *A*(*n*) ∈ **R**( \$*n*+<sup>1</sup> *j*=*n*−*<sup>r</sup>* \$*j i*=0 *ni*)×( \$*n j*=*n*−*<sup>r</sup>* \$*j i*=0 *ni*). Now, consider the symmetric matrices:

$$
\overline{\mathcal{M}}^{(n)} = \overline{\mathcal{A}}^{(n)^{\top}} \overline{\mathcal{A}}^{(n)} = \begin{bmatrix}
\overline{\mathcal{A}}^{(n)^{\top}} \overline{\mathcal{A}}^{(n)} & \overline{\mathcal{A}}^{(n)^{\top}} \overline{\mathcal{B}}^{(n)} \\
\overline{\mathcal{B}}^{(n)^{\top}} \overline{\mathcal{A}}^{(n)} & \overline{\mathcal{B}}^{(n)^{\top}} \overline{\mathcal{B}}^{(n)}
\end{bmatrix}
$$

$$
\mathcal{M}^{(n)} = \begin{bmatrix}
A^{0(n+1)^{\top}} A^{0(n+1)} & A^{0(n+1)^{\top}} B^{0(n+1)} \\
B^{0(n+1)^{\top}} A^{0(n+1)} & B^{0(n+1)^{\top}} B^{0(n+1)}
\end{bmatrix}
$$

$$
\in \mathbb{R}^{\left(\sum\_{i=0}^{n+1} n\_{i} + \sum\_{j=n-r}^{n} \left(\sum\_{i=0}^{j} n\_{i} + n\_{j}\right)\right) \times \left(\Sigma\_{i=0}^{n+1} n\_{i} + \Sigma\_{j=r-r}^{n} \left(\Sigma\_{i=0}^{j} n\_{i} + n\_{j}\right)\right) \\
\vdots
\end{bmatrix} \tag{28}
$$

where the relations between the a priori dynamics of the new iteration after the aggregation of a new substate to the whole dynamics with the a posteriori dynamics of the former iteration are given by:

$$A^{0(n+1)} = \begin{bmatrix} A^{(n)} \ 0\_{\left(\sum\_{i=0}^{n+1} n\_i\right) \times n\_{n+1}} \\ A^{0(n)} + B\_0^{(n)} K\_0^{(n)} C^{(n)} & A^{0(n)} & 0 \\ A^{(n+1)} + B\_0^{(n)} K\_0^{(n)} C^{(n)} & D^{(n)} + \overline{D}^{(n)} + B\_0^{(n)} K\_0^{(n)} C^{(n)} & 0 \end{bmatrix} \in \mathbf{R}^{(\sum\_{i=0}^{n+1} n\_i) \times (\sum\_{i=0}^{n+1} n\_i)} \tag{29}$$

$$\begin{aligned} \boldsymbol{B}^{0(n+1)} - \boldsymbol{B}^{(n)} &= \begin{bmatrix} \left(\boldsymbol{B}\_{1}^{(n)} + \boldsymbol{B}\_{0}^{(n)}\boldsymbol{K}\_{1}^{(n)}\right)\mathbb{C}^{(n-1)} & \boldsymbol{0} & \dots & \left(\boldsymbol{B}\_{j}^{(n)} + \boldsymbol{B}\_{0}^{(n)}\boldsymbol{K}\_{r}^{(n)}\right)\mathbb{C}^{(n-r)} & \boldsymbol{0} \\ \left(\boldsymbol{B}\_{1}^{n(n)} + \boldsymbol{B}\_{0}^{(n)}\boldsymbol{K}\_{1}^{(n)}\right)\mathbb{C}^{(n-1)} & \left(\boldsymbol{B}\_{1}^{(n)} + \boldsymbol{B}\_{0}^{(n)}\boldsymbol{K}\_{1}^{(n)}\right)\mathbb{C}^{(n-1)} & \left(\boldsymbol{B}\_{r}^{(n)} + \boldsymbol{B}\_{0}^{(n)}\boldsymbol{K}\_{r}^{(n)}\right)\mathbb{C}^{(n-r)} & \left(\boldsymbol{B}\_{j}^{(n)} + \boldsymbol{B}\_{0}^{(n)}\boldsymbol{K}\_{r}^{(n)}\right)\mathbb{C}^{(n-r)} \end{bmatrix} \end{aligned} \tag{30}$$

∀*n* ∈ **Z**0+. which are built in order to complete a square a priori matrix of dynamics of the (*n* + <sup>1</sup>)- the aggregated system which was obtained after the aggregation of the (*n* + 1)-th subsystem.

The stability of the aggregation dynamic system (19) to (24) under discrete delays is now discussed via the modified extended system (26), subject to (27), which can be obtained via Lemmas 7,8 from the convergence of the symmetric matrix (28), subject to (29),(30). The following result holds:

**Theorem 1.** *The following properties hold:*


It is of interest to now discuss how the stability properties of the aggregation system of Theorem 1 can be guaranteed or addressed by the synthesis of the basic controller (21) on the current aggregated system, and how its updated rule (24) can be applied to the new aggregated subsystem to generate the aggregated system for the next iteration step. This discussion invokes conditions to guarantee that the equation of dimensionally compatible real matrices

$$BKC = A\_{\rm ll} - A \tag{31}$$

is solvable in *K* for a given quadruple (*<sup>A</sup>*, *B* , *C*, *Am*) with *A* and *Am* being square, *Am* being convergen<sup>t</sup> (basically stable in the discrete context) and defining the closed-loop system dynamics after linear output-feedback control *u* = *Ky* = *KCx* via the linear stabilizing controller of gain *K*; *A*, *B* and *C* are the

open-loop dynamics (i.e., the one being go<sup>t</sup> for *K* = 0) and *B* and *C* are the control and output matrices. Equation (31) is written in equivalent vector form for the unknown *K* as follows:

$$\left(B \otimes \mathbb{C}^{T}\right) \text{vec}\left(K\right) = \text{vec}\left(A\_{\mathfrak{m}} - A\right) \tag{32}$$

It turns out that (31) is solvable in *K* if and only if (32) is solvable in *vec* (*K*), that is, if*rankB* ⊗ *CT*= *rank*" *B* ⊗ *C<sup>T</sup>* , *vec*(*Am* − *A*)# according to the Rouché-Froebenius theorem for solvability of linear systems of algebraic equations. Note that if *Am* satisfies the constraint *Am* = *EA*, for some square matrix *E* of the same order as *A*, then *Am*2 < 1 (so that *Am* is convergent) if *E*2 < 1/*A*2. In particular, if *Am* = ρ*A* with ρ ∈ **R** then *Am* is convergen<sup>t</sup> if ρ < 1/*A*2. A preliminary technical result concerning the solvability if the concerned algebraic system (31), or equivalently (32), is (either indeterminate or determinate) compatible to be then used follows:

**Lemma 9.** *Assume that A* ∈ *R<sup>n</sup>*<sup>×</sup>*n, B* ∈ *R<sup>m</sup>*×*<sup>n</sup> and C* ∈ *Rp*<sup>×</sup>*n. Then, the following properties hold:*

*(i) linear output-feedback controller exists which stabilizes the closed-loop matrix of dynamics Am* = *EA for some E with E*2 < 1/*A*<sup>2</sup>*, [5,7], which satisfies the rank constraint:*

$$\text{rank}\left(\boldsymbol{B}\otimes\boldsymbol{C}^{T}\right) = \text{rank}\left[\boldsymbol{B}\otimes\boldsymbol{C}^{T}, \left(\boldsymbol{I}\otimes\boldsymbol{A}^{T}\right)\text{vec}\left(\boldsymbol{E}\right) - \text{vec}\left(\boldsymbol{A}\right)\right] \tag{33}$$

*If (33) holds, then the set of stabilizing linear-output feedback controllers of gains K which solve (32), equivalently (31), which is a compatible algebraic linear system, for Am* = *EA, are given by*

$$\text{vec}\left(K\right) = \left(\boldsymbol{B}\otimes\boldsymbol{C}^{T}\right)^{\dagger}\left[\left(\boldsymbol{I}\otimes\boldsymbol{A}^{T}\right)\text{vec}\left(\boldsymbol{E}\right) - \text{vec}\left(\boldsymbol{A}\right)\right] + \left[\boldsymbol{I} - \left(\boldsymbol{B}\otimes\boldsymbol{C}^{T}\right)^{\dagger}\left(\boldsymbol{B}\otimes\boldsymbol{C}^{T}\right)\right]k\_{w}\tag{34}$$

*with kw being any arbitrary real vector of the same dimension as vec* (*K*)*. Assume that B* ⊗ *C<sup>T</sup>*† = *B* ⊗ *C<sup>T</sup>*−<sup>1</sup> *(a necessary condition being* min(*<sup>m</sup>*, *p*) ≥ *n). Then (32) for Am* = *EA is a compatible determinate, and the unique solution to (33) is*

$$\text{vec}\left(K\right) = \left(B \otimes \mathbb{C}^{T}\right)^{\dagger} \left[\left(I \otimes A^{T}\right)\text{vec}\left(E\right) - \text{vec}\left(A\right)\right] \tag{35}$$

*If Am* = ρ*A with* ρ < 1/*A*2 *then (33), (34) and (35) become, in particular,*

$$\operatorname{rank}\big(\mathcal{B}\otimes\mathbb{C}^{T}\big)=\operatorname{rank}\big[\mathcal{B}\otimes\mathbb{C}^{T},\,(\rho-1)\operatorname{vec}(A)\big]\tag{36}$$

$$\text{vec}\,(K) = (\rho - 1) \Big(B \otimes \mathbb{C}^T\Big)^\dagger \text{vec}(A) + \left[I - \left(B \otimes \mathbb{C}^T\right)^\dagger \Big(B \otimes \mathbb{C}^T\Big)\right]k\_w\tag{37}$$

*and*

$$\text{vec}\left(K\right) = \left(B \otimes \mathbb{C}^{T}\right)^{\dagger} \left(B \otimes \mathbb{C}^{T}\right)^{\dagger} \text{vec}\left(A\right)\tag{38}$$

*(ii) Assume that Am* = *EA and*

$$\text{rank}\left[B\otimes\mathbb{C}^{T}, \left(I\otimes A^{T}\right)\text{vec}(E) - \text{vec}(A)\right] = \text{rank}\left(B\otimes\mathbb{C}^{T}\right) + 1\tag{39}$$

Then, (32), equivalently (31), is an algebraically incompatible system of equations, and

$$\text{vec}\left(K\right) = \left(B \otimes \mathbb{C}^{T}\right)^{\dagger} \left[\left(I \otimes A^{T}\right)\text{vec}\left(E\right) - \text{vec}\left(A\right)\right], \text{ i.e.}\tag{40}$$

*i.e., Equation (34) for kw* = 0*, is the best least-squares approximated solution to (32) in the sense that the corresponding controller gain minimizes the norm error BKC* + *A* − *Am*22*. If (39) holds for any Am of the form Am* = *EA then there is no solution to (31) in K; only best approximation solutions exist.*

Particular cases of interest which are well-known from basic Control Theory (see e.g., [13]) are:


Lemma 9 is useful to guarantee the relevant results of Theorem 1 in terms of the controller gains choices under certain algebraic solvability conditions. This feature is addressed in the subsequent result:
