On the unconditional robust stability for the multidelays interval coefficient control system (Q2748227)

Consider the linear differential systems with interval coefficients NEWLINENEWLINE\[NEWLINE\dot x_i(t)=\sum^n_{j=1} a_{1ij}^{(0)}x_j(t)+ \sum^N_{r=1} \sum^n_{j=1} a_{1ij}^{(r)}(t-\tau_{1ij}^{(r)})+ \sum^m_{f=1} b_{1if}^{(0)}u_j(t), \quad i=1,\dots,n\tag{1}NEWLINE\]NEWLINE where \(\tau_{1ij}^{(r)} \geq 0\), \(p_{1ij}^{(r)}\leq a_{1ij}^{(r)}\leq q_{1ij}^{(r)}\), \(p_{2if}\leq b_{1if}^{(0)}\leq q_{2if}\), \(i,j=1,\dots,n\), \(f=1,\dots,m\), \(r=0, \dots, N\), Let NEWLINE\[NEWLINEA=\{a_{ij}\},\;B=\{b_{if}\},\;a_{ij}=\frac 12\sum^N_{r=0} (p_{1ij}^{(r)}+q^{(r)}_{1ij}),\;b_{if}=\frac 12(p_{2if}+q_{2if}).NEWLINE\]NEWLINE Suppose that the matrix pair \((A,B)\) is completely controllable. Then there exists a control in the form NEWLINE\[NEWLINEu(t)=-Kx(t),\tag{2}NEWLINE\]NEWLINE for which the quadratic form NEWLINE\[NEWLINEI=\int^\infty_0 [x^T(t)Qx(t)+u^T(t)Ru(t)]\,dtNEWLINE\]NEWLINE has a minimal value on closed-loop system trajectories. Here \(Q\) and \(R\) are positive defined matrices, NEWLINE\[NEWLINEK=R^{-1}B^TP,\tag{3}NEWLINE\]NEWLINE where \(P\) is solution to the Riccati equation \(PA+A^TP-PBR^{-1}B^TP+Q=0\).NEWLINENEWLINE In the article, the following statement is obtained:NEWLINENEWLINE Theorem. Suppose that the matrix \((A,B)\) is completely controllable and \([-\sum^N_{r=0} A^{(r)}]\) is an \(M\)-matrix, then the multigroup multidelays interval coefficient constant linear continuous control system (1) is unconditionally robustly stabilized by (2) and (3).