Stability analysis of interval systems with time-delay (Q2748348)

The authors obtain some sufficient conditions for robust stability of a class of delay interval differential systems of the form NEWLINE\[NEWLINE\dot x(t)= [\underline A,\overline A] x(t)+[\underline B,\overline B] x(t-\tau),\;t\geq 0,\tag{1}NEWLINE\]NEWLINE where \([\underline A,\overline A], [\underline B, \overline B]\) are interval matrices and \(\tau>0\) denotes the delay. Let \(\underline C=(\underline c_{ij})\), \(\overline C=(\overline c_{ij})\) be real constant matrices with \(\underline c_{ij}\leq \overline c_{ij}\), \(i,j=1,2, \dots, n\), then the interval matrix \([\underline C,\overline C]\) and the symmetry interval matrix \([\underline C,\overline C]_s\) are defined, respectively, by \([\underline C,\overline C]: =\{C=(c_{ij}) \mid \underline c_{ij}\leq c_{ij}\leq \overline c_{ij}\}\) and \([\underline C,\overline C]_s:= \{C=(c_{ij}) \mid \underline c_{ij}\leq c_{ij}= c_{ji}\leq\overline c_{ij}\}\).NEWLINENEWLINENEWLINEThe interval system (1) is called robustly (or symmetrically robustly) stable if for arbitrary constant matrices \(A\in[\underline A,\overline A]\), \(B\in[\underline B,\overline B]\) (or \(A\in[\underline A,\overline A]_s\), \(B\in[\underline B,\overline B]_s)\), the zero solution of system (1) is asymptotically stable.NEWLINENEWLINENEWLINEBy applying the relationship between the system (1) and a certain complex ordinary differential equation, some sufficient conditions for the robust stability (or symmetric robust stability) of system (1) are obtained by the construction of some suitable Lyapunov functions. These sufficient conditions can easily be verified and can be viewed as extensions of some known results established for interval differential systems without delay. Two illustrating examples are also given.