ISSN:
1618-3932
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract In this paper, we discuss the problem of stability of Volterra integrodifferential equation (1) $$\dot x = A(t)x + \int_0^t {C(t,s)} x(s)ds$$ with the decomposition (2) $$\begin{gathered} \dot x_i = A_{ii} (t)x_i + \sum\limits_{\mathop {i = 1}\limits_{j \ne i} }^r {A_{ij} (t)x_j + \int_0^t {\left[ {C_{ii} (t,s)x_i (s) + \sum\limits_{\mathop {j = 1}\limits_{j \ne i} }^r {C_{ij} (t,s)x_j (s)} } \right]} ds,} \hfill \\ i = 1,2, \cdot \cdot \cdot ,r, \hfill \\ \end{gathered} $$ wherex ∈R n,x T=(x 1 T , ...,x r/T),x i ∈R ni and ∑ i=1 r n i=n;A(t)=(A ij(t)) in whichA ij(t) (i,j=1, ...,r) is ann i ×n j matrix of functions continuous for 0 ≤t〈∞;C(t,s)=(C ij (t,s,)) in whichC ij(t,s)(i,j=1, ...,r) is ann i ×n j matrix of functions continuous for 0≤s≤t〈∞. According to the decomposition theory of large scale system and with the help of Liapunov functional, we give a criterion for concluding that the zero solution of (2) (i.e. large scale system (1)) is uniformly asymptotically stable. We also discuss the large scale system (3) $$\dot x = A(t)x + \int_0^t {C(t,s)} x(s)ds + f(t)$$ with the decomposition (4) $$\begin{gathered} \dot x_i = A_{ii} (t)x_i + \sum\limits_{\mathop {i = 1}\limits_{j \ne i} }^r {A_{ij} (t)x_j + \int_0^t {\left[ {C_{ii} (t,s)x_i (s) + \sum\limits_{\mathop {j = 1}\limits_{j \ne i} }^r {C_{ij} (t,s)x_j (s)} } \right]} ds + f_i (t),} \hfill \\ i = 1,2, \cdot \cdot \cdot ,r, \hfill \\ \end{gathered} $$ and give a criterion for determining that the solutions of (4) (i.e. large scale system (3)) are uniformly bounded and uniformly ultimately bounded. Those criteria are of simple forms, easily checked and applied.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF02006075
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