On exponential stability of linear difference-differential systems
The authors consider uniform exponential stability of the neutral equation \[ x'(t) = \sum_{g\in G_1} A_g(t) x'(t-g)+ \sum_{h\in H} B_h(t) x(t-h),\;t\in \mathbb{R}, \tag{*} \] where \(G_1\) is a finite set of numbers \(g>0\), \(H\) is a finite set of numbers \(h\geq 0\), the matrix functions \(A_g: \mathbb{R}\to M^n\) and \(B_h: \mathbb{R}\to M^n\) are Lebesgue measurable and satisfy the conditions \[ {\underset t\in \mathbb{R} {\text{Vrai sup}}} \| A_t(t) \|< \infty\quad \text{and} \quad \sup_{t\in \mathbb{R}} \int^{t+1}_t\bigl \| B_h(s) \bigr\| ds <\infty \] for all \(g\in G_1\) and \(h\in H\). The following result is obtained: Equation (*) is uniformly exponentially stable if and only if there is a stable equation \[ y(t)=\sum_{g\in G_1} A_g(t) y(t-g) +p(t),\;t\geq 0, \] \[ y(\xi)=0 \quad \text{if} \quad \xi<0, \] with \(p\in L_1\) \(([0,\infty))\), and \[ \bigl\| X(t,s) \bigr\|\leq Me^{-\alpha (t-s)}, \quad t\geq s\geq t_0\geq 0, \] with \(M,\alpha>0\) and \(X(t,s)\) is the fundamental matrix of the equation \[ x'(t)= \sum_{g\in G} \sum_{h\in H} R_{gh} (t)x(t-g-h)+f(t),\;t\geq t_0\geq 0 \] \[ x(\xi)=0 \quad \text{for} \quad \xi< t_0. \] Here, \(G\) is the set of all possible sums \(g=g_0 +\cdots +g_{m-1}\), \(g_i\in G_1\), \(R_{gh} (t)=P_g (t)B_h(t-g)\), and \(P_g(t)= \sum_{g_0 +\cdots+ g_{m-1} =g} \prod^{m-1}_{k=0} A_{g_k} (t-\sum^k_{i=0} g_i)\) .
scientific article; zbMATH DE number 1175193
scientific article