In the present paper a linear system with saturating controls is considered \[ \dot x(t)= Ax(t)+ A_d x(t-\tau)+ B\text{ sat}(u(t)),\quad t\geq t_0\geq 0,\tag{1} \]
\[ x(t+ \Theta)= \Phi(\Theta),\quad \Theta\in [-\tau, 0],\quad \Phi\in C([-\tau, 0],\mathbb{R}^n), \] where \[ u(t)= Kx(t),\quad K= (K_i)^m_{i=1}\in \mathbb{R}^{mn} \] and \[ \text{sat}(K_i x(t))= \text{sign}(K_i x(t))\min([K_i x(t)], u_{0i}), \]
\[ u_{0i}> 0,\quad i= 1,\dots, m. \] When saturations do not occur, then system (1) admits the form \[ \dot x= Ax(t)+ A_dx(t-\tau)+ Bu(t).\tag{2} \] If a linear matrix inequality is fulfilled, then the following result is valid: there exists a matrix \(K\) and \(\text{sat}D_0\) such that
1. for each \(\Phi\in C([-\tau, D_0])\) the system (1) is asymptotically stable;
2. the trajectories of the system (2) contained in the region \[ \{x\in\mathbb{R}^n: -u_0\leq Kx\leq u_0\},\quad u_0\geq 0, \] are stable with a decay rate \(\beta\) (\(\beta\)-stable).
An approach to a practical realization of the result is suggested.