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Feedback stabilization of linear systems with point delays in state and control variables. (Chinese. English summary) Zbl 0859.93049
The authors consider the linear point time-delay system $$\dot x(t)= A_0x(t)+ A_1x(t-r)+ B_0u(t)+ B_1u(t-h),\tag *$$ with the initial conditions $x(\theta)= x_0(\theta)$, $\theta\in [-r,0]$, $u(\sigma)= u_0(\sigma)$, $\sigma\in [-h,0]$, where $r>0$, $h>0$, $x(t)\in \bbfR^n$, $u(t)\in \bbfR^m$, $A_0,A_1\in \bbfR^{n\times n}$, $B_0,B_1\in \bbfR^{m\times n}$, $x_0(\theta)\in C([-r,0],\bbfR^n)$, $u_0(\tau)\in C([-h,0],\bbfR^m)$. Matrix A is called the characteristic matrix of system $(*)$ if $A$ satisfies $A=A_0+ e^{-Ar}A_1$. Solving characteristic matrix equations of linear system $(*)$ is the key problem for stabilizing the system $(*)$. This paper gives the solution of characteristic matrix equations under the more general condition that the eigenvalues occur in the spectrum of system $(*)$ with algebraic and geometric multiplicities being greater or equal than one. The main idea is to transform the characteristic equations into a linear algebraic system. Sufficient conditions for the existence of the solution of the linear algebraic system and for the independence of the solution vectors of the linear algebraic system corresponding to a given eigenvalue are established. An algorithm for dealing with the stabilization problem in the case that the left eigenvectors of the system corresponding to different eigenvalues are linearly dependent is presented. Finally, an example is given to show the design procedure.

##### MSC:
 93D15 Stabilization of systems by feedback 93B52 Feedback control 34K35 Functional-differential equations connected with control problems