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Zbl 0943.93007
Akhmetov, M.; Sejilova, R.
(Akhmet, Marat)
The control of the boundary value problem for linear impulsive integro-differential systems.
(English)
[J] J. Math. Anal. Appl. 236, No.2, 312-326 (1999). ISSN 0022-247X

The impulse control system of integro-differential equations defined on the interval $[a,b]$ is of the form: $$\dot x(t)= A(t)x(t)+ C(t)u(t)+ f(t)+ \int^t_a K(t,s) x(s) dx,\quad t\ne\theta_i,$$ $$\Delta x(\theta_i)= B_i x(\theta_i)+ \sum_{a<\theta_j\le \theta_i} D_{i,j} x(\theta_j)+ Q_i\nu_i+ I_i,$$ $$x(a)= \alpha,\quad x(b)= \beta,\quad \alpha,\beta\in \bbfR^n,$$ where $\{\theta_i, i= 1,2,\dots,p\}$ is a strictly increasing sequence of real numbers belonging to the interval $[a,b]$, $\Delta x(\theta_j)= x(\theta_j+)- x(\theta_j)$, $A(t)$, $K(t,s)$, $D_{i,j}$, $B_i$, $i,j\in \{1,\dots, p\}$ are matrices of size $n\times n$, $C(t)$, and $Q_i$, $i= 1,2,\dots, p$ are matrices of size $n\times m$, $f$ is a vector function, and $I_i\in \bbfR^n$, $i= 1,2,\dots, p$. The pair $(u,\nu)$ where $u\in L^m_2(a,b)$, $\nu= \{\nu_i\}$, $\nu_i\in \bbfR^m$ denotes a control.\par Under some additional assumptions necessary and sufficient conditions for controllability of the system are given.
[R.Rempała (Warszawa)]
MSC 2000:
*93B05 Controllability
34B37 Boundary value problems with impulses

Keywords: impulse control system; integro-differential equations; controllability

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