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A note on controllability of impulsive systems. (English) Zbl 0965.93015

This article investigates complete controllability of the control system with impulse effects

x ' (t)=A(t)x(t)+B(t)u(t)+f(t,x(t)),tt k ,t[t 0 ,T],
x(t k +)=(I+D k u(t k ))x(t k ),x(t 0 )=x 0 ,

where, for each t[t 0 ,T], the state x(t) is an n-vector, the control u(t) is an m-vector, A(t) and B(t) are n×n and n×m matrices, respectively, with piecewise continuous entries, and 0<t 1 <t 2 <<t ρ <T are the time points when the impulsive controls u(t k ) act. For each k=1,2,,ρ, D k u(t k ) is an n×n diagonal matrix such that

D k u(t k )= i=1 m d i k u i (t k )I,

where I is the identity matrix on n and d i k .

34A37Differential equations with impulses