A certain periodic control problem for differential equations with impulses in the space of bounded number sequences (Q750701)

The following problem is considered: find a pair \((\mu_ 1,\mu_ 2)\) ensuring a solution of \[ dx/dt=\epsilon f(t,x)-u_ 1+\sum (\epsilon H_ i(x)-\mu_ 2)\delta (t-t_ i),\quad x(\tau,x_ 0)=x_ 0 \] to be T-periodical; here \(x=(x_ 1,x_ 2,...)\), \(x_ i\in (-\infty,\infty)\), \(\| x\| =\sup \{| x_ 1|,| x_ 2|,...\}\), f is bounded, continuous, Lipschitz in x and T-periodical in t, \(H_ i's\) are bounded and Lipschitz, \(H_{i+k}=H_ i\), \(t_{i+k}=t_ i+T,\delta (t- t_ i)\) is the unit impulse at time \(t_ i\), and \(\epsilon >0\). An iterational process \(x_ m\) converging to a desired solution \(x_ 0\) for any sufficiently small \(\epsilon\) is described (the proof is given). An upper bound for \(\| x_ m-x_ 0\|\), and expressions for the corresponding \(\mu_ 1\) and \(\mu_ 2\) are written out. The statement on uniqueness of \(\mu_ 1\) and \(\mu_ 2\) is proved for a special dependence of these parameters on \(x_ 0\) (the latter is, however, omitted in the formulation).