Approximate controllability of non-autonomous second order impulsive functional evolution equations in Banach spaces. (English) Zbl 1519.34088
The authors investigate the approximate controllability of an abstract control system whose behavior is alternately governed by the second-order equation
\[
\ddot{x}(t) = A x(t) + B u(t) + f(t, x_t), \; t \in [t_{2_i}, t_{2i+1}].
\]
(where \(x_t\) is the memory of the state variable \(x\) on some finite time span) and the ”non-instantaneous impulse” dynamics
\[
x(t) = \phi(t, x(t_{2i+1}^-)), \; t \in [t_{2i+1}, t_{2_i+2}]
\]
The authors first determine the conditions under which the abstract linear dynamics \(\ddot{x} = A(t) x(t) + B u(t)\) is approximately controllable. Then, using some extra assumptions, they build for each target state \(x_T\) a family of controls for which they show the existence of a mild solution of the original problem. Finally, they extract from this family a sequence \(u_n\) such that the corresponding final state \(x_n(T)\) converges to \(x(T)\), proving the approximate controllability.
Two detailled applications of this result are given; in both cases, the linear dynamics is the wave equation.
Two detailled applications of this result are given; in both cases, the linear dynamics is the wave equation.
Reviewer: Sébastien Boisgérault (Paris)
MSC:
| 34K30 | Functional-differential equations in abstract spaces |
| 34K35 | Control problems for functional-differential equations |
| 34K45 | Functional-differential equations with impulses |
| 34K06 | Linear functional-differential equations |
| 93B05 | Controllability |
| 37C60 | Nonautonomous smooth dynamical systems |
Keywords:
abstract functional evolution equations; non-instantaneous impulses; approximate controllability; evolution operator; cosine familyReferences:
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