Exact null controllability of semilinear integrodifferential systems in Hilbert spaces. (English) Zbl 1061.93056
The authors remove the bounded invertibility condition replacing it by the exact null controllability of the associated linear system with additive term. Exact null controllability of this system does not guarantee the boundedness of \((L_0)^{-1}\), but it guarantees the boundedness of the operator \((L_0)^{-1}N^T_0\) which is defined in Lemma 3. Using this operator they transform the controllability problem into a fixed point problem for some operator and use the Schauder fixed point theorem to show that the operator has a fixed point. An application to partial integro-differential equations, is given.
Reviewer: Jong Yeoul Park (Pusan)
MSC:
| 93C30 | Control/observation systems governed by functional relations other than differential equations (such as hybrid and switching systems) |
| 45K05 | Integro-partial differential equations |
| 45G10 | Other nonlinear integral equations |
| 47H10 | Fixed-point theorems |
| 93B05 | Controllability |
| 93B28 | Operator-theoretic methods |
Keywords:
exact null controllability; semilinear integro-differential equations; Schauder fixed point theorem; bounded invertibility condition; partial integro-differential equationsReferences:
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| [3] | Dauer, J. P.; Balasubramaniam, P., Null controllability of semilinear integrodifferential systems in Banach space, Appl. Math. Lett., 10, 117-123 (1997) · Zbl 0892.93011 |
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