| Abstract: |
| In this talk, we investigate the controllability of second-order problems in Banach spaces, when the nonlinear term also depends on the first derivative. In the existing literature, the most commonly adopted notion of exact controllability for second-order equations ensures only that the state function attains the desired target value, while neglecting the behavior of its derivative at the final time, thereby violating the concept of controllability.
The primary aim of this work is to introduce a new definition of controllability for second-order problems in Banach spaces, which simultaneously accounts for both the solution and its derivative at the final time, through a single control function. We then establish sufficient conditions guaranteeing this controllability. Our approach yields results under easily verifiable and non-restrictive assumptions on the cosine family generated by the associated linear operator, as well as on the nonlinear term, without requiring any compactness conditions. Finally, we illustrate the applicability of our results by considering a system governed by the one-dimensional Klein-Gordon equation. |
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