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Existence and Optimal Control Results for Second-Order Semilinear System in Hilbert Spaces

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Abstract

The article provides sufficient conditions for the existence of optimal control for second-order semilinear control system in Hilbert spaces. We consider the integral cost function as

$$\begin{aligned} J(z,v):=\int _{0}^{T}L(\tau ,z^{v}(\tau ),v(\tau ))\mathrm{d}t, \end{aligned}$$

subject to the equations

$$\begin{aligned} z''(\tau )= & {} Az(\tau )+Bv(\tau )+g(\tau ,z(\tau ));\,0<\tau \le T\\ z(0)= & {} z_0\\ z'(0)= & {} z_1. \end{aligned}$$

Next, we discuss the existence and the uniqueness of mild solutions for the above proposed problem using Banach fixed point theorem. The stated Lagrange’s problem admits at least one optimal control pair under certain assumptions. Finally, the validation of theoretical results is provided through an example.

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Data Availability Statement

Data sharing is not applicable to this article as no data sets were generated or analyzed during the current study.

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Acknowledgements

Authors are thankful to the reviewers for the valuable suggestions. Authors are also thankful to Dr. Chhaya Singh (Assistant Professor, English, Rajkiya Engineering College Kannauj, India-209732), who helped us to improve the language of the article.

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  1. Department of Applied Science, Rajkiya Engineering College Kannauj, Kannauj, 209732, India

    Anurag Shukla & Rohit Patel

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  1. Anurag Shukla
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  2. Rohit Patel
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Correspondence to Anurag Shukla.

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Shukla, A., Patel, R. Existence and Optimal Control Results for Second-Order Semilinear System in Hilbert Spaces. Circuits Syst Signal Process 40, 4246–4258 (2021). https://doi.org/10.1007/s00034-021-01680-2

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  • DOI: https://doi.org/10.1007/s00034-021-01680-2

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