Controllability problem of an evolution equation with singular memory
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908336601956352 |
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| author | Arora, Sumit Ponce, Rodrigo |
| author_facet | Arora, Sumit Ponce, Rodrigo |
| contents | This work addresses control problems governed by a semilinear evolution equation with singular memory kernel $κ(t)=αe^{-βt}\frac{t^{ν-1}}{Γ(ν)}$, where $α>0, β\ge 0$, and $0<ν<1$. We examine the existence of a mild solution and the approximate controllability of both linear and semilinear control systems. To this end, we introduce the concept of a resolvent family associated with the linear evolution equation with memory and develop some of its essential properties. Subsequently, we consider a linear-quadratic regulator problem to determine the optimal control that yields approximate controllability for the linear control system. Furthermore, we derive sufficient conditions for the existence of a mild solution and the approximate controllability of a semilinear system in a super-reflexive Banach space. Additionally, we present an approximate controllability result within the framework of a general Banach space. Finally, we apply our theoretical findings to investigate the approximate controllability of the heat equation with singular memory. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_17566 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Controllability problem of an evolution equation with singular memory Arora, Sumit Ponce, Rodrigo Optimization and Control 34H05, 45J05, 93B05 This work addresses control problems governed by a semilinear evolution equation with singular memory kernel $κ(t)=αe^{-βt}\frac{t^{ν-1}}{Γ(ν)}$, where $α>0, β\ge 0$, and $0<ν<1$. We examine the existence of a mild solution and the approximate controllability of both linear and semilinear control systems. To this end, we introduce the concept of a resolvent family associated with the linear evolution equation with memory and develop some of its essential properties. Subsequently, we consider a linear-quadratic regulator problem to determine the optimal control that yields approximate controllability for the linear control system. Furthermore, we derive sufficient conditions for the existence of a mild solution and the approximate controllability of a semilinear system in a super-reflexive Banach space. Additionally, we present an approximate controllability result within the framework of a general Banach space. Finally, we apply our theoretical findings to investigate the approximate controllability of the heat equation with singular memory. |
| title | Controllability problem of an evolution equation with singular memory |
| topic | Optimization and Control 34H05, 45J05, 93B05 |
| url | https://arxiv.org/abs/2504.17566 |