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Approximate Controllability from the Exterior of Space-Time Fractional Wave Equations

Abstract

We investigate the controllability from the exterior of space-time fractional wave equations involving the Caputo time fractional derivative with the fractional Laplace operator subject to nonhomogeneous Dirichlet or Robin type exterior conditions. We prove that if \(1<\alpha < 2\), \(0<s<1\) and \(\Omega \subset {{\mathbb {R}}}^N\) is a bounded Lipschitz domain, then the system

$$\begin{aligned} {\left\{ \begin{array}{ll} {\mathbb {D}}_t^\alpha u+(-\Delta )^su=0\;\;&{} \text{ in } \;\;\Omega \times (0,T), \\ Bu=g\;&{} \text{ in } \; ({\mathbb {R}}^N\setminus \Omega )\times (0,T), \\ u(\cdot ,0)=u_0, \;\;\partial _tu(\cdot ,0)=u_1\;&{} \text{ in } \;\Omega , \end{array}\right. } \end{aligned}$$

is approximately controllable for any \(T>0\), \((u_0,u_1)\in L^2(\Omega )\times {\mathbb {V}}_B^{-\frac{1}{\alpha }}\) and every \(g\in {\mathcal {D}}({\mathcal {O}}\times (0,T))\), where \({\mathcal {O}}\subset ({\mathbb {R}}^N\setminus \Omega )\) is any non-empty open set in the case of the Dirichlet exterior condition \(Bu=u\), and \({\mathcal {O}}\subseteq {\mathbb {R}}^N\setminus \Omega \) is any open set dense in \({\mathbb {R}}^N\setminus \Omega \) for the Robin exterior conditions \(Bu:=\mathcal N_su+\kappa u\). Here, \({\mathcal {N}}_su\) is the nonlocal normal derivative of u and \({\mathbb {V}}_B^{-\frac{1}{\alpha }}\) denotes the dual of the domain of the fractional power of order \(\frac{1}{\alpha }\) of the realization in \(L^2(\Omega )\) of the operator \((-\Delta )^s\) with the zero (Dirichlet or Robin) exterior conditions \(Bu=0\) in \({{\mathbb {R}}}^N\setminus \Omega \).

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Acknowledgements

The work of the second author is partially supported by the Air Force Office of Scientific Research under the Award No. FA9550-18-1-0242.

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Correspondence to Mahamadi Warma.

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Louis-Rose, C., Warma, M. Approximate Controllability from the Exterior of Space-Time Fractional Wave Equations. Appl Math Optim 83, 207–250 (2021). https://doi.org/10.1007/s00245-018-9530-9

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Keywords

  • Fractional Laplacian
  • Dirichlet and Robin exterior conditions
  • Fractional wave equations
  • Unique continuation principle
  • Existence, regularity and explicit representation of solutions
  • Approximate controllability from the exterior

Mathematics Subject Classification

  • 93B05
  • 93C20
  • 26A33
  • 35R11