On the approximate controllability from the boundary for fractional wave equations

Abstract

We study the boundary controllability of fractional wave (also known as super diffusion) equations associated to the Caputo fractional derivative with a symmetric non-negative uniformly elliptic operator subject to the non-homogeneous Dirichlet- or Robin-type boundary conditions. Our results show that if and is a connected bounded open set with smooth boundary , then the system in , on , , , is approximately controllable for any , , any non-empty open set and any , where in the case of the Dirichlet boundary condition and for the Robin boundary conditions. Here denotes the dual of the domain of the fractional power of order of the realization in of the operator A with the zero boundary conditions on . The results obtained in the article are optimal given that such a system cannot be null controllable.

Keywords:

• Fractional wave equations
• Dirichlet and Robin boundary conditions
• Mittag–Leffler functions
• existence
• regularity and explicit representation of solutions
• boundary controllability

AMS Subject Classifications:

• 93B05
• 26A33
• 35R11

Acknowledgements

We would like to thank the referee for the careful reading of the manuscript and for his helpful comments and suggestions.

Notes

No potential conflict of interest was reported by the author.

Additional information

Funding

The work of the author is partially supported by the Air Force Office of Scientific Research grant under the Award Number [FA9550-15-1-0027].