We investigate in this manuscript an optimal control problem for a fractional wave equation involving the fractional Riemann-Liouville derivative and with missing initial condition. For this purpose, we use the concept of no-regret and low-regret controls. Assuming that the missing datum belongs to a certain space we show the existence and the uniqueness of the low-regret control. Besides, its convergence to the no-regret control is discussed together with the optimality system describing the no-regret control.

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In the present paper, we are concerned with a fractional wave equation of Sturm-Liouville type in a general star graph. We first give several existence, uniqueness and regularity results of weak solutions for the one-dimensional case using the spectral theory; we prove the existence and uniqueness of solutions to a quadratic boundary optimal control problem and provide a characterization of the optimal control via the Euler-Lagrange first order optimality conditions. We then investigate the analogous problems for a fractional Sturm-Liouville problem in a general star graph with mixed Dirichlet and Neumann boundary conditions and controls of the velocity. We show the existence and uniqueness of minimizers, and by using the first order optimality conditions with the Lagrange multipliers, we are able to characterize the optimal controls.