On the cost of fast controls for some families of dispersive or parabolic equations in one space dimension

Pierre Lissy (1)

1 LJLL - Laboratoire Jacques-Louis Lions

In this paper, we consider the cost of null controllability for a large class of linear equations of parabolic or dispersive type in one space dimension in small time. By extending the work of Tenenbaum and Tucsnak in ''New blow-up rates for fast controls of Schrödinger and heat equations'', we are able to give precise upper bounds on the time-dependance of the cost of fast controls when the time of control T tends to 0. We also give a lower bound of the cost of fast controls for the same class of equations, which proves the optimality of the power of T involved in the cost of the control. These general results are then applied to treat notably the case of linear KdV equations and fractional heat or Schrödinger equations.

Mots clés: moment method; fast controls; linear dispersive and parabolic equations

Domaines: Optimisation et contrôle [math.OC]

Pierre Lissy. On the cost of fast controls for some families of dispersive or parabolic equations in one space dimension. SIAM Journal on Control and Optimization, 2014, 52 (4), http://epubs.siam.org/doi/abs/10.1137/140951746. ⟨10.1137/140951746⟩. ⟨hal-00859458⟩