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

2013/09/08 by Pierre Lissy, Lissy, Pierre
Mathematics · #FOS: Mathematics #Optimization and Control (math.OC) #math.OC

paper · pdf · doi:10.48550/arxiv.1309.1943

arxiv created 2013/09/08 · arxiv updated 2013/09/10

Abstract

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.

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