Two results on exact boundary control of parabolic equations

Abstract

Abstract.(1) The known nullcontrollability result for boundary control of u t = uxx + q(x)u is generalized to consider a time-dependent coefficient q. (2) For boundary control of u t = Au (where it is known that CT: (initial data) (optimal nullcontrol for time T) exists for all T>0) it is shown that logllCrll = O(1/T) as T---> 0.

Closed-Form Boundary State Feedbacks for a Class of 1-D Partial Integro-Differential Equations

“…The kernel well posedness for was shown by Liu [23], whose result has inspired the efforts that have led to this paper. Integral operator transformations for linear parabolic PDEs can be traced as far back as the papers of Colton [14] and Seidman [30] who were studying solvability and open-loop controllability of the problem with . More recent considerations include [12].…”

Geometric bounds on the growth rate of null-controllability cost for the heat equation in small time

“…[Rus78]). In [Sei84], Seidman applied Russell's method to obtain an upper bound which, taking [BLR92] into account, corresponds to theorem 2.3 with α * = β * ≈ 42.86. Theorem 2.3 improves Seidman's result beyond this slight improvement of the rate α * insofar as the complex analysis multiplier method he uses does not necessarily allow to reach the optimal α * in theorem 2.2.…”

“…Seidman obtained lemma 4.4 for α * = β * with β * ≈ 42.86 in the proof of Theorem 3.1 in [Sei84]. His later Theorem 1 in [Sei86] improves the rate to α * = 2β * with β * ≈ 4.17.…”

Explicit lower bounds for the cost of fast controls for some 1-D parabolic or dispersive equations, and a new lower bound concerning the uniform controllability of the 1-D transport–diffusion equation

“…The first results concerning the cost of fast boundary controls have been obtained in the case of heat and Schrödinger equations. Concerning the one-dimensional heat equation on (0, T ) × (0, L) with boundary control on one side, the time-dependence of the cost of the boundary control is exp (β + /T ) for some constant β > 0 (see [7] for the lower bound and [18] for the upper bound), where the notation β + means that we simultaneously have that the cost of the control is exp(β/T ) and exp(K/T ) for every K > β as close as β as we want (the implicit constant in front of the exponential may explode when we get closer to β because it seems to be a fraction of some power of T ). The constant β verifies…”