Sharp estimates and homogenization of the control cost of the heat equation on large domains

Abstract

We prove new bounds on the control cost for the abstract heat equation, assuming a spectral inequality or uncertainty relation for spectral projectors. In particular, we specify quantitatively how upper bounds on the control cost depend on the constants in the spectral inequality. This is then applied to the heat flow on bounded and unbounded domains modeled by a Schrödinger semigroup. This means that the heat evolution generator is allowed to contain a potential term. The observability/control set is assumed to obey an equidistribution or a thickness condition, depending on the context. Complementary lower bounds and examples show that our control cost estimates are sharp in certain asymptotic regimes. One of these is dubbed homogenization regime and corresponds to the situation that the control set becomes more and more evenly distributed throughout the domain while its density remains constant.

Smoothing properties of fractional Ornstein-Uhlenbeck semigroups and null-controllability

“…Observability estimates for this semigroup in any positive times have already been established in the work [30] (Theorem 3.8), where the authors proved that there exist some positive constants…”

A uniform bound on costs of controlling semilinear heat equations on a sequence of increasing domains and its application

“…The question whether or not a given control system is null controllable and obtaining upper bounds for the associated null control costs, with respect to the time interval or control regions, are important topics in control theory, both for linear partial differential control equations and abstract linear control systems. We refer the reader to [1,3,9,10,17,18,25,26,28] and references therein for a wider discussion on this activated research field.…”

“…We say a set as an equidistributed set in R N if it contains a union of suitably distributed balls of fixed radius. Recently, there are many beautiful existing results on the quantitative unique continuation for general elliptic operators on equidistributed sets (c.f., [10,17,23] and references therein). Meanwhile, we will consider the varying domains Ω n as an approximation of R N , for the simplicity.…”

Null-controllability and control cost estimates for the heat equation on unbounded and large bounded domains

“…An inequality of the above form is usually called a spectral inequality in the context of control theory and unique continuation principle in the theory of (random) Schrödinger operators. We refer the reader to [8,25] and [15,24,31,35], respectively, and the works cited therein for an overview of the different uses of (1.3) in both these fields.…”

“…The latter can also be deduced directly from Corollary 1.5 via the transformation formula. Such a reasoning has previously been used in [25,Theorem 4.6]; see also [8,Remark 5.9]. With regard to control theory and the particular form (1.4) of a spectral inequality, it should, however, be mentioned that this is useful only if 1{p2sq ă 1, that is, s ą 1{2.…”