Bang-bang property for time optimal control of semilinear heat equation

Quantitative unique continuation for the heat equation with Coulomb potentials

“…Moreover, quantitative estimates of strong unique continuation for second order parabolic equations, such as the doubling property and the two-ball one-cylinder inequality, have been well understood (see, e.g., [10,22,23,25]). We refer to [28] for a more extensive review on this subject.…”

“…(1.2) This kind of Hölder-type quantitative estimate of unique continuation was first established in [22] for the heat equation with bounded potentials in a bounded convex domain. Later on, it has been extended in [25] to the case of bounded domain with a C 2 -smooth boundary (see also [2,23]). Using sharp analyticity estimates for solutions to general parabolic equations or systems with analytic coefficients, such a kind of quantitative estimate have been established in a series of recent works [1,6,7].…”

Observability from measurable sets for a parabolic equation involving the Grushin operator and applications

“…It also has independent interest in control theory. Indeed, by making use of such observability inequality, we can derive different kinds of bang-bang properties for both time optimal and norm optimal control problems (see, for instance, [13,[15][16][17][18][19][20]). In [15], the authors obtained an observability inequality from a measurable set in time for parabolic equations with spacetime dependent potentials.…”

An Observability Inequality On A Kind of Linear Parabolic Equation

“…Let E be a subset of positive measure in (0, ) T ,let  be a density point of E, using proposition 2.1 [9],for each 1…”

“…To obtain the observability estimate, with the similar method in [9][10], we first derive the inequalities of the solution and the gradient ) ( 2 Ω L norms, then reduce an inequality lemma about the initial and the final state from a corollary [1] directly. At last, combing the these inequalities with some other inequalities such as Nash inequality and Poincare inequality so on, we give the proof of the observability estimate.…”