Carleman estimates for a stochastic degenerate parabolic equation and applications to null controllability and an inverse random source problem

Abstract

In this paper, we establish two Carleman estimates for a stochastic degenerate parabolic equation. The first one is for the backward stochastic degenerate parabolic equation with singular weight function. Combining this Carleman estimate and an approximate argument, we prove the null controllability of the forward stochastic degenerate parabolic equation with the gradient term. The second one is for the forward stochastic degenerate parabolic equation with regular weighted function, based on which we obtain the Lipschitz stability for an inverse problem of determining a random source depending only on time in the forward stochastic degenerate parabolic equation.

Null controllability and inverse source problem for stochastic Grushin equation with boundary degeneracy and singularity

“…In recent years, many efforts have been devoted to studying the Carleman estimate for stochastic partial differential equations, for example [3,24,34,40] for stochastic heat equation, [42] for stochastic wave equation, [15] for stochastic Korteweg-de Vries equation, [17] for stochastic Kuramoto-Sivashinsky equation, [27] for stochastic Schrödinger equation, [18] for the stochastic Kawahara equation, and so on. To the best of our knowledge, there are only two papers about Carleman estimates for one dimensional stochastic degenerate operator du − (x α u x ) x dt [25,38], which is very different from the degenerate Grushin operator du − u xx dt − x 2γ u yy dt. In these works, Carleman estimates were mainly applied to deal with stochastic control problems.…”

Determination the Solution of a Stochastic Parabolic Equation by the Terminal Value

“…Recently, Carleman estimates are also introduced to solve inverse problems for stochastic partial differential equations. Particularly, we refer the readers to [13,15,18,20,21] for some recent works on inverse problems of stochastic parabolic equations via Carleman estimates. In all the above mentioned papers, Carleman estimates are established by using two-layer weight functions.…”

A Concise Introduction to Control Theory for Stochastic Partial Differential Equations

“…• In this notes, we only give a very brief introduction to controllability problems for three kinds of SPDEs. Recently, there are also some works for controllability problems for other types of SPDEs, such as [16] for stochastic complex parabolic equations, [21] for stochastic Kuramoto-Sivashinsky equations, [33,72] for degenerate stochastic parabolic equations, [32] for coupled stochastic parabolic equations, [38] for stochastic Schrödinger equations and [55] for a refined stochastic wave equation.…”

“…• By means of the tools that we developed for solving the stochastic controllability problems, in [35,36,50] we initiated the study of inverse problems for SPDEs (See also [72,77,78] for further interesting progress), in which the point is that, unlike most of the previous works in this topic, the problem is genuinely stochastic and therefore it cannot be reduced to any deterministic one. Especially, it was found in [50] that both the formulation of stochastic inverse problems and the tools to solve them differ considerably from their deterministic counterpart.…”

“…3.3 Clearly, if F is stochastic, then one cannot use the above argument to solve the observability problems for SEEs. For establishing the observability for SPDEs, a powerful method is the global Carleman estimate (e.g.,[16,21,33,35,36,37,72,77,78,80,81]) Remark 3.4 In this section, for simplicity, we assume that B ∈ L(U ; H) and C ∈ L(H; U ). Some results for unbounded B and C can be found in[40,53].…”