Exponential stabilization of the stochastic Burgers equation by boundary proportional feedback

Abstract

In the present paper it is designed a simple, finite-dimensional, linear deterministic stabilizing boundary feedback law for the stochastic Burgers equation with unbounded time-dependent coefficients. The stability of the system is guaranteed no matter how large the level of the noise is.2010 Mathematics Subject Classification. 60H15, 76F70, 93B52, 60H20.

Deep Reinforcement Learning for Online Control of Stochastic Partial Differential Equations

“…Due to the presence of nonlinear convective terms, solution of the 1D Burgers' equation is prone to exhibit a chaotic behaviour. Here, we test our proposed RL framework to control the fluid flow governed by the SBE to damp the developed shock wave and stabilize the system in an online manner, and compare it against previous non-RL approaches (Choi et al [1993], Munteanu [2019]). A distributed control method is used by introducing an external forcing term, f (x, t), as a control input of the system.…”

Design of boundary stabilizers for the non-autonomous cubic semilinear heat equation driven by a multiplicative noise

“…The method to design the feedback controller u relies on the ideas in [20], [17], [18]. In [17], a proportional type law was proposed to stabilize, in mean, the stochastic heat equation, while in [18] a stabilizer is constructed for the stochastic 1-D Burgers equation. We shall follow the approach in [18].…”

“…In [17], a proportional type law was proposed to stabilize, in mean, the stochastic heat equation, while in [18] a stabilizer is constructed for the stochastic 1-D Burgers equation. We shall follow the approach in [18]. Roughly speaking, we design a feedback controller which is: linear, of finite-dimensional structure, given in a very simple form, being easy to manipulate from the computational point of view, involving only the eigenfunctions of the Neumann-Laplace operator (see relations (24)-(26) below).…”

“…i.e., we have µ 1 < µ 2 < ... < µ N . Now, since we are set with the theoretical results and the hypotheses of the paper, we may proceed to apply the approach from [17], [20], [18]. Firstly, in order to lift the boundary control into the equations (to obtain an internal control-type problem), we introduce the so-called Neumann operator as: given g ∈ L 2 (Γ 1 ) and γ > 0, we denote by D γ g := y, the solution to the equation…”

“…In comparison to [17], [19], [18], in this work we managed to pass from the 1 − D case to the 2 − D case domain O, based on the L ∞ -estimates and on L 2 −estimates of the eigenfunctions of the Laplacean. As a future work, we intend to solve the 3 − D case as-well.…”