Stochastic degenerate Sobolev equations: well posedness and exact controllability

Abstract

In this paper, we revisit the recently proposed results for a general class of linear stochastic degenerate Sobolev systems with additive noise by using a different approach keeping, however, the main assumptions unchanged for the purpose of comparison. In particular, the mild and strong well‐posedness for the initial and final value problems are presented and studied by applying a suitable transformation that formulates the degenerate stochastic system as a pseudoparabolic one. Based on the derived results for the forward and backward cases, under this new framework, the conditions for the exact controllability are revisited for a particular class of degenerate Sobolev systems. Copyright © 2016 John Wiley & Sons, Ltd.

Towards sample path estimates for fast–slow stochastic partial differential equations

“…First note that we have the following estimate for one single mode. Proof Let k ≥ k * , the equation for the kth component reads 20) and using Duhamel's principle, its solution can be represented as…”

“…It would be very desirable to have a generalisation of this theory to fast-slow SPDEs. Examples of such systems arising in applications are the FitzHugh-Nagumo SPDE [7,13], slowly driven amplitude/modulation equations [8,14], and degenerate controlled SPDEs [19,20]. There are certainly many other important examples as most PDEs arising in applications have parameters, which quite often are slow variables, and those PDEs should frequently have noise terms, e.g., due to finite-size effects or external forces.…”

Controllability and observability of stochastic implicit systems and stochastic GE-evolution operator

“…Stochastic implicit systems play an important role in many application fields, for example, input-output economics, the problem of protein folding, the modelling of multi-body mechanisms and so on (e.g., [19], [23], [15], [16], [17]). Many results have been obtained concerning stochastic implicit systems in finite dimensional case (e.g., [3], [5], [4], [6], [7], [28], [30], [24], [26], [27], [29], [14], [25], [11], [12] ), but there are few results in infinite dimensional case (see [19], [23], [15], [16], [17]). Classical and distributional solutions for a class of abstract stochastic implicit equations were studied by degenerate strongly continuous semigroup in Hilbert spaces in [19].…”

An exact null controllability of stochastic singular systems

“…Dear editor, The stochastic singular system is also called the stochastic differential algebraic system, stochastic descriptor system, generalized stochastic system, and stochastic degenerate system (e.g., [1][2][3][4]). This type of system is found in numerous fields of application, which include fluid dynamics, the modeling of multi-body mechanisms, finance, inputoutput economics and the problem of protein folding.…”

“…However, theorem 5.2.1 of [8] was incorrectly applied, leading to inappropriate conclusion regarding impulse solution. The solution and exact controllability of the degenerate Sobolev equation have been discussed [4]. The impulse terms may exist in the solution for a stochastic singular system.…”