Linear stochastic degenerate Sobolev equations and applications^†

Abstract

In this paper, a general class of linear stochastic degenerate Sobolev equations with additive noise is considered. This class of systems is the infinite dimensional analogue of linear descriptor systems in finite dimensions. Under appropriate assumptions, the mild and strong well posedness for the initial value problem are studied using elements of the semigroup theory and properties of the stochastic convolution. The final value problem is also examined and it is proved that this is uniquely strongly solvable and the solution is continuously dependent on the final data. Based on the results of the forward and backward problem, the conditions for the exact controllability are investigated for a special but important class of these equations. The abstract results are illustrated by applications in complex media electromagnetics, in the one dimensional stochastic Dirac equation in the non-relativistic limit and in a potential application in input-output analysis in economics.

Stochastic degenerate Sobolev equations: well posedness and exact controllability

“…For the definition of the strong solution, we refer to . Theorem Assume that ( A 1 ) − ( A 6 ) hold.…”

“…Following the analysis made in section 2 of , the initial value problem for the deterministic analogue of takes the form: in the space Q ⊥ H = Ran ( L ), where M 0 = M X and X : P ⊥ D → D is the inverse of the restriction of the projection P ⊥ | D . It is assumed that (see ( A 2 )( i i )) is dense in P ⊥ H , M 0 and ( λ L − M ) X = λ L ⊥ − M 0 are closed operators (see and ) in P ⊥ D .…”

“…From , the following result is derived. Theorem Let the assumptions of Theorem 1 in hold. Then, the initial value problem for the deterministic analogue of has a unique strict solution x ( t ), t ∈[0, T ], with x ( t ) = X y ( t ), where y ( t ) = ( L ⊥ ) −1 ψ ( t ) and ψ ( t ) is the unique strict solution of problem , that is, and …”

“…The equivalence of the two cases is provided whenever L ⊥ :( KerL ) ⊥ → Ran ( L ) is bounded and boundedly invertible. In , the pseudoparabolicity of the stochastic equation is provided by a multiplication with the bounded operator ( L ⊥ ) −1 , in view of the first methodology. In the present paper, the assumptions on the deterministic coefficients L , M and λ L − M are the same and the two methodologies are equivalent.…”

“…The aim of this paper is to study the forward, backward and exact controllability problems under the same framework with but applying the second methodology of , that is, using the transformation ψ ( t ) = L ⊥ y ( t ) (which is briefly presented in Section 2). It is shown that the equivalence of the two methods is valid not only for the deterministic case but also for the stochastic problems.…”