Backward stochastic partial differential equations with jumps and application to optimal control of random jump fields

We prove an existence and uniqueness result for a general class of backward stochastic partial differential equations (SPDE) with jumps. This is a type of equations, which appear as adjoint equations in the maximum principle approach to optimal control of systems described by SPDE driven by Lévy processes.     

Optimal control for two-dimensional stochastic second grade fluids

This article deals with a stochastic control problem for certain fluids of non-Newtonian type. More precisely, the state equation is given by the two-dimensional stochastic second grade fluids perturbed by a multiplicative white noise. The control acts through an external stochastic force and we search for a control that minimizes a cost functional. We show that the Gâteaux derivative of the control to state map is a stochastic process being the unique solution of the stochastic linearized state equation. The well-posedness of the corresponding stochastic backward adjoint equation is also established, allowing to derive the first order optimality condition.     

A direct approach to the finite element solution of elliptic optimal control problems

A general framework is developed for the finite element solution of optimal control problems governed by elliptic nonlinear partial differential equations. Typical applications are steady‐state problems in nonlinear continuum mechanics, where a certain property of the solution (a function of displacements, temperatures, etc.) is to be minimized by applying control loads. In contrast to existing formulations, which are based on the “adjoint state,” the present formulation is a direct one, which does not use adjoint variables. The formulation is presented first in a general nonlinear setting, then specialized to a case leading to a sequence of quadratic programming problems, and then specialized further to the unconstrained case. Linear governing partial differential equations are also considered as a special case in each of these categories. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15:371–388, 1999     

Existence and global attractiveness of a square‐mean ‐pseudo almost automorphic solution for some stochastic evolution equation driven by Lévy noise

In this work, we introduce the concept of μ‐pseudo almost automorphic processes in distribution. We use the μ‐ergodic process to define the spaces of μ‐pseudo almost automorphic processes in the square mean sense. We establish many interesting results on the functional space of such processes like a composition theorem. Under some appropriate assumptions, we establish the existence, the uniqueness and the stability of the square‐mean μ‐pseudo almost automorphic solutions in distribution to a class of abstract stochastic evolution equations driven by Lévy noise. We provide an example to illustrate our results.     

Exponential decay of solution to the viscoelastic porous‐thermoelastic system of type III with boundary time‐varying delay

In this paper, we are concerned with a one‐dimensional porous‐thermoelastic system of type III with a viscoelastic damping and boundary time‐varying delay. Under suitable assumptions on relaxation function and time delay, we establish the exponential decay result of the system in which the damping is strong enough to stabilize the thermoelastic system in the presence of time delay. Copyright © 2015 John Wiley & Sons, Ltd.