In this paper, we consider the Cauchy–Ventcel problem in an inhomogeneous medium with dynamic boundary conditions subject to a nonlinear damping distributed around a neighborhood [Formula: see text] of the boundary according to the Geometric Control Condition. Uniform decay rates of the associated energy are established and, in addition, the exact internal controllability for the linear problem is also proved. For this purpose, refined microlocal analysis arguments are considered by exploiting ideas due to Burq and Gérard [Contrôle Optimal des équations aux dérivées partielles. (2001); http://www.math.u-psud.fr/burq/articles/coursX.pdf ].

This article studies the stochastic evolution of incompressible non-Newtonian fluids of differential type. More precisely, we consider the equations governing the dynamic of a third grade fluid filling a three-dimensional bounded domain O, perturbed by a multiplicative white noise. Taking the initial condition in the Sobolev space H2(O), and supplementing the equations with a Navier slip boundary condition, we establish the existence of a global weak stochastic solution with sample paths in L∞(0,T;H2(O)).

We study an optimal control problem with a quadratic cost functional for non-Newtonian fluids of differential type. More precisely, we consider the system governing the evolution of a second grade fluid filling a two-dimensional bounded domain, supplemented with a Navier slip boundary condition. Under certain assumptions on the size of the initial data and parameters of the model, we prove second-order sufficient optimality conditions. Furthermore, we establish a global uniqueness result for the solutions of the first-order optimality system.

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