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We consider the equations governing the motion of incompressible second grade fluids in a bounded two-dimensional domain with Navier-slip boundary conditions. We first prove that the corresponding solutions are uniformly bounded with respect to the normal stress modulus α in the L∞-H1 and the L2-H2 time–space norms. Next, we study their asymptotic behavior when α tends to zero, prove that they converge to regular solutions of the Navier–Stokes equations and give the rate of convergence in terms of α.

We study optimal control problems of systems describing the flow of incompressible shear-thinning fluids. Taking advantage of regularity properties of the flows, we derive necessary optimality conditions under a restriction on the optimal control.

We consider a non-Newtonian fluid governed by stationary, incompressible Navier–Stokes equations with shear-dependent viscosity. Using a fixed point argument in an appropriate functional setting, we establish the existence of a strong solution for small and suitably regular data. Uniqueness results are obtained under similar conditions.

The aim of this paper is to establish necessary optimality conditions for optimal control problems governed by steady, incompressible Navier-Stokes equations with shear-dependent viscosity. The main difficulty derives from the fact that equations of this type may exhibit non-uniqueness of weak solutions, and is overcome by introducing a family of approximate control problems governed by well posed generalized Stokes systems and by passing to the limit in the corresponding optimality conditions.

This article is concerned with the equations governing the steady motion of a viscoelastic incompressible second-order fluid in a bounded domain. A new proof of existence and uniqueness of strong solutions is given. In addition, using appropriate finite element methods to approximate a coupled equivalent problem, sharp error estimates are obtained using a fixed point argument. The method is applied to the two-dimensional lid-driven cavity problem, at low Reynolds number and in a certain range of values of the viscoelastic parameters, to analyze the combined effects of inertia and viscoelasticity on the flow. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007

This work is concerned with the study of steady flows of an incompressible viscoelastic fluid of Oldroyd type, with viscosity depending on the second invariant of the rate of deformation tensor in an exterior domain. We establish a result of existence and uniqueness of strong solutions for sufficiently small data and give estimates relating these solutions to those of the corresponding generalized Newtonian fluid.

This work is concerned with the study of steady flows of an incompressible quasi-Newtonian fluid, with viscosity depending on the second invariant of the rate of deformation tensor, in bounded domains. We first establish a result of existence of measure-valued solutions. Next, we prove that under some natural monotonicity conditions, these solutions are weak.

We study a system of nonlinear partial differential equations governing the motion of an incompressible viscoelastic fluid of Oldroyd type in a bounded domain, with a non-Newtonian viscosity depending on the second invariant of the rate of deformation tensor. Considering the equations in a suitably decomposed form, we establish, for small and suitably regular data, existence of a unique solution using a fixed point argument in an appropriate functional setting. This model includes the classical Oldroyd-B fluid as a particular case.

We consider time optimal control problems governed by semilinear parabolic equations with Dirichlet boundary controls in the presence of a target state constraint. To establish optimality conditions for the terminal time $T$, we define a new Hamiltonian functional. Due to regularity results for the state and the adjoint state variables, this Hamiltonian belongs to $L_{l o c}^r(0,T)$ for some $r>1$. By proving that it satisfies a differential equation corresponding to an optimality condition for $T$, we deduce that it belongs to $W^{1,1}(0,T)$. This result answers to the question: how to define Hamiltonian functionals for infinite dimensional problems with variable endpoints (see [10], p. 282 and p. 595).

This paper is concerned with steady, fully developed motion of a Navier--Stokes fluid with shear-dependent viscosity in a curved pipe under a given axial pressure gradient. We establish existence and uniqueness results, derive appropriate estimates, and prove a characterization of the secondary flows. The approximation, with respect to the curvature ratio, of the full governing systems by some Dean-like equation is studied.

This paper is concerned with distributed and Dirichlet boundary controls of semilinear parabolic equations, in the presence of pointwise state constraints. The paper is divided into two parts. In the first part we define solutions of the state equation as the limit of a sequence of solutions for equations with Robin boundary conditions. We establish Taylor expansions for solutions of the state equation with respect to perturbations of boundary control (Theorem 5.2). For problems with no state constraints, we prove three decoupled Pontryagin's principles, one for the distributed control, one for the boundary control, and the last one for the control in the initial condition (Theorem 2.1). Tools and results of Part 1 are used in the second part to derive Pontryagin's principles for problems with pointwise state constraints.

We study the numerical approximation of distributed nonlinear optimal control problems governed by semilinear elliptic partial differential equations with pointwise constraints on the control. Our main result are error estimates for optimal controls in the maximum norm. Characterization results are stated for optimal and discretized optimal control. Moreover, the uniform convergence of discretized controls to optimal controls is proven under natural assumptions.

An augmented Lagrangian SQP method is discussed for a class of nonlinear optimal control problems in Banach spaces with constraints on the control. The convergence of the method is investigated by its equivalence with the generalized Newton method for the optimality system of the augmented optimal control problem. The method is shown to be quadratically convergent, if the optimality system of the standard non-augmented SQP method is strongly regular in the sense of Robinson. This result is applied to a test problem for the heat equation with Stefan-Boltzmann boundary condition. The numerical tests confirm the theoretical results.

We consider control problems governed by semilinear parabolic equations with pointwise state constraints and controls in an Lp -space (p < ∞). We construct a correct relaxed problem, prove some relaxation results, and derive necessary optimality conditions.

We consider systems governed by a nonlinear parabolic equation, with a distributed control and a disturbance in the initial condition. We prove the existence of solutions to a corresponding minimax problem, and we obtain necessary optimality conditions in the form of Pontryagin's principles.

\noindent We consider control problems governed by semilinear parabolic equations in the presence of pointwise mixed control-state constraints. We obtain optimality conditions with finitely additive measures as multipliers associated to the mixed constraints. We study the regularity of the multipliers for different problems. In particular, when a monotonicity or a separation condition is satisfied, we prove that multipliers are bounded measurable functions.

We study the approximation of control problems governed by elliptic partial differential equations with pointwise state constraints. For a finite dimensional approximation of the control set and for suitable perturbations of the state constraints, we prove that the corresponding sequence of discrete control problems converges to a relaxed problem. A similar analysis is carried out for problems in which the state equation is discretized by a finite element method.

In this paper, we prove that the combination of representation theorems for additive measures introduced in Ref. 1, together with a Lagrange multiplier theorem, leads to a short and direct proof of the optimality conditions for Dirichlet control problems with pointwise state constraints.

This paper deals with optimal control problems of parabolic equations in the presence of pointwise state constraints. We consider bounded controls which act in the initial condition of the state equation. The state variable is a bounded continuous function on the domain $Q=\Omega\times ]0,T[$ but is not continuous on $\overline Q$. In this case, the multiplier associated with state constraints is a regular bounded finitely additive measure on Q (but not a $\sigma$-additive one). Using some properties of the Stone--\u{C}ech compactification, we prove a decomposition theorem for this measure which allows us to interpret the adjoint equation in a classical sense. We obtain new optimality conditions for these kinds of problems, and we apply these results to the case of bilateral constraints.

In this paper we study a minimax control problem for parabolic equations in the presence of pointwise state constraints. The terminology minimax here refers to a cost functional defined with a $L^{\infty}$-norm. The directional derivatives of the $L^{\infty}$-norm are elements of $(L^{\infty})'$. Therefore, the adjoint equation may involve finitely additive measures in place of Radon measures. To overcome this difficulty, we introduce a compactification (of Stone--\u{C}ech type). We prove necessary optimality conditions which are new, both in the case with no state constraints and in the case with state constraints. Under some convexity conditions, these optimality conditions are also sufficient.

In this paper we study optimal control problems governed by semilinear elliptic equations in the presence of pointwise state constraints. Since no convexity condition is assumed on data of the problem, we define a relaxed control problem, prove the existence of relaxed solutions, and give some relaxation results. By adapting the penalty method of Berkovitz, we prove a Pontryagin's minimum principle for relaxed solutions in nonqualified form and in qualified form under a stability condition.

In this paper we study a minimax boundary control problem with pointwise state constraints for parabolic equations under unknown disturbances in the initial condition. We establish necessary optimality conditions.