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We are interested in the necessary optimality conditions for optimal control problems of systems describing the unsteady flow of incompressible shear-thickening fluids. To overcome the difficulties related to the differentiability of the control-to-state mapping, consequence of the lack of regularity of the state variable, we first establish some intermediate optimality conditions and then pass to the limit.

We study optimal control problems of systems describing the flow of incompressible shear-thickening fluids. We prove existence of solutions and derive necessary optimality conditions under precise restrictions on the optimal control.

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.

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 is related to the differentiability of the control-to-state mapping and is overcome by introducing a family of smooth approximate control problems and by passing to the limit in the corresponding optimality conditions.

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.