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In his book on convex polytopes [2] A. D. Aleksandrov raised a general question of finding variational formulations and solutions to geometric problems of existence of convex polytopes in , n ≥ 2, with prescribed geometric data. Examples of such problems for closed convex polytopes for which variational solutions are known are the celebrated Minkowski problem [2] and the Gauss curvature problem [20]. In this paper we give a simple variational proof of existence for the A. D. Aleksandrov problem...
We present an a posteriori error analysis of adaptive finite element approximations of distributed control problems for second order elliptic boundary value problems under bound constraints on the control. The error analysis is based on a residual-type a posteriori error estimator that consists of edge and element residuals. Since we do not assume any regularity of the data of the problem, the error analysis further invokes data oscillations. We prove reliability and efficiency of the error estimator...
We present an a posteriori error analysis of adaptive finite
element approximations of distributed control problems for second
order elliptic boundary value problems under bound constraints on
the control. The error analysis is based on a residual-type a posteriori error estimator that consists of edge and element
residuals. Since we do not assume any regularity of the data of
the problem, the error analysis further invokes data oscillations.
We prove reliability and efficiency of the error estimator...
Local Lipschitz continuity of minimizers of certain integrals of the Calculus of Variations is obtained when the integrands are convex with respect to the gradient variable, but are not necessarily uniformly convex. In turn, these regularity results entail existence of minimizers of variational problems with non-homogeneous integrands nonconvex with respect to the gradient variable. The -dependence, explicitly appearing in the integrands, adds significant technical difficulties in the proof.
Local Lipschitz continuity of minimizers of certain integrals of the
Calculus of Variations is obtained when the integrands are convex with
respect to the gradient variable, but are not necessarily uniformly
convex. In turn, these regularity results entail existence of minimizers of
variational problems with non-homogeneous integrands nonconvex with
respect to the gradient variable. The x-dependence, explicitly appearing
in the integrands, adds significant technical difficulties in the proof.
In this paper, a sub-optimal boundary control strategy for a free boundary problem is investigated. The model is described by a non-smooth convection-diffusion equation. The control problem is addressed by an instantaneous strategy based on the characteristics method. The resulting time independent control problems are formulated as function space optimization problems with complementarity constraints. At each time step, the existence of an optimal solution is proved and first-order optimality conditions...
This paper presents a new observability estimate for parabolic equations in , where is a convex domain. The observation region is restricted over a product set of an open nonempty subset of and a subset of positive measure in . This estimate is derived with the aid of a quantitative unique continuation at one point in time. Applications to the bang-bang property for norm and time optimal control problems are provided.
Dans cet article, nous étudions la sensibilité d’un problème de contrôle optimal de type bilinéaire. Le coût est différentiable, quadratique et strictement convexe. Le système est gouverné par un opérateur parabolique du quatrième ordre et présente une perturbation additive dans l’équation d’état, ainsi qu’une partie bilinéaire, relativement au contrôle et à l’état , de la forme . Sous des conditions de petitesse de l’état initial et de la perturbation, nous exploitons les propriétés de régularité...
We consider a time optimal control problem arisen from the optimal management of a bioreactor devoted to the treatment of eutrophicated water. We formulate this realistic problem as a state-control constrained time optimal control problem. After analyzing the state system (a complex system of coupled partial differential equations with non-smooth coefficients for advection-diffusion-reaction with Michaelis-Menten kinetics, modelling the eutrophication processes) we demonstrate the existence of,...
We consider a time optimal control problem arisen from the optimal
management of a bioreactor devoted to the treatment of
eutrophicated water. We formulate this realistic problem as a
state-control constrained time optimal control problem. After
analyzing the state system (a complex system of coupled partial
differential equations with non-smooth coefficients for
advection-diffusion-reaction with Michaelis-Menten kinetics,
modelling the eutrophication processes) we demonstrate the
existence of,...
The main goal of this paper is to show some applications of the optimal control theory to the wastewater elimination problem. Firstly, we deal with the numerical simulation of a given situation. We present a suitable mathematical model, propose a method to solve it and show the numerical results for a realistic situation in the ría of Arousa (Spain). Secondly, in the same framework of wastewater elimination problem, we pose two economic-environmental problems which can be formulated as constrained...
We consider a class of age-structured control problems with nonlocal dynamics and boundary conditions. For these problems we suggest Arrow-type sufficient conditions for optimality of problems defined on finite as well as infinite time intervals. We examine some models as illustrations (optimal education and optimal offence control problems).
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