On the singularities of Polar curves.
L2 Estimates for the Finite Element Method for the Dirichlet Problem with Singular Data.
Error estimates for the numerical approximation of semilinear elliptic control problems with finitely many state constraints
The goal of this paper is to derive some error estimates for the numerical discretization of some optimal control problems governed by semilinear elliptic equations with bound constraints on the control and a finitely number of equality and inequality state constraints. We prove some error estimates for the optimal controls in the norm and we also obtain error estimates for the Lagrange multipliers associated to the state constraints as well as for the optimal states and optimal adjoint states....
Necessary and sufficient optimality conditions for elliptic control problems with finitely many pointwise state constraints
The goal of this paper is to prove the first and second order optimality conditions for some control problems governed by semilinear elliptic equations with pointwise control constraints and finitely many equality and inequality pointwise state constraints. To carry out the analysis we formulate a regularity assumption which is equivalent to the first order optimality conditions. Though the presence of pointwise state constraints leads to a discontinuous adjoint state, we prove that the optimal...
Exact penalization of pointwise constraints for optimal control problems
Necessary and sufficient optimality conditions for elliptic control problems with finitely many pointwise state constraints
The goal of this paper is to prove the first and second order optimality conditions for some control problems governed by semilinear elliptic equations with pointwise control constraints and finitely many equality and inequality pointwise state constraints. To carry out the analysis we formulate a regularity assumption which is equivalent to the first order optimality conditions. Though the presence of pointwise state constraints leads to a discontinuous adjoint state, we prove that the optimal control...
Error Estimates for the Numerical Approximation of Semilinear Elliptic Control Problems with Finitely Many State Constraints
The goal of this paper is to derive some error estimates for the numerical discretization of some optimal control problems governed by semilinear elliptic equations with bound constraints on the control and a finitely number of equality and inequality state constraints. We prove some error estimates for the optimal controls in the norm and we also obtain error estimates for the Lagrange multipliers associated to the state constraints as well as for the optimal states and optimal...
Base points of polar curves
The base points of the system of polar curves of an irreducible algebroid plane curve with general moduli are determined. As consequences a lower bound for the Tjurina number and many continuous analytic invariants of the curve are found.
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Sobre el cálculo efectivo del género de las curvas algebraicas.
Acerca del genero virtual de la superficies algebráicas.
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Acerca del género virtual de las superficies algebráicas.
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Some optimal control problems of multistate equations appearing in fluid mechanics
Numerical analysis of some optimal control problems governed by a class of quasilinear elliptic equations
In this paper, we carry out the numerical analysis of a distributed optimal control problem governed by a quasilinear elliptic equation of non-monotone type. The goal is to prove the strong convergence of the discretization of the problem by finite elements. The main issue is to get error estimates for the discretization of the state equation. One of the difficulties in this analysis is that, in spite of the partial differential equation has a unique solution for any given control, the uniqueness...
Recent advances in the analysis of pointwise state-constrained elliptic optimal control problems
Optimal control problems for semilinear elliptic equations with control constraints and pointwise state constraints are studied. Several theoretical results are derived, which are necessary to carry out a numerical analysis for this class of control problems. In particular, sufficient second-order optimality conditions, some new regularity results on optimal controls and a sufficient condition for the uniqueness of the Lagrange multiplier associated with the state constraints are presented.
Numerical analysis of some optimal control problems governed by a class of quasilinear elliptic equations
In this paper, we carry out the numerical analysis of a distributed optimal control problem governed by a quasilinear elliptic equation of non-monotone type. The goal is to prove the strong convergence of the discretization of the problem by finite elements. The main issue is to get error estimates for the discretization of the state equation. One of the difficulties in this analysis is that, in spite of the partial differential equation has a unique solution for any given control, the uniqueness...
New regularity results and improved error estimates for optimal control problems with state constraints
In this paper we are concerned with a distributed optimal control problem governed by an elliptic partial differential equation. State constraints of box type are considered. We show that the Lagrange multiplier associated with the state constraints, which is known to be a measure, is indeed more regular under quite general assumptions. We discretize the problem by continuous piecewise linear finite elements and we are able to prove that, for the case of a linear equation, the order of convergence...
Perturbing plane cruve singularities.
We describe the singularity of all but finitely-many germs in a pencil generated by two germs of plane curve sharing no tangent.
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