Papers by Domingo Alberto Tarzia
Mecánica Computacional, 2005
Los modelos anteriores de toma de agua por raíces de cultivos no toman en cuenta el crecimiento d... more Los modelos anteriores de toma de agua por raíces de cultivos no toman en cuenta el crecimiento de las mismas resolviendo el conjunto de ecuaciones que gobiernan el sistema
Numerical Functional Analysis and Optimization, 2020
We consider an optimal control problem Q governed by an elliptic quasivariational inequality with... more We consider an optimal control problem Q governed by an elliptic quasivariational inequality with unilateral constraints. The existence of optimal pairs of the problem is a well known result, see , for instance. We associate to Q a new optimal control problem Q, obtained by perturbing the state inequality (including the set of constraints and the nonlinear operator) and the cost functional, as well. Then, we provide sufficient conditions which guarantee the convergence of solutions of Problem Q to a solution of Problem Q. The proofs are based on convergence results for elliptic quasivariational inequalities, obtained by using arguments of compactness, lower semicontinuity, monotonicity, penalty and various estimates. Finally, we illustrate the use of the abstract convergence results in the study of optimal control associated with two boundary value problems. The first one describes the equilibrium of an elastic body in frictional contact with an obstacle, the so-called foundation. The process is static and the contact is modeled with normal compliance and unilateral constraint, associated to a version of Coulomb's law of dry friction. The second one describes a stationary heat transfer problem with unilateral constraints. For the two problems we prove existence, uniqueness and convergence results together with the corresponding physical interpretation.
We study the asymptotic behavior of an optimal distributed control problem where the state is giv... more We study the asymptotic behavior of an optimal distributed control problem where the state is given by the heat equation with mixed boundary conditions. The parameter α intervenes in the Robin boundary condition and it represents the heat transfer coefficient on a portion Γ 1 of the boundary of a given regular n-dimensional domain. For each α, the distributed parabolic control problem optimizes the internal energy g. It is proven that the optimal control ĝα with optimal state u ĝαα and optimal adjoint state p ĝαα are convergent as α → ∞ (in norm of a suitable Sobolev parabolic space) to ĝ, u ĝ and p ĝ , respectively, where the limit problem has Dirichlet (instead of Robin) boundary conditions on Γ 1 . The main techniques used are derived from the parabolic variational inequality theory.
Se utiliza una condicion convectiva, en el borde fijo de un material semi-infinito, en el modelo ... more Se utiliza una condicion convectiva, en el borde fijo de un material semi-infinito, en el modelo de region pastosa de Solomon-Wilson-Alexiades (Letters Heat Mass Transfer, 9 (1982), 319-324) a una fase para obtener soluciones explicitas en el correspondiente problema de cambio de fase. Se obtiene la condicion necesaria y suficiente para la existencia de la solucion explicita la cual esta dada por una desigualdad que debe verificar el coeficiente de transferencia de calor en funcion de los datos del modelo. Mas aun, dicho problema es equivalente al problema con condicion de temperatura en el borde fijo obteniendose para este caso una desigualdad para el coeficiente que caracteriza una de las fronteras libres.
Se consideran dos sistemas estacionarios de conduccion del calor, S y Sa, en un dominio multidime... more Se consideran dos sistemas estacionarios de conduccion del calor, S y Sa, en un dominio multidimensional D acotado para la ecuacion de Poisson con fuente de energia g. En uno de ellos se proponen condiciones de contorno mixtas (temperatura b en la porcion de frontera F1, flujo de calor q sobre el borde F2 y una condicion adiabatica sobre la restante porcion de frontera F3). En el otro sistema se reemplaza la condicion sobre F1 por una condicion de flujo de calor convectivo con coeficiente de conveccion a. A su vez, para S y Sa, se establecen problemas de control optimo (P) y (Pa) donde la variable de control sera el flujo de calor q. Si el dominio D es rectangular, se conocen de manera explicita el control optimo continuo y el estado correspondiente de los sistemas. En el presente trabajo, utilizando un esquema de diferencias finitas, se discretizan los sistemas S y Sa obteniendose Sh y Sah ademas de (Ph) y (Pah), siendo h el paso espacial en la discretizacion. El objetivo del traba...
We study the determination of unknown thermal coefficient of a semi-infinite material through a p... more We study the determination of unknown thermal coefficient of a semi-infinite material through a phase-change process with an overspecified condition on the fixed face with temperature-dependent thermal conductivity. We determine necessary and sufficient conditions on data in order to obtain the existence of the solution. We also give formulae for the unknown coefficients.
En (M. Boukrouche and D.A. Tarzia, Comput. Optim. Appl., 53, 375-393 (2012)) se estudia una famil... more En (M. Boukrouche and D.A. Tarzia, Comput. Optim. Appl., 53, 375-393 (2012)) se estudia una familia de problemas de control optimo continuos Pa gobernados por inecuaciones variacionales elipticas Sa donde el parametro a de la familia es el coeficiente de transferencia de calor sobre una porcion de la frontera del dominio n-dimensional del material. En ese trabajo se mostro la existencia y unicidad de su solucion, es decir que, dado un control le queda asociada una unica funcion estado y ademas hay solo un control que minimiza al funcional costo. El objetivo de este trabajo consiste en analizar numericamente la antes mencionada familia de problemas de control optimo usando el metodo de elementos finitos con triangulos de Lagrange de tipo 1. Entonces, para cada valor del parametro a se discretiza la inecuacion variacional eliptica que determina el estado del sistema. Se obtiene existencia de una solucion del estado discreto del sistema Sa y la convergencia global fuerte del estado discr...
Communications in Nonlinear Science and Numerical Simulation, 2021
In this paper, we study optimal control problems on the internal energy for a system governed by ... more In this paper, we study optimal control problems on the internal energy for a system governed by a class of elliptic boundary hemivariational inequalities with a parameter. The system has been originated by a steady-state heat conduction problem with non-monotone multivalued subdifferential boundary condition on a portion of the boundary of the domain described by the Clarke generalized gradient of a locally Lipschitz function. We prove an existence result for the optimal controls and we show an asymptotic result for the optimal controls and the system states, when the parameter, like a heat transfer coefficient, tends to infinity on a portion of the boundary.
Proceedings of the American Mathematical Society, 2000
We consider a one-phase Stefan problem for the heat equation with a nonlinear reaction term. We f... more We consider a one-phase Stefan problem for the heat equation with a nonlinear reaction term. We first exhibit an energy condition, involving the initial data, under which the solution blows up in finite time in L ∞ norm. We next prove that all global solutions are bounded and decay uniformly to 0, and that either: (i) the free boundary converges to a finite limit and the solution decays at an exponential rate, or (ii) the free boundary grows up to infinity and the decay rate is at most polynomial. Finally, we show that small data solutions behave like (i).
System Modeling and Optimization, 2016
We consider a bounded domain n whose regular boundary 12 consists of the union ... more We consider a bounded domain n whose regular boundary 12 consists of the union of two disjoint portions 1 and 2 with meas 1 ( ) 0 . The convergence of a family of continuous distributed mixed elliptic optimal control problems ( P ), governed by elliptic variational equalities, when the parameter of the family (the heat transfer coefficient on the portion of the boundary 1 ) goes to infinity was studied in Gariboldi -Tarzia, Appl. Math. Optim., 47 (2003), 213-230. It has been proved that the optimal control, and their corresponding system and adjoint system states are strongly convergent, in adequate functional spaces, to the optimal control, and the system and adjoint states of another distributed mixed elliptic optimal control problem ( P ) governed also by an elliptic variational equality with a different boundary condition on the portion of the boundary 1 We consider the discrete approximations ( h P ) and ( h P ) of the optimal control problems ( P ) and ( P ) respectively, for each 0 h and for each 0 , through the finite element method with Lagrange's triangles of type 1 with parameter h (the longest side of the triangles). We also discretize the elliptic variational equalities which define the system and their adjoint system states, and the corresponding cost functional of the distributed optimal control problems ( P ) and ( P ). The goal of this paper is to study the double convergence of this family of discrete distributed mixed elliptic optimal control problems ( h P ) when the parameters goes to infinity and the parameter h goes to zero simultaneously. We prove the convergence of the discrete optimal controls, the discrete system and adjoint states of the family ( h P ) to the corresponding to the discrete distributed mixed elliptic optimal control problem ( h P ) when , for each 0 h , in adequate functional spaces. We study the convergence of the discrete distributed optimal control problems ( h P ) and ( h P ) when 0 h obtaining a commutative diagram which relates the continuous and discrete distributed mixed elliptic optimal control problems ,, hh P P P and ( P ) by taking the limits 0 h and respectively. We also study the double convergence of ( h P ) to ( P ) when ( , ) (0, ) h which represents the diagonal convergence in the above commutative diagram.
Journal of Applied Mathematics and Computing, 2020
We consider a steady-state heat conduction problem in a multidimensional bounded domain Ω for the... more We consider a steady-state heat conduction problem in a multidimensional bounded domain Ω for the Poisson equation with constant internal energy g and mixed boundary conditions given by a constant temperature b in the portion Γ 1 of the boundary and a constant heat flux q in the remaining portion Γ 2 of the boundary. Moreover, we consider a family of steady-state heat conduction problems with a convective condition on the boundary Γ 1 with heat transfer coefficient α and external temperature b. We obtain explicitly, for a rectangular domain in R 2 , an annulus in R 2 and a spherical shell in R 3 , the optimal controls, the system states and adjoint states for the following optimal control problems: a distributed control problem on the internal energy g, a boundary optimal control problem on the heat flux q, a boundary optimal control problem on the external temperature b and a distributed-boundary simultaneous optimal control problem on the source g and the flux q. These explicit solutions can be used for testing new numerical methods as a benchmark test. In agreement with theory, it is proved that the system state, adjoint state, optimal controls and optimal values corresponding to the problem with a convective condition on Γ 1 converge, when α → ∞, to the corresponding system state, adjoint state, optimal controls and optimal values that arise from the problem with a temperature condition on Γ 1 . Also, we analyze the order of convergence in each case, which turns out to be 1/α being new for these kind of elliptic optimal control problems.
MAT Serie A, 2007
An alternative method to compute the Michaelis-Menten parameters for nutrient uptake kinetics is ... more An alternative method to compute the Michaelis-Menten parameters for nutrient uptake kinetics is presented. The method uses the explicit integration in time of the differential equation obtained from the relationship between decreasing nutrient concentration in a given volume of solution culture and nutrient uptake by roots. Thus, the depletion curve C = C(t) is obtained. Then, by using a non linear least squares fit method we obtain good estimations of the initial first derivative (rate of nutrient uptake) and the initial second derivative of the concentration. With these values and the average concentration at large times, explicit formulas for J m (maximum influx at infinity concentrations), K M (the Michaelis-Menten constant) and E (the efflux) are obtained. The method is applied to the pioneer depletion curve obtained by Claassen and Barber. Results obtained by Claassen-Barber, linearization Hanes method, non linear fit method applied to C´(t) vs. C and our method are compared. The results shown that the error obtained by our method is significantly smaller for the parameter E in contrast to the results obtained by the other methods.
MAT Serie A, 2005
Un caso de determinación de coeficientes térmicos desconocidos de un material semiinfinito poroso... more Un caso de determinación de coeficientes térmicos desconocidos de un material semiinfinito poroso a través de un problema de desublimación con acoplamiento de temperatura y humedad", 17-22.
MAT Serie A, 2004
A two-phase Stefan problem with heat source terms in both liquid and solid phases for a semi-in¯n... more A two-phase Stefan problem with heat source terms in both liquid and solid phases for a semi-in¯nite phase-change material is considered. The internal heat source functions are given by g ´(j = 1 solid phase; j = 2 liquid phase), ½ is the mass density, l is the fusion latent heat by unit of mass; a 2 j is the di®usion coe±cient, x is spatial variable, t is the temporal variable and d j 2 R. A similarity solution is obtained for any data when a temperature boundary condition is imposed at the ¯xed face x = 0; when a °ux condition of the type ¡q 0 = p t (q 0 > 0) is imposed on x = 0 then there exists a similarity solution if and only if a restriction on q 0 is satis¯ed.
MAT Serie A, 2004
We study a one-phase Stefan problem for a semi-in¯nite material with temperaturedependent thermal... more We study a one-phase Stefan problem for a semi-in¯nite material with temperaturedependent thermal conductivity with a constant temperature or a heat °ux condition of the type ¡q 0 = p t (q 0 > 0) at the ¯xed face x = 0. We obtain in both cases su±cient conditions for data in order to have a parametric representation of the solution of the similarity type for t ¸t0 > 0 with t 0 an arbitrary positive time. These explicit solutions are obtained through the unique solution of an integral equation with the time as a parameter.
Quarterly of Applied Mathematics, 1992
We consider a slab, represented by the interval 0 > x > x 0 0 > x > {x_0} , at the in... more We consider a slab, represented by the interval 0 > x > x 0 0 > x > {x_0} , at the initial temperature θ 0 = θ 0 ( x ) ≥ 0 ( o r ϕ 0 = ϕ 0 ( x ) ≥ 0 ) {\theta _0} = {\theta _0}\left ( x \right ) \ge 0\left ( {or {\phi _0} = {\phi _0}\left ( x \right ) \ge 0} \right ) having a heat flux q = q ( t ) > 0 q = q\left ( t \right ) > 0 (or convective boundary condition with a heat transfer coefficient h h ) on the left face x = 0 x = 0 and a temperature condition b ( t ) > 0 b\left ( t \right ) > 0 on the right face x = x 0 x = {x_0} ( x 0 x_{0} could be also + ∞ + \infty , i.e., a semi-infinite material). We consider the corresponding heat conduction problem and assume that the phase-change temperature is 0 ∘ C {0^ \circ }C .
Advances in Differential Equations and Control Processes, 2016
The numerical analysis of a family of distributed mixed optimal control problems governed by elli... more The numerical analysis of a family of distributed mixed optimal control problems governed by elliptic variational inequalities (with parameter α > 0) is obtained through the finite element method when its parameter h → 0. We also obtain the limit of the discrete optimal control and the associated state system solutions when α → ∞ (for each h > 0) and a commutative diagram for two continuous and two discrete optimal control and its associated state system solutions is obtained when h → 0 and α → ∞. Moreover, the double convergence is also obtained when (h, α) → (0, ∞).
Journal of Applied Mathematics, 2015
We complete the Solomon-Wilson-Alexiades’s mushy zone model (Solomon, 1982) for the one-phase Lam... more We complete the Solomon-Wilson-Alexiades’s mushy zone model (Solomon, 1982) for the one-phase Lamé-Clapeyron-Stefan problem by obtaining explicit solutions when a convective or heat flux boundary condition is imposed on the fixed face for a semi-infinite material. We also obtain the necessary and sufficient condition on data in order to get the explicit solutions for both cases which is new with respect to the original model. Moreover, when these conditions are satisfied, the two phase-change problems are equivalent to the same problem with a temperature boundary condition on the fixed face and therefore an inequality for the coefficient which characterized one of the two free interfaces of the model is also obtained.
International Journal of Differential Equations, 2015
The objective of this work is to make the numerical analysis, through the finite element method w... more The objective of this work is to make the numerical analysis, through the finite element method with Lagrange’s triangles of type 1, of a continuous optimal control problem governed by an elliptic variational inequality where the control variable is the internal energyg. The existence and uniqueness of this continuous optimal control problem and its associated state system were proved previously. In this paper, we discretize the elliptic variational inequality which defines the state system and the corresponding cost functional, and we prove that there exist a discrete optimal control and its associated discrete state system for each positiveh(the parameter of the finite element method approximation). Finally, we show that the discrete optimal control and its associated state system converge to the continuous optimal control and its associated state system when the parameterhgoes to zero.
IFIP Advances in Information and Communication Technology, 2013
I) We consider a system governed by a free boundary problem with Tresca condition on a part of th... more I) We consider a system governed by a free boundary problem with Tresca condition on a part of the boundary of a material domain with a source term g through a parabolic variational inequality of the second kind. We prove the existence and uniqueness results to a family of distributed optimal control problems over g for each parameter h > 0, associated to the Newton law (Robin boundary condition), and of another distributed optimal control problem associated to a Dirichlet boundary condition. We generalize for parabolic variational inequalities of the second kind the Mignot's inequality obtained for elliptic variational inequalities (Mignot, J. Funct. Anal., 22 (1976), 130-185), and we obtain the strictly convexity of a quadratic cost functional through the regularization method for the non-differentiable term in the parabolic variational inequality for each parameter h. We also prove, when h → +∞, the strong convergence of the optimal controls and states associated to this family of optimal control problems with the Newton law to that of the optimal control problem associated to a Dirichlet boundary condition. II) Moreover, if we consider a parabolic obstacle problem as a system governed by a parabolic variational inequalities of the first kind then we can also obtain the same results of Part I for the existence, uniqueness and convergence for the corresponding distributed optimal control problems. III) If we consider, in the problem given in Part I, a flux on a part of the boundary of a material domain as a control variable (Neumann boundary optimal control problem) for a system governed by a parabolic variational inequality of second kind then we can also obtain the existence and uniqueness results for Neumann boundary optimal control problems for each parameter h > 0, but in this case the convergence when h → +∞ is still an open problem.
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Papers by Domingo Alberto Tarzia