The dual weighted residuals approach to optimal control of ordinary differential equations

The methodology of dual weighted residuals is applied to an optimal control problem for ordinary differential equations. The differential equations are discretized by finite element methods. An a posteriori error estimate is derived and an adaptive algorithm is formulated. The algorithm is implemented in Matlab and tested on a simple model problem from vehicle dynamics

Improved approximation rates for a parabolic control problem with an objective promoting directional sparsity

We discretize a directionally sparse parabolic control problem governed by a linear equation by means of control approximations that are piecewise constant in time and continuous piecewise linear in space. By discretizing the objective functional with the help of appropriate numerical quadrature formulas, we are able to show that the discrete optimal solution exhibits a directional sparse pattern alike the one enjoyed by the continuous solution. Error estimates are obtained and a comparison with the cases of having piecewise approximations of the control or a semilinear state equation are discussed. Numerical experiments that illustrate the theoretical results are included.The first two authors were partially supported by the Spanish Ministerio de Economía y Competitividad under projects MTM2014-57531-P and MTM2017-83185-P

Error estimates for the discretization of the velocity tracking problem

In this paper we are continuing our work (Casas and Chrysafinos, SIAM J Numer Anal 50(5):2281–2306, 2012), concerning a priori error estimates for the velocity tracking of two-dimensional evolutionary Navier–Stokes flows. The controls are of distributed type, and subject to point-wise control constraints. The discretization scheme of the state and adjoint equations is based on a discontinuous time-stepping scheme (in time) combined with conforming finite elements (in space) for the velocity and pressure. Provided that the time and space discretization parameters, t and h respectively, satisfy t = Ch2, error estimates of order O(h2) and O(h 3/2 – 2/p ) with p > 3 depending on the regularity of the target and the initial velocity, are proved for the difference between the locally optimal controls and their discrete approximations, when the controls are discretized by the variational discretization approach and by using piecewise-linear functions in space respectively. Both results are based on new duality arguments for the evolutionary Navier–Stokes equations

Sparse initial data indentification for parabolic pde and its finite element approximations

We address the problem of inverse source identification for parabolic equations from the optimal control viewpoint employing measures of minimal norm as initial data. We adopt the point of view of approximate controllability so that the target is not required to be achieved exactly but only in an approximate sense. We prove an approximate inversion result and derive a characterization of the optimal initial measures by means of duality and the minimization of a suitable quadratic functional on the solutions of the adjoint system. We prove the sparsity of the optimal initial measures showing that they are supported in sets of null Lebesgue measure. As a consequence, approximate controllability can be achieved efficiently by means of controls that are activated in a finite number of pointwise locations. Moreover, we discuss the finite element numerical approximation of the control problem providing a convergence result of the corresponding optimal measures and states as the discretization parameters tend to zero.The first author was supported by Spanish Ministerio de Economía y Competitividad under project MTM2011-22711. The third author was supported by the Advanced Grant NUMERIWAVES/FP7-246775 of the European Research Council Executive Agency, FA9550-14-1-0214 of the EOARD-AFOSR, FA9550-15-1-0027 of AFOSR, the BERC 2014-2017 program of the Basque Government, the MTM2011-29306 and SEV-2013-0323 Grants of the MINECO, the CIMI-Toulouse Excellence Chair in PDEs, Control and Numerics and a Humboldt Award at the University of Erlangen-Nürnberg

A model describing photosynthesis in terms of gas diffusion and enzyme kinetics

A model predicting net photosynthesis of individual plant leaves for a variety of environmental conditions has been developed. It is based on an electrical analogue describing gas diffusion from the free atmosphere to the sites of CO 2 fixation and a Michaelis-Menten equation describing CO 2 fixation. The model is presented in two versions, a simplified form without respiration and a more complex form including respiration. Both versions include terms for light and temperature dependence of CO 2 fixation and light control of stomatal resistance. The second version also includes terms for temperature, light, and oxygen dependence of respiration and O 2 dependence of CO 2 fixation.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/47495/1/425_2004_Article_BF00387066.pd

Densidade, tamanho e distribuição estomática em 35 espécies de árvores na Amazônia Central

Stomata are turgor-operated valves that control water loss and CO2 uptake during photosynthesis, and thereby water relation and plant biomass accumulation is closely related to stomatal functioning. The aims of this work were to document how stomata are distributed on the leaf surface and to determine if there is any significant variation in stomatal characteristics among Amazonian tree species, and finally to study the relationship between stomatal density (S D) and tree height. Thirty five trees (>17 m tall) of different species were selected. Stomatal type, density (S D), size (S S) and stomatal distribution on the leaf surface were determined using nail polish imprints taken from both leaf surfaces. Irrespective of tree species, stomata were located only on the abaxial surface (hypostomaty), with large variation in both S D and S S among species. S D ranged from 110 mm-2 in Neea altissima to 846 mm-2 in Qualea acuminata. However, in most species S D ranges between 271 and 543 mm-2, with a negative relationship between S D and S S. We also found a positive relationship between S D and tree height (r² = 0.14, p 17 m de altura) de diferentes espécies foram selecionadas. Tipo de complexo estomático, S D, tamanho (S S) e distribuição na superfície foliar foram determinados utilizando impressões de ambas as superfícies foliares com esmalte incolor. Independente da espécie, os estômatos foram encontrados apenas na superfície abaxial (hipoestomatia) com ampla variação na S D e no S S entre espécies. A densidade estomática variou de 110 mm-2 em Neea altissima a 846 mm-2 em Qualea acuminata. Entretanto, a maioria das espécies apresentou S D entre 271 e 543 mm-2, com uma relação negativa entre S D e S S. Observou-se uma relação positiva entre S D e altura arbórea (r² = 0.14, p < 0.01), não havendo relação entre S D e espessura foliar. Os tipos estomáticos mais comuns foram: anomocíticos (37%), seguidos de paracíticos (26%) e anisocíticos (11%). Concluiu-se que em espécies da Amazônia, a distribuição de estômatos na superfície foliar está mais relacionada a fatores genéticos de cada espécie do que a variações ambientais. Entretanto, S D é fortemente influenciada por fatores ambientais concernentes à altura da árvore

Trust region methods with hierarchical finite element models for PDE-constrained optimization

In this paper, a Hierarchical Trust Region Algorithm for solving PDE-constrained optimization problems is developed. A hierarchy of finite element meshes is used to define a hierarchy of quadratic models for the approximation of the discrete reduced cost functional on the finest mesh. The proposed algorithm simultaneously controls the choice of the model and the size of the trust region radius. Application of the trust region convergence theory allows for proving that every accumulation point of the sequence produced by the algorithm is a stationary point of the discretized problem. Numerical examples illustrate the behavior of the method and show a considerable reduction of computation time compared to the standard Newton trust region scheme