80 research outputs found
Does Newton's method for set-valued maps converges uniformly in mild differentiability context?
No full textIn this article, we study the existence of Newton-type sequence for solving the equation where y is a small parameter, f is a function whose Fréchet derivative satisfies a Hölder condition of the form and F is a set-valued map between two Banach spaces X and Y . We prove that the Newton-type method , is locally convergent to a solution of if the set valued map is Aubin continuous at (0; x*) where x* is a solution of . Moreover, we show that this convergence is superlinear uniformly in the parameter y and quadratic when d = 1
High Order Discrete Approximations to Mayer's Problems for Linear Systems
No full textThis paper presents a discretization scheme for Mayer's type optimal control problems of linear systems. The scheme is based on second order Volterra--Fliess approximations, and on an augmentation of the control variable in a control set of higher dimension. Compared with the existing results, it has the advantage of providing a higher order accuracy, which may make it more efficient when aiming for a certain precision. Error estimations (depending on the controllability index of the system at the solution) are proved by using a recent result about stability of the optimal solution with respect to disturbances. Numerical results are provided which show the sharpness of the error estimations.
Read More: http://epubs.siam.org/doi/abs/10.1137/16M107914
On the Discretization of Switched Linear Systems
No full textA bilinear single-control system which can be viewed as a control formulation of a linear switched system is considered. The control is restricted to take values either (i) in f0; 1g (switched system), or (ii) in [0; 1] (relaxed system). In more practical considerations the control is often allowed to change only at the points of a given
time-net with a step length h. The paper investigates what is the approximation error in terms of the reachable set in the two cases (i) and (ii). The error estimates that follow directly from known results are of order ph and h, respectively. In the present paper estimations of order h and h1:5 are proved in a constructive way. The
second one makes use of the e®ect of non-accumulation of errors established earlier by the second author
Does Newton's method converges uniformly in mild differentiability context?
No full textInternational audienc
No differentiable perturbed Newton's method for functions with values in a cone
No full textInternational audienceThis paper deals with variational inclusions of the form 0 2 f(x) + g(x) K where f is smooth function from a re exive Banach space X into a Banach space Y , g is a Lipschitz function from X into Y and K is a nonempty closed convex cone in the space Y . We show that the previous problem can be solved by an extension of the Zincenko's method which can be seen as a perturbed Newton's method. Numerical results are given at the end of the paper
Modelos matemáticos para o estudo da aderência ao tratamento da tuberculose levando em conta os efeitos do HIV/AIDS e diabetes
No full textIn this work, we propose a new mathematical model for the study of the effectiveness of TB treatment taking into account the vulnerable subpopulations, HIV/AIDS and diabetic patients. Our model studies the different types of treatment resistance, multidrug-resistant (MDR TB) and extensively drug-resistant (XDRTB). We use two modeling techniques, ordinary differential equations (ODE) and fractional-order derivatives equations (FDE) in the Caputo sense. The main mathematical and epidemiological properties of the model are investigated. The basic reproduction number (0) in the different subpopulations (diabetics, HIV/AIDS, and those who do not suffer from these diseases) was studied. We present results that allow us to know how the basic reproductive number is affected when we vary the parameters of resistance and recovery together. We performed a sensitivity analysis of the parameters associated with TB. We proved the persistence of tuberculosis in a subpopulation showing the need to apply a control strategy. We formulated and studied an optimal control problem with the objective of reducing resistance to tuberculosis treatment. The controls are focused on reinfection/reactivation, MDR-TB and XDR-TB differentiated into subpopulations. We use the models with ODE and FDE in the formulation of the control problems. In order to study our models, we performed computational simulations. Among the results obtained, we have that drug-sensitive TB reported a greater number of cases with respect to MDR-TB and XDR-TB cases, and MDR-TB cases surpass XDR-TB cases, except in the diabetes subpopulation, which has a growth of XDR-TB cases that surpasses the other compartments of resistant of all the subpopulations. We show the need to pay differentiated attention to these vulnerable subpopulations due to the behavior of resistant cases. Regarding the control study, we obtained that the most effective strategy is to activate all controls and start with a high control. With this strategy we reduced the number of resistant cases significantly and prevented the growth of cases. This work helps health policies on how to act in this disease and these ideas can be applied to other epidemics of respiratory transmission.Neste trabalho, propomos um novo modelo matemático para o estudo da eficácia do tratamento da tuberculose, tendo em conta as subpopulações vulneráveis, o HIV/AIDS e doentes diabéticos. O nosso modelo estuda os diferentes tipos de resistência ao tratamento, multirresistente (MDR-TB) e extensivamente resistente aos fármacos (XDR-TB). Utilizamos duas técnicas de modelagem, equações diferenciais ordinárias (EDO) e derivadas de ordem fracional (EDF) no sentido de Caputo. As principais características matemáticas e epidemiológicas do modelo são investigadas. Foi obtido o número básico de reprodução (0) nas diferentes subpopulações (diabéticos, HIV/AIDS, e aqueles que não sofrem destas doenças). Apresentamos resultados que nos permitem saber como o número básico de reprodução é afetado quando variamos os parâmetros de resistência e recuperação conjuntamente. Realizamos uma análise de sensibilidade dos parâmetros associados à tuberculose. Demonstramos a persistência da tuberculose numa subpopulação num caso particular, mostrando a necessidade de aplicar uma estratégia de controle. Formulamos e estudamos um problema de controle ótimo com o objetivo de reduzir a resistência ao tratamento da tuberculose. Os controles se concentram na reinfecção/reactivação, MDR-TB e XDR-TB diferenciados em subpopulações. Para formular estes problemas, utilizamos os modelos ODE e FDE. A fim de estudar o nosso modelo, realizamos simulações computacionais. Entre os resultados obtidos, temos que o maior número de casos de infectados foram os TB sensíveis, e os casos de MDR-TB ultrapassam os casos de XDR-TB, exceto na subpopulação de diabéticos, que tem um crescimento de casos de XDR-TB que ultrapassa os outros compartimentos de todas as subpopulações. Mostramos a necessidade de prestar uma atenção diferenciada a estas subpopulações vulneráveis devido ao comportamento de casos resistentes. Em relação ao estudo de controle, obtivemos que a estratégia mais eficaz é quando ativamos todos os controles e começamos com um controle elevado. Com esta estratégia, reduzimos significativamente o número de casos resistentes e impedimos o crescimento de casos ao longo do tempo. Este trabalho ajuda as políticas de saúde sobre como agir nesta doença e estas ideias podem ser aplicadas a outras epidemias de transmissão respiratória
Solving triangular algebraic systems by means of simultaneous iterations
No full textInternational audienc
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