Regularization and Bang-bang conjugate times in optimal control

Consideramos o problema de controlo óptimo de tempo mínimo para sistemas de controlo mono-entrada e controlo afim num espaço de dimensão finita com condições inicial e final fixas, onde o controlo escalar toma valores num intervalo fechado. Quando aplicamos o método de tiro a este problema, vários obstáculos podem surgir uma vez que a função de tiro não é diferenciável quando o controlo é bang-bang. No caso bang-bang os tempos conjugados são teoricamente bem definidos para este tipo de sistemas de controlo, contudo os algoritmos computacionais directos disponíveis são de difícil aplicação. Por outro lado, no caso suave o conceito teórico e prático de tempos conjugados é bem conhecido, e ferramentas computacionais eficazes estão disponíveis. Propomos um procedimento de regularização para o qual as soluções do problema de tempo mínimo correspondente dependem de um parâmetro real positivo suficientemente pequeno e são definidas por funções suaves em relação à variável tempo, facilitando a aplicação do método de tiro simples. Provamos, sob hipóteses convenientes, a convergência forte das soluções do problema regularizado para a solução do problema inicial, quando o parâmetro real tende para zero. A determinação de tempos conjugados das trajectórias localmente óptimas do problema regularizado enquadra-se na teoria suave conhecida. Provamos, sob hipóteses adequadas, a convergência do primeiro tempo conjugado do problema regularizado para o primeiro tempo conjugado do problema inicial bang-bang, quando o parâmetro real tende para zero. Consequentemente, obtemos um algoritmo eficiente para a computação de tempos conjugados no caso bang-bang.

Nanosensor for assessing the risk of a cardiovascular disease

The incidence of cardiovascular diseases (CVD) has been increasing according to the European and global statistics. Thus, the development of new analytical devices, such as biosensors for assessing the risk of CVD could become a valuable approach for the improvement of healthcare service. In latest years, the nanotechnology has provided new materials with improved electronic properties which have an important contribution in the transduction mechanism of biosensors. Thus, in this thesis, biosensors based on field effect transistors with single-walled carbon nanotubes (NTFET) were developed for the detection of C-reactive protein (CRP) in clinical samples, that is, blood serum and saliva from a group of control patients and a group of CVD risk patients. CRP is an acute-phase protein, which is commonly known as the best validated biomarker for the assessment of CVD, the single-walled carbon nanotubes (SWCNT) were applied as transduction components, and the immunoreaction (interaction between the CRP antigen and the antibodies specific to CRP) was used as the mechanism of molecular recognition for the label-free detection of CRP. After the microfabrication of field effect transistors (FET), the screening of the most important variables for the dispersion of SWCNT, the assemblage of NTFET, and their application on standard solutions of CRP, it was found that NTFET respond accurately to CRP both in saliva and in serum samples, since similar CRP levels were found with the NTFET and the traditional methodology (ELISA technique). On the other hand, a strong correlation between salivary and serum CRP was found with NTFET, which means that saliva could be used, based on non-invasive sampling, as an alternative fluid to blood serum. It was also shown that NTFET could discriminate control patients from CVD risk patients, allowing the determination of a cut-off value for salivary CRP of 1900 ng L-1, which corresponds to the well established cut-off of 3 mg L-1 for CRP in serum, constituting an important finding for the possible establishment of a new range of CRP levels based on saliva. According to the data provided from the volunteer patients regarding their lipoprotein profile and lifestyle factors, it was concluded that the control and the CVD risk patients could be separated taking into account the various risk factors established in literature as strong contributors for developing a CVD, such as triglycerides, serum CRP, total cholesterol, LDL cholesterol, body mass index, Framingham risk score, hypertension, dyslipidemia, and diabetes mellitus. Thus, this work could provide an additional contribution to the understanding of the association of biomarkers levels in serum and saliva samples, and above all, cost-effective, rapid, label-free, and disposable NTFET were developed, based on a noninvasive sampling, for the assessment of CVD risk, thus constituting a potential point-of-care technology.