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SUMMARYThe problem of designing strategies for optimal feedback control of non-linear processes, specially for regulation and set-point changing, is attacked in this paper. A novel procedure based on the Hamiltonian equations associated to a bilinear approximation of the dynamics and a quadratic cost is presented. The usual boundary-value situation for the coupled state-costate system is transformed into an initial-value problem through the solution of a generalized algebraic Riccati equation. This allows to integrate the Hamiltonian equations on-line, and to construct the feedback law by using the costate solution trajectory. Results are shown applied to a classical non-linear chemical reactor model, and compared against suboptimal bilinear-quadratic strategies based on power series expansions. Since state variables calculated from Hamiltonian equations may differ from the values of physical states, the proposed control strategy is suboptimal with respect to the original plant.

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The stochastic sensitivity analysis problem in chemical kinetics is defined as determining the probability density function (pdf) of the concentrations given probability density functions for the parameters and initial conditions. The joint concentration parameter pdf is found to satisfy the equation (∂p/∂t)+div(Fp) = 0, where the system dynamics are given by ? = F(x). The properties of the solution of this equation are studied, and the approach is applied to analyze the sensitivity of the kinetics of the photolysis of a mixture of carbon monoxide, nitrogen dioxide, nitric oxide, and water in air to uncertainties in the initial concentrations of the nitrogen oxides and in the values of two photolysis rate constants. Comparisons to other sensitivity analysis approaches are discussed.

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This review shows the potential ground-breaking impact that mathematical tools may have in the analysis and the understanding of the HIV dynamics. In the first part, early diagnosis of immunological failure is inferred from the estimation of certain parameters of a mathematical model of the HIV infection dynamics. This method is supported by clinical research results from an original clinical trial: data just after 1 month following therapy initiation are used to carry out the model identification. The diagnosis is shown to be consistent with results from monitoring of the patients after 6 months. In the second part of this review, prospective research results are given for the design of individual anti-HIV treatments optimizing the recovery of the immune system and minimizing side effects. In this respect, two methods are discussed. The first one combines HIV population dynamics with pharmacokinetics and pharmacodynamics models to generate drug treatments using impulsive control systems. The second one is based on optimal control theory and uses a recently published differential equation to model the side effects produced by highly active antiretroviral therapy therapies. The main advantage of these revisited methods is that the drug treatment is computed directly in amounts of drugs, which is easier to interpret by physicians and patients.