In this work we introduce a type of hybrid equation closely related to those that appear when a digital controller or a hybrid controller of a certain class is interconnected with a full non-linear continuous-time system. For these equations we obtain results concerning existence, uniqueness and noncontinuability of solutions. This results are used to study the behaviour of the hybrid dynamical systems associated with those equations. We establish relationships between the stability of these hybrid dynamical systems and the ones corresponding to their associated discretized systems on one hand, and to their associated linearized systems on the other. We also characterize their stability, uniform asymptotic stability and exponential stability. These results are applied to the exponential stabilization of a non-linear control system by digital or hybrid controllers.

Fractional differential equations can describe the dynamics of several complex and nonlocal systems with memory. They arise in many scientific and engineering areas such as physics, chemistry, biology, biophysics, economics, control theory, signal and image processing, etc. Particularly, nonlinear systems describing different phenomena can be modeled with fractional derivatives. Chaotic behavior has also been reported in some fractional models. There exist theoretical results related to existence and uniqueness of solutions to initial and boundary value problems with fractional differential equations; for the nonlinear case, there are still few of them. In this work we will present a summary of the different definitions of fractional derivatives and show models where they appear, including simple nonlinear systems with chaos. Existing results on the solvability of classical fractional differential equations and numerical approaches are summarized. Finally, we propose a numerical scheme to approximate the solution to linear fractional initial value problems and boundary value problems.

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A wide number of inverse problems consist in selecting best parameter values of a given mathematical model based fits to measured data. These are usually formulated as optimization problems and the accuracy of their solutions depends not only on the chosen optimization scheme but also on the given data. The problem of collecting data in the "best way" in order to assure a statistically efficient estimate of the parameter is known as Optimal Design. In this work we consider the problem of finding optimal locations for source identification in the 3D unit sphere from data on its boundary. We apply three different optimal design criteria to this 3D problem: the Incremental Generalized Sensitivity Function (IGSF), the classical D-optimal criterion and the SE-criterion recently introduced in [3]. The estimation of the parameters is then obtained by means of the Ordinary Least Square procedure on the resulting optimal observation points and compared to that for a uniform observation mesh.. In order to analyze the performance of each strategy, the data are numerically simulated and the estimated values are compared with the values used for simulation.

The fundamental problem of collecting data in the ``best way'' in order to assure statistically efficient estimation of parameters is known as Optimal Experimental Design. Many inverse problems consist in selecting best parameter values of a given mathematical model based on fits to measured data. These are usually formulated as optimization problems and the accuracy of their solutions depends not only on the chosen optimization scheme but also on the given data. We consider an electromagnetic interrogation problem, specifically one arising in an electroencephalography (EEG) problem, of finding optimal number and locations for sensors for source identification in a 3D unit sphere from data on its boundary. In this effort we compare the use of the classical D-optimal criterion for observation points as opposed to that for a uniform observation mesh. We consider location and best number of sensors and report results based on statistical uncertainty analysis of the resulting estimated parameters.