Stochastic modeling in hydrology and reliability theory, and optimal control of dynamical systems

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We mainly study three interrelated problems for dynamical systems in engineering.1) Filtered renewal processes in hydrology. We have successfully used filtered Poisson processes to model river flows. However, our model implies that the time between events that significantly increase the river flow is a random variable having an exponential distribution. In reality, this hypothesis is generally false. Consequently, in order to improve the hydrological forecasts obtained from this model, we will generalize it by assuming that the flow evolves according to a filtered renewal process. We will have to derive new formulas to estimate the river flow. Furthermore, with an appropriate response function, filtered renewal processes are related to models used in queuing theory. We will make use of this fact to propose models that should yield even more accurate forecasts of river flows. In the case when analytical expressions cannot be derived, we will resort to simulations. The results that we will obtain are of great importance to dam managers, in particular.2) Reliability. A basic problem in reliability theory is the determination of the distribution of the lifetime of a given product. To do so realistically, when the system is not repairable, the model used must be such that the remaining lifetime decreases with time. We want to use two-dimensional diffusion processes for this remaining lifetime, defined in such a way that the first component of the vectors is a deterministic decreasing function of the second component, which is a diffusion process. We will also consider other models for which the remaining lifetime decreases with time, such as conditioned one-dimensional diffusion processes and processes for which there is a reflecting boundary. We want to determine, in particular, the mean time required to reach a given boundary, that is, the mean-time-to-failure (MTTF). To do so, we must solve partial differential equations under the appropriate boundary conditions. Companies are interested in the MTTF in order to estimate the cost of the warranties they offer on their products. In the case of repairable systems, filtered renewal processes similar to the ones that we will use in hydrology, but with a different response function, can be proposed.3) Stochastic optimal control. It is sometimes possible to obtain the optimal control of continuous stochastic processes until a random final time by making use of a theorem due to Whittle that enables us to express this optimal control in terms of a mathematical expectation computed for the corresponding uncontrolled processes. However, in order to apply Whittle's theorem, a certain relation between the control and noise matrices must be satisfied, which is rarely true in the most interesting applications. We want to solve this type of problems, sometimes called LQG homing, in the case when the relation in question is not satisfied. This entails solving the appropriate dynamic programming equation. Depending on the sign of a parameter in the cost function, the aim can be to minimize the time spent by the controlled process in the continuation region, or to maximize survival time instead. An application of these problems consists in computing the control that enables an aircraft to land optimally. Moreover, we want to extend LQG homing problems to the discrete-time case, which is often more realistic for the applications considered. Then, we will have to deal with nonlinear difference equations. Finally, we will also consider the problem of optimally controlling the filtered renewal processes used in hydrology and in reliability theory. In particular, we will determine the control that enables a dam manager to optimally release some water when the risk of flooding becomes too high.