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

我们主要研究工程中动力系统的三个相互关联的问题。 1)水文学中的过滤更新过程。我们已经成功地使用过滤的泊松过程来模拟河流流动。然而，我们的模型表明，显著增加河流流量的事件之间的时间是一个具有指数分布的随机变量。在现实中，这种假设通常是错误的。因此，为了改进从该模型得到的水文预报，我们将假定水流按照过滤的更新过程演变，从而对其进行推广。我们将不得不推导出新的公式来估算河流流量。此外，在适当的响应函数下，过滤的更新过程与排队论中使用的模型相关联。我们将利用这一事实来提出模型，这些模型应该能够对河流流量做出更准确的预测。在无法导出解析表达式的情况下，我们将求助于模拟。我们将获得的结果对大坝管理人员尤其重要。 2)可靠性。可靠性理论中的一个基本问题是确定给定产品的寿命分布。要实际地做到这一点，当系统不可修复时，所使用的模型必须使剩余寿命随时间减少。我们希望在剩余的寿命中使用二维扩散过程，以这样的方式定义，即向量的第一个分量是第二个分量的确定性递减函数，这是一个扩散过程。我们还将考虑其他剩余寿命随时间减少的模型，例如条件化的一维扩散过程和存在反射边界的过程。我们特别想确定达到给定边界所需的平均时间，即平均无故障时间(MTTF)。为此，我们必须在适当的边界条件下求解偏微分方程组。公司对MTTF感兴趣，是为了估计他们为产品提供保修的成本。在可修复系统的情况下，可以提出过滤更新过程，类似于我们将在水文学中使用的过程，但具有不同的响应函数。 3)随机最优控制。有时可以通过利用惠特尔的一个定理来获得连续随机过程的最优控制，直到随机最后时刻，该定理使我们能够用为相应的未被控过程计算的数学期望来表示这种最优控制。然而，为了应用惠特尔定理，必须满足控制矩阵和噪声矩阵之间的某种关系，这在最有趣的应用中很少是正确的。我们想要解决这种类型的问题，有时称为LQG归位，在所讨论的关系不满足的情况下。这需要求解适当的动态规划方程。根据成本函数中参数的符号，目标可以是最小化受控过程在连续区域中花费的时间，或者是最大化生存时间。这些问题的一个应用在于计算使飞机最佳着陆的控制。此外，我们希望将LQG寻的问题扩展到离散时间的情况，这对于所考虑的应用通常更现实。然后，我们将不得不处理非线性差分方程组。最后，我们还将考虑在水文学和可靠性理论中使用的过滤更新过程的最优控制问题。特别是，我们将确定使大坝管理人员能够在洪水风险变得太高时以最佳方式释放一些水的控制措施。

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.