### 统计代写|随机控制代写Stochastic Control代考|Generalized Probability Density Evolution Equation

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## 统计代写|随机控制代写Stochastic Control代考|Generalized Probability Density Evolution Equation

Without loss of generality, a stochastic dynamical system under the random excitation can be represented by
$$\dot{\mathbf{Z}}(t)=\mathbf{g}[\mathbf{Z}(t), \mathbf{F}(\boldsymbol{\Theta}, t), t], \mathbf{Z}\left(t_{0}\right)=\mathbf{z}_{0}$$
where $\mathbf{F}(\cdot)$ is a column vector denoting the nonstationary and non-Gaussian random excitation; $\boldsymbol{\Theta}$ is a random parameter vector denoting the randomness inherent in the excitation.

As to the quantity of interest such as the system state or the control force $\mathbf{Z}^{\mathrm{T}}=$ $\left{Z_{i}\right}_{i=1}^{m}$, the formal solution can be given by
$$\mathbf{Z}(t)=\mathbf{H}\left(\boldsymbol{\Theta}, \mathbf{Z}{0}, t\right)$$ where $\mathbf{H}$ is an $m$-dimensional column vector denoting arithmetic operator. It is indicated in Eq. (2.3.61) that the randomness inherent in the random process $\mathbf{Z}(t)$ is completely represented by the random parameter vector $\boldsymbol{\Theta}$. In view of the probability preservation principle, the augmented system consisting of the quantity of interest and the random parameter vector, i.e., $(\mathbf{Z}(t), \mathbf{\Theta})$, thus sustains a preservative probability, that is $$\frac{\mathrm{D}}{\mathrm{D} t} \int{\Omega_{t} \times \Omega_{\Theta}} p_{\mathbf{Z \Theta}}(\mathbf{z}, \theta, t) \mathrm{d} \mathbf{z} d \theta=0$$
where $p_{\mathbf{Z} \Theta}(\mathbf{z}, \boldsymbol{\theta}, t)$ denotes the joint probability density function of the augmented system $(\mathbf{Z}(t), \boldsymbol{\Theta}) ; \Omega_{t}$ denotes the time domain; $\Omega_{\boldsymbol{\Theta}}$ denotes the sample domain of random parameter vector $\Theta ; \mathrm{D}(\cdot) / \mathrm{D} t$ denotes the total derivative.
Extending Eq. (2.3.62), we have (Li and Chen 2009)
$\frac{\mathrm{D}}{\mathrm{D} t} \int_{\Omega_{l} \times \Omega_{\boldsymbol{\Theta}}} p_{\mathbf{Z} \Theta}(\mathbf{z}, \boldsymbol{\theta}, t) \mathrm{d} \mathbf{z} \mathrm{d} \theta$
$=\frac{\mathrm{D}}{\mathrm{D} t} \int_{\Omega_{t_{0}} \times \Omega_{\Theta}} p_{\mathbf{Z} \Theta}(\mathbf{z}, \theta, t)|J| \mathrm{d} \mathbf{z} d \theta$
$=\int_{\Omega_{t_{0}} \times \Omega_{\Theta}}\left(|J| \frac{\mathrm{D} p_{\mathbf{Z \Theta}}}{\mathrm{D} t}+p_{\mathbf{Z \Theta}} \frac{\mathrm{D}|J|}{\mathrm{D} t}\right) \mathrm{d} \mathbf{z} d \theta$
$=\int_{\Omega_{t_{0}} \times \bar{\Omega}{\Theta}}\left{|J|\left(\frac{\partial p{\mathbf{Z} \Theta}}{\partial t}+\sum_{j=1}^{m} \dot{Z}{j} \frac{\partial p{\mathbf{Z} \Theta}}{\partial z_{j}}\right)+|J| p_{\mathbf{Z} \Theta} \sum_{j=1}^{m} \frac{\partial \dot{Z}{j}}{\partial z{j}}\right} \mathrm{d} \mathbf{z} d \boldsymbol{\theta}$

## 统计代写|随机控制代写Stochastic Control代考|Historic Notes

It is generally recognized that the random vibration discipline originates from the research and application of stochastic dynamics that involves two logical clues ( $\mathrm{Li}$ and Chen 2009). Einstein first explored the Brownian motion from a phenomenological perspective using the random process theory in 1905 (Einstein 1905), which was later developed by Fokker (Fokker 1914), Planck (Planck 1917), and mathematized by Kolmogorov (Kolmogorov 1931) that formed into the associated theory and methods with the FPK equation. From an almost coinstantaneous physical perspective, Langevin investigated the motion of a Brownian particle by the Newtonian equation (Langevin 1908), which was later developed by Wiener (Wiener 1923), Itô (Itô 1942) and Stratonovich (Stratonovich 1963) that underlined the formulation and solution schemes of the stochastic differential equation. Although the probabilistic description of structural vibration was pioneered in Rayleigh’s investigation on the random flight in the early of twentieth century (Rayleigh 1919), the random vibration theory was widely applied in the engineering fields and gradually became to a new discipline until the middle of twentieth century. Since then, it has gained extensive progress from the primary linear random vibration analysis such as the random vibration with initial random conditions, the random vibration simultaneously involving the randomness inherent in external excitations and in structural parameters, to the nonlinear random vibration analysis (Crandall 1958; Crandall and Mark 1963; Lin 1967; Nigam 1983; Roberts and Spanos 1990; Lin and Cai 1995; Lutes and Sarkani 2004; Li and Chen 2009).

As to the classical linear random vibration analysis, an elegant theoretical formula and the pertinent numerical schemes have been formed by virtue of the statistical relation between the input and the output in temporal and frequency domains (Crandall 1958), e.g., the spectral transfer matrix method (Lutes and Sarkani 2004), the modal superposition method such as the complete quadratic combination (CQC) (Der Kiureghian 1981; Der Kiureghian and Neuenhofer 1992), the pseudo-excitation method (Lin et al. 2001; Li et al. 2004). However, the principle of superposition is not suitable for the nonlinear system. The temporal and frequency-domain methods prevailing in the linear random vibration analysis cannot deal with the problem of essentially nonlinear random vibrations. The classical Markov process method accommodates a few specific nonlinear systems but encounters the challenge as well in dealing with the general multi-degree-of-freedom and multidimensional systems. It is thus a preferable choice of deriving the approximate solution or the accurate stationary solution for the nonlinear random vibration analysis. In the past over 50 years, a collection of methods for nonlinear random vibration analysis were proposed, e.g., the statistical linearization technique (Caughey 1963) and the moment closure method (Stratonovich 1963) suitable for the weakly nonlinear systems; the extended statistical linearization technique (Beaman and Hedrick 1981), the equivalent nonlinear equation (Caughey 1986), and the Monte Carlo simulation (Shinozuka 1972) suitable for the strongly nonlinear systems. Meanwhile, the attempt of classical stochastic structure theory to the application of random vibration analysis was

carried out. For instance, the perturbation expansion method was applied to deal with the random vibration analysis of low-order systems (Crandall 1963); the orthogonal function expansion was applied to deal with the random vibration analysis of white noise-driven Duffing oscillatory systems (Orabi and Ahmadi 1987); the polynomial chaos expansion was applied to deal with the random vibration analysis of stationary excitation-driven Duffing systems (Li and Ghanem 1998). One might realize that the mentioned methods cannot solve the problem of nonlinear random vibration of highdimensional structural systems, not even to gain the complete probability density. In theory, the FPK equation is the most rigorous and most elegant method for the nonlinear random vibration analysis. As far as steady solution of stochastic dynamical systems is concerned, the method for solving high-dimensional FPK equation in Hamiltonian framework attained systematic progress since 1990 s (Soize 1994; Zhu and Huang 1999; Er 2011). However, when the unsteady solution of stochastic dynamical systems is concerned, the computational complexity will increase exponentially with the dimension of systems. In this case, the solution is still hard to be derived even employing efficiently numerical schemes and advanced computational platforms. In recent years, the dimension reduction of FPK equation has been paid extensive attention (Chen and Yuan 2014; Chen and Rui 2018).

The probability density evolution method (PDEM) with the kernel generalized probability density evolution equation (GDEE) provided an efficient means for solving the stochastic dynamical system from a physical perspective. This method has been applied into the stochastic response analysis and reliability assessment of general nonlinear stochastic systems (Li and Chen 2004a, b, 2005, 2006a; Chen and Li 2005 ; Li and Chen 2008). The progress underlies the development of the physically based stochastic optimal control of structures.

## 统计代写|随机控制代写Stochastic Control代考|Dynamic Reliability of Structures

The primary goal of structural analysis aims at the performance-based design and control of structures. If the random factors involved in the basic physical background are concerned, the logical manner of structural analysis is to carry out the reliabilitybased structural design and control.

As regards the assessment of dynamic reliability of structures as the first-passage failure criterion, the primary methods include the level-crossing process theory, the diffusion process theory and the probability density evolution method. Two families of criteria are usually applied in the probability density evolution method, i.e., the absorbing boundary condition criterion and the equivalent extreme-value event criterion. Herein, the level-crossing process theory and the equivalent extreme-value event criterion-based probability density evolution methở are intrơuucè sincé thé two methods are both widely used in practice.

The level-crossing process theory originated from Rice’s researches on the digital noise process in 1940 s (Rice 1944,1945 ). As to the $b$-level-crossing process, shown in Fig. 2.4, the probability of occurring once upward level-crossing during the time interval $t<\tau \leqslant t+\Delta t$ is given by \begin{aligned} &\operatorname{Pr}\left{N^{+}(t+\Delta t)-N^{+}(t)=1\right} \ &=\operatorname{Pr}{X(t+\Delta t)>b, X(t)b, X(t) b, xb, X(t)b, xb-\dot{x} \Delta t, x<b} p_{X \dot{X}}(x, \dot{x}, t) \mathrm{d} x \mathrm{~d} \dot{x} \ &=\int_{0}^{\infty} \mathrm{d} \dot{x} \int_{b-\dot{x} \Delta t}^{b} p_{X \dot{X}}(x, \dot{x}, t) \mathrm{d} x \end{aligned}

## 统计代写|随机控制代写Stochastic Control代考|Generalized Probability Density Evolution Equation

DD吨∫Ω吨×Ωθp从θ(和,θ,吨)d和dθ=0

DD吨∫Ωl×Ωθp从θ(和,θ,吨)d和dθ
=DD吨∫Ω吨0×Ωθp从θ(和,θ,吨)|Ĵ|d和dθ
=∫Ω吨0×Ωθ(|Ĵ|Dp从θD吨+p从θD|Ĵ|D吨)d和dθ
=\int_{\Omega_{t_{0}} \times \bar{\Omega}{\Theta}}\left{|J|\left(\frac{\partial p{\mathbf{Z} \Theta}} {\partial t}+\sum_{j=1}^{m} \dot{Z}{j} \frac{\partial p{\mathbf{Z} \Theta}}{\partial z_{j}}\右）+|J| p_{\mathbf{Z} \Theta} \sum_{j=1}^{m} \frac{\partial \dot{Z}{j}}{\partial z{j}}\right} \mathrm{d } \mathbf{z} d \boldsymbol{\theta}=\int_{\Omega_{t_{0}} \times \bar{\Omega}{\Theta}}\left{|J|\left(\frac{\partial p{\mathbf{Z} \Theta}} {\partial t}+\sum_{j=1}^{m} \dot{Z}{j} \frac{\partial p{\mathbf{Z} \Theta}}{\partial z_{j}}\右）+|J| p_{\mathbf{Z} \Theta} \sum_{j=1}^{m} \frac{\partial \dot{Z}{j}}{\partial z{j}}\right} \mathrm{d } \mathbf{z} d \boldsymbol{\theta}

## 统计代写|随机控制代写Stochastic Control代考|Dynamic Reliability of Structures

\begin{aligned} &\operatorname{Pr}\left{N^{+}(t+\Delta t)-N^{+}(t)=1\right} \ &=\operatorname{Pr}{X( t+\Delta t)>b, X(t)b, X(t)b, xb, X(t)b, xb-\dot{x} \Delta t, x<b} p_{X \dot {X}}(x, \dot{x}, t) \mathrm{d} x \mathrm{~d} \dot{x} \ &=\int_{0}^{\infty} \mathrm{d} \dot{x} \int_{b-\dot{x} \Delta t}^{b} p_{X \dot{X}}(x, \dot{x}, t) \mathrm{d} x \结束{对齐}\begin{aligned} &\operatorname{Pr}\left{N^{+}(t+\Delta t)-N^{+}(t)=1\right} \ &=\operatorname{Pr}{X( t+\Delta t)>b, X(t)b, X(t)b, xb, X(t)b, xb-\dot{x} \Delta t, x<b} p_{X \dot {X}}(x, \dot{x}, t) \mathrm{d} x \mathrm{~d} \dot{x} \ &=\int_{0}^{\infty} \mathrm{d} \dot{x} \int_{b-\dot{x} \Delta t}^{b} p_{X \dot{X}}(x, \dot{x}, t) \mathrm{d} x \结束{对齐}

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