分数维系统的随机动力学与控制

The present project is going to study the stochastic dynamics and control of fractional systems, including the dynamics and stochastic fractional optimal control of quasi Hamiltonian system with fractional derivative damping under various kinds of stochastic excitations, as well as the dynamics and control of quasi Hamiltonian system subject to fractional noise excitation. The averaged It? stochastic differential equation, FPK equation, backward Kolmogorov equation, and generalized Pontryagin equation of quasi Hamiltonian system with fractional derivative damping under various kinds of stochastic excitations will be derived. Through solving these equations, the response, reliability function, first passage time of system can be obtained. The stability of quasi Hamiltonian system with fractional derivative damping under various kinds of stochastic excitations will be investigated by calculating the largest Lyapunov exponent of the averaged It? stochastic equation. Based on the averaged It? stochastic differential equation and stochastic dynamic programming principle, the stochastic fractional optimal control is obtained by establishing and solving the HJB equation. The stochastic averaging method for quasi Hamiltonian systems under fractional noise excitation will be developed. The averaged It? stochastic differential equation is derived. The response, reliability function, first passage time of quasi Hamiltonian system under fractional noise excitation will be obtained by solving the averaged FPK equation, backward Kolmogorov equation, and generalized Pontryagin equation associated with the averaged It? stochastic equation. By computing the largest Lyapunov exponent of averaged It? stochastic equation, the stability of quasi Hamiltonian system under fractional noise excitation will be studied. Based on the averaged It? stochastic differential equation and stochastic dynamic programming principle, the stochastic optimal control of quasi Hamiltonian systems under fractional noise excitation will be investigated. The proposed method will be applied to typical systems in engineering and science.

本项目研究分数维系统的随机动力学与控制，包括具有分数阶导数型阻尼的拟哈密顿系统在各种随机激励下的动力学与分数阶随机最优控制，及拟哈密顿系统在分数阶噪声激励下的动力学与控制。推导具有分数阶导数型阻尼的拟哈密顿系统在各种随机激励下的平均It?随机微分方程、FPK方程、后向Kolmogorov方程及HJB方程，通过求解上述方程得到系统的响应，可靠性及分数阶随机最优控制。计算平均It?随机微分方程的最大Lyapunov指数，研究具有分数阶导数型阻尼的拟哈密顿系统在各种随机激励下的稳定性。发展拟哈密顿系统在分数阶噪声激励下的随机平均法，推导拟哈密顿系统在分数阶噪声激励下的平均It?随机微分方程、FPK方程、后向Kolmogorov方程及HJB方程。通过求解上述方程得到系统的响应，可靠性及随机最优控制。将上述理论方法应用于工程和科学中的典型动力学系统。

传统的非线性随机动力学是基于整数型微积分的。在系统方面，惯量和阻尼是位移和速度对时间的整数型导数；在噪声方面，布朗运动具有独立增量性，其一阶导数过程即为白噪声过程；在理论方法方面，各类随机微分方程都是整数型微分方程。最近二十多年来，随着分数阶微积分的理论发展，及其向许多工程领域的渗透，分数阶概念被引入到了非线性随机动力学领域，并带来许多课题。. 本项目研究工作按系统与噪声的差别可分为两大类，第一类研究工作即是与系统分数阶相关的非线性随机动力学，第二类研究工作即是与噪声分数阶相关的非线性随机动力学。在第一类研究工作中，本项目主要涉及了分数阶导数型阻尼。把分数阶引入阻尼本构关系是近年来的新思路，它在描述粘弹性阻尼方面有特殊的优势，体现了粘弹性阻尼同时具有线性阻尼和刚度的属性。本项目研究各型噪声激励下含分数阶导数型阻尼的非线性系统的响应和稳定性等动力学特性，通过等效处理，把分数阶导数型阻尼分成阻尼与刚度两部分，并使它们分别与原系统的阻尼与刚度相结合，以形成新的等效系统。据此，一方面可以应用哈密顿系统理论体系框架内的非线性随机动力学与控制的理论方法，另一方面又保留了分数阶导数型阻尼对系统性质的影响。. 在第二类研究工作中，本项目主要涉及了分数阶高斯噪声。此类噪声具有历史长相关性，受其激励的系统响应不再具有马尔科夫性，传统的基于马尔科夫扩散过程的理论方法受到了挑战。本项目结合随机平均法的优点，把拟哈密顿系统随机平均法被推广到分数阶高斯噪声的激励情形，平均后分数阶随机微分方程保留了原系统的动力学性质，方程维数大大降低，且模拟时间远小于原系统的模拟时间。通过对分数阶高斯噪声激励下线性系统响应的研究，以精确解的形式得到了均方响应，更重要的是可以解析地分析响应的长相关性，研究结果指出，线性系统对分数阶高斯噪声的位移响应保留了噪声的长相关性，长相关指数仍然为2-2H，速度响应则不再具有长相关性。