Numerical and Analytical Analysis of Nonlinear Implicit Differential Equations Arising in Control and Mechanics Problems

奇异隐式微分方程是形式为F(y‘，y，u，t)=0的隐式常微分方程，其中y是向量值函数，雅可比Fy’是奇异矩阵，u是向量值函数，t是时间。这种隐式系统经常出现在飞行控制、机器人和力学中的接触或约束问题中的各种控制问题中。此外，在用直线法求解偏微分方程组和许多电路模型时，都会出现隐式微分方程组。奇异系统通常被称为微分代数方程(DAE)或描述系统。建议发展分析和计算非线性隐式微分方程解的方法。当状态方程存在时，这些结果也是有意义的，但稀疏性、非线性或对方程参数的依赖性，使得原始隐式系统优于低阶但非稀疏的(或难以计算局部)显式状态系统。作为拟议研究的一部分，将研究直接从原始方程中获取控制理论信息的数值计算。

A singular implicit differential equation is an implicit ordinary differential equation of the form F(y',y,u,t) = 0 where y is a vector valued function, the Jacobian Fy' is a singular matrix, u is a vector valued function, and t is time. Such implicit systems frequently arise in a variety of control problems in flight control, robotics, and contact or constrained problems in mechanics. In addition, implicit differential equations occur in the solution of partial differential equations by the method of lines and in many circuit models. Singular systems are frequently called differential algebraic equations (DAEs) or discriptor systems. It is proposed to develop methods for analyzing nonlinear implicit differential equations and computing their solutions. These results will also be of interest when the state equations exist, but the sparsity, nonlinearity, or dependence on parameters of the equations, make the original implicit system preferable to a lower order, but non-sparse, (or difficult to compute local) explicit state system. The numerical computation of control theoretic information directly from the original equations will be investigated as part of the proposed research.