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

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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.