![]() Hence, the scope of numerical methods for nonsmooth equation-solving and local optimization is extended to systems with nonsmooth parametric ODEs embedded. Directional derivatives and lexicographic derivatives of solutions of parametric ordinary differential equation (ODE) systems are expressed as the unique solutions of corresponding ODE systems, under Carathéodory-style assumptions. It is argued that in applications, and in several established results in nonsmooth analysis, elements of the plenary hull of the generalized Jacobian of a locally Lipschitz continuous function are no less useful than elements of the generalized Jacobian itself. Difference algebra as a separate area of mathematics was born in the 1930s when J.F. ![]() In this article, for any locally Lipschitz continuous mapping between finite-dimensional Euclidean spaces, Nesterov’s lexicographic derivatives are shown to be elements of the plenary hull of the (Clarke) generalized Jacobian whenever they exist. However, obtaining useful sensitivity information for systems with nonsmooth dynamic systems embedded is a challenging task. Sensitivity analysis provides useful information for equation-solving, optimization, and post-optimality analysis. Linear differential equation is an equation which is defined as a linear system in terms of unknown variables and their derivatives. Systems of differential equations can be converted to matrix form and this is the form that we usually use in solving systems. Generalized Jacobians, nonsmooth analysis, ordinary differential equations, sensitivity analysis We can also write this system of equations with matrix-vector notation as follows: introduce the matrix A 2 1 1 2 (2) and the vector y. Journal of Optimization Theory and Applications Here’s a simple example of a system of differential equations: solve the coupled equations dy 1 dt 2y 1 +y2 dy2 dt y 1 2y2 (1) for y 1 (t)and y2 (t)given some initial values y 1 (0)and y2 (0). Generalized Derivatives for Solutions of Parametric Ordinary Differential Equations with Non-differentiable Right-Hand Sides
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