Generalized derivatives for solutions of parametric ordinary differential equations with non-differentiable right-hand sides.

*(English)*Zbl 1304.49035Summary: Sensitivity analysis provides useful information for equation-solving, optimization, and post-optimality analysis. However, obtaining useful sensitivity information for systems with nonsmooth dynamic systems embedded is a challenging task. 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. 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. Directional derivatives and lexicographic derivatives of solutions of systems of parametric Ordinary Differential Equations (ODEs) are expressed as the unique solutions of the corresponding ODE systems under Carathéodory-style assumptions. Hence, the scope of numerical methods for nonsmooth equation-solving and local optimization is extended to systems with nonsmooth parametric ODEs embedded.

##### MSC:

49J52 | Nonsmooth analysis |

34A12 | Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations |

90C31 | Sensitivity, stability, parametric optimization |

##### Keywords:

generalized Jacobians; sensitivity analysis; nonsmooth analysis; ordinary differential equations##### Software:

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\textit{K. A. Khan} and \textit{P. I. Barton}, J. Optim. Theory Appl. 163, No. 2, 355--386 (2014; Zbl 1304.49035)

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##### References:

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