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## Nonsmooth Dynamic Sensitivity Analysis

Chemical process models are often nonsmooth dynamic systems, which are considered here to be parametric ordinary differential equation (ODE) systems with right-hand side functions that are locally Lipschitz continuous but not differentiable everywhere. Nonsmooth dynamic systems pose challenges for simulation, sensitivity analysis, equation-solving, and optimization. Except in certain special cases, little is known a priori about the behavior of the state variables as functions of system parameters. Moreover, nonsmoothness of the ODE right-hand side function can lead to nonsmoothness of these state variables with respect to parameters, hindering the application of numerical methods developed for smooth functions.

Established numerical methods for nonsmooth equation-solving and nonsmooth optimization include semismooth Newton methods and bundle methods. Such methods typically require local sensitivity information in the form of generalized derivatives such as the Clarke Jacobian or the B-subdifferential. However, existing methods for evaluating elements of these generalized derivatives for nonsmooth dynamic systems are limited either in scope or in the quality of the obtained derivatives. This presentation summarizes the main contributions of my Ph.D. work on nonsmooth dynamic sensitivity analysis.

Firstly, sufficient conditions are presented under which the state variables of a nonsmooth dynamic system are in fact smooth functions of system parameters. These conditions are computationally inexpensive and simple to verify a posteriori during a simulation run.

Secondly, new and old theoretical results are combined to show that Nesterovâ€™s lexicographic derivatives are no less useful than Clarke Jacobian elements in numerical methods. A variant of the vector forward mode of algorithmic differentiation (AD) is presented, which computes lexicographic derivatives for general finite compositions of simple smooth and nonsmooth functions. Like the standard vector forward AD mode, this method is accurate, fully automatable, and computationally tractable relative to the cost of a function evaluation.

It is shown that lexicographic derivatives of the unique solution of a nonsmooth parametric ODE system are described by the unique solution of an auxiliary ODE system, under minimal assumptions. To my knowledge, this is the first description of a useful generalized derivative for a general nonsmooth dynamic system. Unlike in the smooth case, however, the auxiliary ODE system is more difficult to solve numerically than the original ODE system, with discontinuities emerging in its right-hand side function. A proposed numerical method for handling these discontinuities when computing nonsmooth dynamic sensitivities is outlined. These nonsmooth dynamic sensitivity results are generalized to certain hybrid discrete/continuous systems, in which nonsmoothness is permitted in functions describing timing and handling of discrete events, as well as continuous evolution of the systems between events.