Generalized Derivatives for Solutions of Parametric Ordinary Differential Equations with Non-differentiable Right-Hand Sides

Abstract

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 parametric ordinary differential equation (ODE) systems are expressed as the unique solutions of 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.

Appendix

The following example shows that the unique solution of a nonsmooth parametric ODE is not necessarily differentiable with respect to the ODE parameters.

Example 5.1

Consider the following parametric ODE, with $c\in \mathbb {R}$ denoting a scalar parameter:

$$\begin{aligned} \frac{\mathrm {d} x}{\mathrm {d}t}(t,c) = |x(t,c)|, \quad x(0,c) = c. \end{aligned}$$

By inspection, this ODE is uniquely solved by the mapping:

$$\begin{aligned} x:(t,c)\mapsto \left\{ \begin{array}{ll} c\mathrm {e}^t, &{}\quad \text {if }c\ge 0, \\ c\mathrm {e}^{-t}, &{}\quad \text {if }c<0. \end{array}\right. \end{aligned}$$

Hence, for any fixed $t\ne 0$, the mapping $x(t,\cdot )$ is continuous but not differentiable at $0$.

The following example illustrates the properties of linear Newton approximations described in Sect. 1.

Example 5.2

Consider the mappings $f:\mathbb {R}\rightarrow \mathbb {R}:x\mapsto x$, $g:\mathbb {R}\rightarrow \mathbb {R}:x\mapsto \max (x,0)$, and $h:\mathbb {R}\rightarrow \mathbb {R}:x\mapsto \min (x,0)$. Using [1, Theorem 2.5.1], the Clarke generalized Jacobians of $g$ and $h$ are evaluated as:

$$\begin{aligned} \partial g(x) = \left\{ \begin{array}{ll} \{0\}, &{}\quad \text {if }x<0, \\ \left[ 0,1\right] , &{}\quad \text {if }x=0, \\ \{1\}, &{}\quad \text {if }x>0, \end{array} \right. \quad \partial h(x) = \left\{ \begin{array}{ll} \{1\}, &{}\quad \text {if }x<0, \\ \left[ 0,1\right] , &{}\quad \text {if }x=0, \\ \{0\}, &{}\quad \text {if }x>0. \end{array} \right. \end{aligned}$$

Now, $g$ and $h$ are each piecewise linear, and are therefore semismooth [14]. Since $f\equiv g+h$ on $\mathbb {R}$, it follows from [14, Corollary 7.5.18] that the following set-valued mapping is a linear Newton approximation for F:

$$\begin{aligned} \Gamma f:x\mapsto \partial g(x) + \partial h(x) = \left\{ \begin{array}{ll} \{1\}, &{}\quad \text {if }x<0, \\ \left[ 0,2\right] , &{}\quad \text {if }x=0, \\ \{1\}, &{}\quad \text {if }x>0. \end{array}\right. \end{aligned}$$

By inspection, F is convex and continuously differentiable on its domain, and has a derivative of $\mathbf {J} f(x) = 1$ for each $x\in \mathbb {R}$. In addition, F does not have any local minima on $\mathbb {R}$. However, although $\mathbf {J} f(0)\ne 0$, $0\in \Gamma f(0)$.