Unified framework for the propagation of continuous-time enclosures for parametric nonlinear ODEs

Abstract

This paper presents a framework for constructing and analyzing enclosures of the reachable set of nonlinear ordinary differential equations using continuous-time set-propagation methods. The focus is on convex enclosures that can be characterized in terms of their support functions. A generalized differential inequality is introduced, whose solutions describe such support functions for a convex enclosure of the reachable set under mild conditions. It is shown that existing continuous-time bounding methods that are based on standard differential inequalities or ellipsoidal set propagation techniques can be recovered as special cases of this generalized differential inequality. A way of extending this approach for the construction of nonconvex enclosures is also described, which relies on Taylor models with convex remainder bounds. This unifying framework provides a means for analyzing the convergence properties of continuous-time enclosure methods. The enclosure techniques and convergence results are illustrated with numerical case studies throughout the paper, including a six-state dynamic model of anaerobic digestion.

Appendix: Technical Lemmata

The following two lemmata are used in the proof of Theorem 3. Although variants of these results can be found in the literature [23], we provide short proofs for the sake of completeness.

Lemma 1

Let $\varphi : \mathbb {R}^{n} \rightarrow \mathbb {R}^{m}$ be a continuous function. For any compact set $D\subset \mathbb {R}^n$ and any finite tolerance $\varepsilon > 0$, there exists a smooth function $\varphi _\varepsilon : \mathbb {R}^{n} \rightarrow \mathbb {R}^{m}$ such that

$$\begin{aligned} \forall x \in D\, , \quad \Vert \varphi _\varepsilon (x) - \varphi (x) \Vert \, \le \, \alpha (\varepsilon ) \, , \end{aligned}$$

(74)

for some continuous function $\alpha : \mathbb {R}_+ \rightarrow \mathbb {R}_+$ with $\alpha (0) = 0$.

Proof

The proof follows by applying well-known standard analysis techniques [23], and we only summarize the main idea here. Let $\sigma _\varepsilon : \mathbb {R}^{n} \rightarrow \mathbb {R}$, $\varepsilon > 0$ be a family of smooth functions parameterized in $\varepsilon >0$, such that $\sigma _\varepsilon (x) = 0$ for all $x$ with $\Vert x \Vert \ge \varepsilon$ and $\int _{\mathbb {R}^n} \sigma _\varepsilon (x)\,{\mathrm{d}}x = 1$. Of the alternatives for constructing such a family of ‘mollifier’ functions, we consider the function

$$\begin{aligned} \sigma _\varepsilon (x) \, {:=} \, \left\{ \begin{array}{ll} C(\varepsilon ) \exp \left( \frac{1}{\Vert x \Vert ^2 - \varepsilon ^2} \right) &{} \, \text {if} \, \Vert x \Vert < \varepsilon \\ 0 &{} \, \text {otherwise} \end{array} \right\} \quad \text {with} \quad C(\varepsilon ) \ {:=} \int _{\Vert x \Vert \!\le \! \varepsilon } \, \exp \left( \frac{1}{\Vert x \Vert ^2 \!-\! \varepsilon ^2} \right) \, {\mathrm{d}}x \,. \end{aligned}$$

In turn, the function $\varphi _{\varepsilon }$ can be defined as the convolution

$$\begin{aligned} \forall x \in \mathbb {R}^{n_x}, \quad \varphi _{\varepsilon }(x) \, {:=} \, \int _{\mathbb {R}^{n_x}} \sigma _\varepsilon (x-y) \varphi (y) \, {\mathrm{d}}y \, , \end{aligned}$$

which is smooth by construction for any $\varepsilon > 0$. Observe that the function $\alpha : \mathbb {R}_+ \rightarrow \mathbb {R}_+$ defined by

$$\begin{aligned} \forall \varepsilon \ge 0, \quad \alpha (\varepsilon ) \, {:=} \, \max _{x \in D} \, \max _{y} \left\{ \Vert \varphi (x) - \varphi (y) \Vert \, \left| \, \Vert y-x \Vert \le \varepsilon \right. \right\} \, , \end{aligned}$$

is continuous since $\varphi$ is itself continuous and $D$ is compact, and such that $\alpha (0) = 0$. In particular, this choice of $\alpha$ satisfies the condition (74). $\square$

Lemma 2

Let $Y: \left[ 0,T\right] \rightarrow \mathbb {K}_{\mathrm{C}}^{n_x}$ be a set-valued function such that $V[Y(\cdot )](c)$ is differentiable and $\dot{V}[Y(\cdot )](c)$ is bounded for all $c \in \mathbb {R}^{n_x}$ with $c^{{{\mathrm{T}}}}c = 1$. Then, there exists a family of functions $g_{\varepsilon }: [0,T] \times \mathbb {R}^{n_x} \rightarrow \mathbb {R}$ parameterized by $\varepsilon \ge 0$, such that $g_{\varepsilon }(t,\cdot )$ is strictly convex and smooth for all $\varepsilon > 0$ and all $t\in [0,T]$, and the associated sets $Y_{\varepsilon }(t) \,{:=}\, \{ x \in \mathbb {R}^{n_x} \mid g_\varepsilon (t,x) \le 0 \}$ satisfy

$$\begin{aligned}&\displaystyle \forall \varepsilon \!\ge \! 0,\, \forall t \!\in \!\! \left[ 0,T\right] , \quad Y(t) \subseteq Y_{\varepsilon }(t) \quad \text {and}\quad d_{\mathrm{H}}( Y(t), Y_{\varepsilon }(t) ) \!\le \! \alpha (\varepsilon )\, ,\nonumber \\&\displaystyle \forall \varepsilon \ge 0,\, \forall t \in \left[ 0,T\right] ,\, \forall c \in \mathbb {R}^{n_x}\ \text {with}\ c^{{{\mathrm{T}}}}c=1, \quad \dot{V}[Y_{\varepsilon }(t)](c) \ge \dot{V}[Y(t)](c) + L \, \alpha (\varepsilon ) \nonumber \\ \end{aligned}$$

(75)

for some continuous function $\alpha : \mathbb {R}_+ \rightarrow \mathbb {R}_+$ with $\alpha (0) = 0$, and any constant $0\le L<\frac{1}{T}$.

Proof

A proof can be obtained by passing through two steps.

• S1 We start with any smooth function $\nu _\varepsilon (t,\cdot ): \mathbb {R}^{n_x} \rightarrow \mathbb {R}$ such that

  $$\begin{aligned}&\forall \varepsilon > 0,\ \forall c \in \mathbb {R}^{n_x}\ \text {with}\ c^{{{\mathrm{T}}}}c = 1,\ \forall t\in [0,T],\\&\quad \nu _\varepsilon (t,c) \ge \dot{V}[Y(t)](c) \quad \text {and} \quad \left\| \nu _\varepsilon (t,c)- \dot{V}(t,c) \right\| \le \alpha _1(\varepsilon ) \, , \end{aligned}$$

  for some continuous function $\alpha _1: \mathbb {R}_+ \rightarrow \mathbb {R}_+$ with $\alpha _1(0) = 0$. Such a function is guaranteed to exist by Lemma 1. Then, we define the set-valued function $Z_\varepsilon :\left[ 0,T\right] \rightarrow \mathbb {K}_{\mathrm{C}}^{n_x}$ such that

  $$\begin{aligned} \forall c \in \mathbb {R}^{n_x}\ \text {with}\ c^{{{\mathrm{T}}}}c = 1 , \quad V[Z_{\varepsilon }(t)](c) \, {:=} \, V[Y(0)](c) + \int _0^t \nu _\varepsilon (\tau ,c) \, {\mathrm{d}}\tau \,. \end{aligned}$$

  The following properties hold by construction of $Z_\varepsilon$, for every $\varepsilon > 0$:

  • a) For all $c\in \mathbb {R}^{n_x}$ with $c^{{{\mathrm{T}}}}c = 1$, the function $V[Z_{\varepsilon }(\cdot )](c)$ is differentiable on $\left[ 0,T\right]$, and we have

    $$\begin{aligned} \forall c \in \mathbb {R}^{n_x}\ \text {with}\ c^{{{\mathrm{T}}}}c = 1 ,\ \forall t \in \left[ 0,T\right] , \quad \dot{V}[Z_{\varepsilon }(t)](c) \, \ge \, \dot{V}[Y(t)](c) \,. \end{aligned}$$

  • b) $d_{\mathrm{H}}( Z_{\varepsilon }(t), Y(t) ) \le T \alpha _1(\varepsilon )$, by property (6).

• S2 We construct the set-valued function

  $$\begin{aligned} Y_{\varepsilon }(t) \, {:=} \, Z_{\varepsilon }(t) \oplus \left[ T \alpha _1(\varepsilon ) + t L \alpha (\varepsilon ) \right] \, \fancyscript{B}^{n_x} \quad \text {with}\quad \alpha (\varepsilon ) \, {:=} \, \frac{2T}{1 - T L} \alpha _1(\varepsilon ) \, , \end{aligned}$$

  with $0\le L< \frac{1}{T}$. Note that the function $\alpha$ is continuous and non-negative and it satisfies $\alpha (0) = 0$ by definition. Therefore, we have $Y_{\varepsilon }(t) \supseteq Y(t)$ since $Y_{\varepsilon }(t) \supseteq Z_\varepsilon (t)$ and $d_{\mathrm{H}}( Y_{\varepsilon }(t), Z_\varepsilon (t) ) \ge d_{\mathrm{H}}( Z_{\varepsilon }(t), Y(t) )$. It follows from Property a) that

  $$\begin{aligned} \dot{V}[Y_{\varepsilon }(t)](c) =&\dot{V}[Z_{\varepsilon }(t)](c) + L \alpha (\varepsilon )\\ \ge&\dot{V}[Y(t)](c) + L \alpha (\varepsilon ) \, , \end{aligned}$$

  for all $t\in \left[ 0,T\right]$, all $c \in \mathbb {R}^{n_x}$ with $c^{{{\mathrm{T}}}}c = 1$, and all $\varepsilon \ge 0$. Moreover, by Property b), we have

  $$\begin{aligned} d_{\mathrm{H}}( Y_{\varepsilon }(t), Y(t) ) \le&d_{\mathrm{H}}( Z_{\varepsilon }(t), Y(t) ) + \alpha _0(\varepsilon ) + T \alpha _1(\varepsilon ) + T L \alpha (\varepsilon )\\ \le&2 \alpha _0(\varepsilon ) + 2 T \alpha _1(\varepsilon ) + T L \alpha (\varepsilon )\\ =&\alpha (\varepsilon ) \, , \end{aligned}$$

  for all $t \in \left[ 0,T\right]$, and all $\varepsilon \ge 0$. Finally, by Theorem 1 in [6], there exists a functions $g_{\varepsilon }: [0,T] \times \mathbb {R}^{n_x} \rightarrow \mathbb {R}$ such that $Y_{\varepsilon }(t) =: \{ x \in \mathbb {R}^{n_x} \mid g_\varepsilon (t,x) \le 0 \}$ and $g_{\varepsilon }(t,\cdot )$ is convex and smooth for all $\varepsilon \ge 0$ and all $t\in \left[ 0,T\right]$. In order for $g_{\varepsilon }(t,\cdot )$ to be strictly convex for all $\varepsilon >0$, one can always add a strictly convex and smooth term of order $O(\varepsilon )$ that is negative on the compact sets $\bigcup _{t\in \left[ 0,T\right] } Y_{\varepsilon }(t)$. $\square$

The following lemma is used in the proof of Theorem 4. The result allows one to bound the solution of a particular parametric differential inequality and can be regarded as a generalization of Gronwall’s lemma [24].

Lemma 3

Let $v\in \mathbb {R}_+$ and let $u: \left[ 0,T\right] \rightarrow \mathbb {R}$ be a Lipschitz-continuous function satisfying the parametric differential inequality

$$\begin{aligned} \text {a.e. }t\in \left[ 0,T\right] \, , \quad \frac{\mathrm{d}}{\mathrm{dt}} [u(t)]^n \, \le \, n \, \sum _{i=0}^n L_i \, u(t)^{n-i} \, v^{im} \quad \text {with} \quad u(0) \le C_0 v^m \, , \end{aligned}$$

(76)

for some integers $m,n\ge 1$ and a set of constants $0 \le L_0, \ldots , L_n < \infty$ and $C_0 \ge 1$. Then, $u(t) \le C_0 \, \exp \left( \sum _{i=0}^n L_i t\right) \, v^m$, for all $t \in \left[ 0,T\right]$.

Proof

The proof proceeds in two steps. It is assumed first that the function $u$ is differentiable on $[0,T]$. Then, it is argued that the result still holds in extending this class of functions to Lipschitz-continuous.

Assuming that $u$ is differentiable on $[0,T]$ and discretizing the differential inequality (76) with a step-size $h\,{:=}\,\frac{T}{N}$ for a large enough $N\in \mathbb {N}$ gives

$$\begin{aligned} \forall k\in \{0,N-1\}\, , \quad \left[ u((k+1) h)\right] ^{n} \le&[u(k h)]^{n} + h \, n \, \sum _{i=0}^n L_i \, (u(kh))^{n-i} \, v^{im} + h\,\alpha (h) \, , \end{aligned}$$

for some continuous function $\alpha : \mathbb {R}_+ \rightarrow \mathbb {R}_+$ with $\alpha (0) = 0$. Now, supposing that $u(kh) \le C_k \,v^{m}$ with $C_k \ge 1$, we have

$$\begin{aligned} \left[ u((k+1) h) \right] ^{n}&\le (C_{k+1})^n \, v^{nm} \, + \, h\,\alpha (h) \quad \text {with} \quad (C_{k+1})^n {:=} \, \left( 1 + h n \sum _{i=0}^n L_i\right) (C_k)^n \ge 1 \,. \end{aligned}$$

In particular, the definition of $(C_{k+1})^n$ uses the result that

$$\begin{aligned} (C_k)^n + h n \, \sum _{i=0}^n L_i (C_k)^{n-i} \, \le \, \left( 1 + h n \sum _{i=0}^n L_i\right) (C_k)^n \, , \end{aligned}$$

for all $C_k \ge 1$. It follows by induction that $u(kh) \le C_{k} \,v^{m} + h\,\alpha (h)$ for each $k=0,\ldots ,N$, with

$$\begin{aligned} C_k \, = \, \left( 1 + h n \sum _{i=0}^n L_i\right) ^{k/n}C_0 \,. \end{aligned}$$

Let $\overline{t}\in [0,T]$ be such that $\bar{t}\,{:=}\,\frac{k_0}{N_0}T$ for given $0\le k_0 \le N_0$, and consider the sequence $\{\overline{C}_j\}$ given by

$$\begin{aligned} \overline{C}_j \, {:=} \, \left( 1 + \frac{n\,T}{j\,N_0} \sum _{i=0}^n L_i\right) ^{jk_0/n}C_0 \, , \end{aligned}$$

so that $u(\overline{t}) \le \overline{C}_j\,v^{m} + \frac{T}{j\,N_0}\,\alpha (\frac{T}{j\,N_0})$ for all $j\ge 1$. It follows from the definition of the exponential function as $\exp (x)\,{:=}\,\lim _{j\rightarrow \infty } (1+\frac{x}{j})^j$ that this sequence is convergent, and we have

$$\begin{aligned} \lim _{j\rightarrow \infty }\overline{C}_j \, = \, \exp \left( \sum _{i=0}^n L_i \overline{t} \right) \,C_0 \,. \end{aligned}$$

As $u$ is continuous on $[0,T]$, and since the rationals are a dense subset of the real numbers, it follows that

$$\begin{aligned} \forall t\in [0,T]\, , \quad u(t) \, \le \, C(t) \, v^{m} \quad \text {with} \quad C(t) \, {:=} \, \exp \left( \sum _{i=0}^n L_i t \right) \,C_0 \,. \end{aligned}$$

In a second step, the assumption of differentiability for $u$ can be relaxed to Lipschitz-continuity, by a similar argument as in part S3 of the proof of Theorem 3, namely that any (locally) Lipschitz-continuous function is differentiable almost everywhere. $\square$