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.
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Acknowledgments
This paper is based upon work supported by the Engineering and Physical Sciences Research Council (EPSRC) under Grant EP/J006572/1. Financial support from Marie Curie Career Integration Grant PCIG09-GA-2011-293953 and from the Centre of Process Systems Engineering (CPSE) of Imperial College is gratefully acknowledged. M.E.V. thanks CONACYT for doctoral scholarship. The authors are grateful to the anonymous reviewers and the associate editor for their thoughtful comments that led to substantial improvement of the article. Special thanks also go to Prof. Joseph K. Scott from Clemson University for fruitful discussions in connection to the proof of the generalized differential inequality in Sect. 4.
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Appendix: Technical Lemmata
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
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
In turn, the function \(\varphi _{\varepsilon }\) can be defined as the convolution
which is smooth by construction for any \(\varepsilon > 0\). Observe that the function \(\alpha : \mathbb {R}_+ \rightarrow \mathbb {R}_+\) defined by
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
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).
-
a)
-
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
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
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
In particular, the definition of \((C_{k+1})^n\) uses the result that
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
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
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
As \(u\) is continuous on \([0,T]\), and since the rationals are a dense subset of the real numbers, it follows that
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. \(\square \)
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Villanueva, M.E., Houska, B. & Chachuat, B. Unified framework for the propagation of continuous-time enclosures for parametric nonlinear ODEs. J Glob Optim 62, 575–613 (2015). https://doi.org/10.1007/s10898-014-0235-6
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DOI: https://doi.org/10.1007/s10898-014-0235-6