Exploiting nonlinear invariants and path constraints to achieve tighter reachable set enclosures using differential inequalities

Abstract

This article presents a new method for computing sharp bounds on the solutions of nonlinear dynamic systems subject to uncertain initial conditions, parameters, and time-varying inputs. Such bounds are widely used in algorithms for uncertainty propagation, robust state estimation, system verification, global dynamic optimization, and more. Recently, it has been shown that bounds computed via differential inequalities can often be made much less conservative by exploiting state constraints that are known to hold for all trajectories of interest (e.g., path constraints that describe feasible trajectories in the context of dynamic optimization, or constraints that explicitly describe invariant sets containing all system trajectories). However, effective bounding algorithms of this type are currently only available for problems with linear constraints. Moreover, the theoretical results underlying these algorithms do not apply to constraints that depend on time-varying inputs and rely on assumptions that prove to be very restrictive for nonlinear constraints. This article contributes a new differential inequalities theorem that permits the use of a very general class of nonlinear state constraints. Moreover, a new algorithm is presented for efficiently exploiting nonlinear constraints to achieve tighter bounds. The proposed approach is shown to produce very sharp bounds for two challenging case studies.

Proof of Theorem 1

The proof of Theorem 1 is based on a general result from [33] that provides sufficient conditions for two functions $\mathbf {v},\mathbf {w}\in \mathscr {AC}(I,{\mathbb {R}}^{n_x})$ to bound an arbitrary function $\varvec{\phi }\in \mathscr {AC}(I,{\mathbb {R}}^{n_x})$ (i.e., $\varvec{\phi }$ need not be associated with a system of ODEs). This result is stated abstractly in terms of interval-valued mappings of the form $\varPi _i^L,\varPi _i^U:D_{\varPi }\subset I \times {\mathbb {R}}^{n_x}\times {\mathbb {R}}^{n_x}\rightarrow \mathbb {IR}$. Eventually, these more general functions will be used to represent the operator ${\mathscr {R}}$ used in Theorem 1. The specific conditions we will require of these functions are given in Hypothesis 1.

Hypothesis 1

For every $i\in \{1,\ldots ,n_x\}$, let $\varPi _i^L,\varPi _i^U:D_{\varPi }\subset I \times {\mathbb {R}}^{n_x}\times {\mathbb {R}}^{n_x}\rightarrow \mathbb {IR}$ satisfy the following conditions:

• 1. If $(t,\mathbf {v},\mathbf {w})\in D_{\varPi }$ satisfies $\mathbf {v}\le \varvec{\phi }(t)\le \mathbf {w}$ and $\phi _i(t)=v_i$ for some $i\in \{1,\dots ,n_x\}$, then ${\dot{\phi }}_i(t) \in \varPi ^L_i(t,\mathbf {v},\mathbf {w})$.

• 2. If $(t,\mathbf {v},\mathbf {w})\in D_{\varPi }$ satisfies $\mathbf {v}\le \varvec{\phi }(t)\le \mathbf {w}$ and $\phi _i(t)=w_i$ for some $i\in \{1,\dots ,n_x\}$, then ${\dot{\phi }}_i(t) \in \varPi ^U_i(t,\mathbf {v},\mathbf {w})$.

• 3. $D_{\varPi }$ is open with respect to the set $A\equiv \{(t,\mathbf {v},\mathbf {w})\in I\times {\mathbb {R}}^{n_x}\times {\mathbb {R}}^{n_x}:\mathbf {v}\le \mathbf {w}\}$. Specifically, for any $({\hat{t}},\hat{\mathbf {v}},\hat{\mathbf {w}})\in D_{\varPi }\cap A$, there exists $\eta >0$ such that $B_{\eta }(({\hat{t}},\hat{\mathbf {v}},\hat{\mathbf {w}}))\cap A$ is a subset of $D_{\varPi }$.

• 4. $\varPi _i^L$ and $\varPi _i^U$ are locally Lipschitz continuous with respect to $\mathbf {v}$ and $\mathbf {w}$, uniformly with respect to t. Specifically, for any $({\hat{t}},\hat{\mathbf {v}},\hat{\mathbf {w}})\in D_{\varPi }$, there exists $\eta >0$ and $\alpha \in L^1(I)$ such that

  $$\begin{aligned} d_H(\varPi _i^L(t,\mathbf {v},\mathbf {w}),\varPi _i^L(t,\tilde{\mathbf {v}},\tilde{\mathbf {w}}))&\le \alpha (t)\max \left( \Vert \mathbf {v}-\tilde{\mathbf {v}}\Vert ,\Vert \mathbf {w}-\tilde{\mathbf {w}}\Vert \right) , \end{aligned}$$

  (46)
  $$\begin{aligned} d_H(\varPi _i^U(t,\mathbf {v},\mathbf {w}),\varPi _i^U(t,\tilde{\mathbf {v}},\tilde{\mathbf {w}}))&\le \alpha (t)\max \left( \Vert \mathbf {v}-\tilde{\mathbf {v}}\Vert ,\Vert \mathbf {w}-\tilde{\mathbf {w}}\Vert \right) , \end{aligned}$$

  (47)

  for every $(t,\mathbf {v},\mathbf {w}),(t,\tilde{\mathbf {v}},\tilde{\mathbf {w}})\in B_{\eta }(({\hat{t}},\hat{\mathbf {v}},\hat{\mathbf {w}}))\cap D_{\varPi }$.

Theorem 3 is the central result we will use to prove Theorem 1. It is proved as Theorem 3.5.1 in [33] under slightly different hypotheses, as discussed below.

Theorem 3

Let $\varvec{\phi },\mathbf {v},\mathbf {w} \in \mathscr {AC}(I,{\mathbb {R}}^{n_x})$ satisfy

• 1. $(t,\mathbf {v}(t),\mathbf {w}(t))\in D_{\varPi }, \forall t\in I$.

• 2. $\mathbf {v}(t_0)\le \varvec{\phi }(t_0)\le \mathbf {w}(t_0)$.

• 3. For a.e. $t\in I$ and each index i,

  • (a) ${\dot{v}}_i(t)\le \sigma _i$ for all $\sigma _i\in \varPi _i^L(t,\mathbf {v}(t),\mathbf {w}(t))$,

  • (b) ${\dot{w}}_i(t)\ge \sigma _i$ for all $\sigma _i\in \varPi _i^U(t,\mathbf {v}(t),\mathbf {w}(t))$.

If Hypothesis 1 holds, then $\mathbf {v}(t)\le \varvec{\phi }(t)\le \mathbf {w}(t), \forall t\in I$.

In [33], Theorem 3 is proved with a modified version of Hypothesis 1, which is stated explicitly as Hypothesis 2. We prefer Hypothesis 1 because it is easier to verify when using Theorem 3 to prove Theorem 1. Moreover, the conditions of Hypothesis 1 are much easier to understand, whereas Hypothesis 2 is very abstract. In Lemma 1, we show that Hypothesis 1 implies Hypothesis 2, so that Theorem 3 follows immediately from Theorem 3.5.1 in [33].

Hypothesis 2

For every $i\in \{1,\ldots ,n_x\}$, let $\varPi _i^L,\varPi _i^U:D_{\varPi }\subset I \times {\mathbb {R}}^{n_x}\times {\mathbb {R}}^{n_x}\rightrightarrows {\mathbb {R}}$ (i.e., $\varPi ^{L}_i(t,\mathbf {v},\mathbf {w})$ and $\varPi ^{U}_i(t,\mathbf {v},\mathbf {w})$ are subsets of ${\mathbb {R}}$, not necessarily intervals). Assume that, given any $({\hat{t}},\hat{\mathbf {v}},\hat{\mathbf {w}})\in D_{\varPi }$ satisfying $\hat{\mathbf {v}}\le \varvec{\phi }({\hat{t}})\le \hat{\mathbf {w}}$ and either $\phi _i({\hat{t}})={\hat{v}}_i$ or $\phi _i({\hat{t}})={\hat{w}}_i$ for at least one $i\in \{1,\dots ,n_x\}$, there exists $\eta >0$ and $\alpha \in L^1(I)$ such that the following conditions hold for every $(t,\mathbf {v},\mathbf {w})\in B_{\eta }(({\hat{t}},\hat{\mathbf {v}},\hat{\mathbf {w}}))\cap D_{\varPi }$:

• 1. If $\phi _i(t)<v_i$, then $\exists \sigma _i \in \varPi ^L_i(t,\mathbf {v},\mathbf {w})$ such that

  $$\begin{aligned} |\sigma _i-{\dot{\phi }}_i(t)|\le \alpha (t)\max (\Vert \max (\mathbf {v}-\mathbf {\varvec{\phi }}(t),\mathbf {0})\Vert ,\Vert \max (\mathbf {\varvec{\phi }}(t)-\mathbf {w},\mathbf {0})\Vert ). \end{aligned}$$

  (48)

• 2. If $\phi _i(t)>w_i$, then $\exists \sigma _i \in \varPi ^U_i(t,\mathbf {v},\mathbf {w})$ such that (48) holds.

Lemma 1

If $\varPi ^L_i$ and $\varPi ^U_i$ satisfy Hypothesis 1, then they also satisfy Hypothesis 2.

Proof

Assume that Hypothesis 1 holds. To verify Hypothesis 2, choose any $({\hat{t}},\hat{\mathbf {v}},\hat{\mathbf {w}})\in D_{\varPi }$ such that $\hat{\mathbf {v}}\le \varvec{\phi }({\hat{t}})\le \hat{\mathbf {w}}$ and either $\phi _i({\hat{t}})={\hat{v}}_i$ or $\phi _i({\hat{t}})={\hat{w}}_i$ for at least one $i\in \{1,\dots ,n_x\}$. Define

$$\begin{aligned}&{\underline{\varvec{\phi }}}(t,\mathbf {v},\mathbf {w})\equiv \min (\mathbf {v},\varvec{\phi }(t)) \quad \text {and}\quad {\overline{\varvec{\phi }}}(t,\mathbf {v},\mathbf {w})\equiv \max (\mathbf {w},\varvec{\phi }(t)),\nonumber \\&\quad \forall (t,\mathbf {v},\mathbf {w})\in I\times {\mathbb {R}}^{n_x}\times {\mathbb {R}}^{n_x}. \end{aligned}$$

(49)

Note that $({\underline{\varvec{\phi }}}({\hat{t}},\hat{\mathbf {v}},\hat{\mathbf {w}}),{\overline{\varvec{\phi }}}({\hat{t}},\hat{\mathbf {v}},\hat{\mathbf {w}}))=(\hat{\mathbf {v}},\hat{\mathbf {w}})$ and ${\underline{\varvec{\phi }}}(t,\mathbf {v},\mathbf {w})\le \varvec{\phi }(t) \le {\overline{\varvec{\phi }}}(t,\mathbf {v},\mathbf {w}),$$\forall (t,\mathbf {v},\mathbf {w})\in I\times {\mathbb {R}}^{n_x}\times {\mathbb {R}}^{n_x}$.

With $({\hat{t}},\hat{\mathbf {v}},\hat{\mathbf {w}})$ as above, let $\eta _{C3}>0$ satisfy Condition 3 of Hypothesis 1, and let $\eta _{C4}>0$ and $\alpha \in L^1(I)$ satisfy Condition 4 of Hypothesis 1. Set $\eta _C=\min (\eta _{C3},\eta _{C4})$. Since $({\underline{\varvec{\phi }}}({\hat{t}},\hat{\mathbf {v}},\hat{\mathbf {w}}),{\overline{\varvec{\phi }}}({\hat{t}},\hat{\mathbf {v}},\hat{\mathbf {w}}))=(\hat{\mathbf {v}},\hat{\mathbf {w}})$ and the functions ${\underline{\varvec{\phi }}}$ and ${\overline{\varvec{\phi }}}$ are continuous, we may choose $\eta \in (0,\eta _C]$ such that

$$\begin{aligned} (t,\mathbf {v},\mathbf {w})\in B_{\eta }(({\hat{t}},\hat{\mathbf {v}},\hat{\mathbf {w}})) \quad \implies \quad (t,{\underline{\varvec{\phi }}}(t,\mathbf {v},\mathbf {w}),{\overline{\varvec{\phi }}}(t,\mathbf {v},\mathbf {w})) \in B_{\eta _C}(({\hat{t}},\hat{\mathbf {v}},\hat{\mathbf {w}})). \end{aligned}$$

(50)

By Condition 3 of Hypothesis 1, it follows that

$$\begin{aligned} (t,\mathbf {v},\mathbf {w})\in B_{\eta }(({\hat{t}},\hat{\mathbf {v}},\hat{\mathbf {w}})) \ \text {and} \ t\in I \quad \implies \quad (t,{\underline{\varvec{\phi }}}(t,\mathbf {v},\mathbf {w}),{\overline{\varvec{\phi }}}(t,\mathbf {v},\mathbf {w})) \in D_{\varPi }. \end{aligned}$$

(51)

We now show that Hypothesis 2 holds with this choice of $\eta$ and $\alpha$. To verify Condition 1 of Hypothesis 2, choose any $(t,\mathbf {v},\mathbf {w})\in B_{\eta }(({\hat{t}},\hat{\mathbf {v}},\hat{\mathbf {w}}))\cap D_{\varPi }$ such that $\phi _i(t)<v_i$. We will apply the Lipschitz condition (46) with this choice of $\mathbf {v}$ and $\mathbf {w}$ and with $\tilde{\mathbf {v}}={\underline{\varvec{\phi }}}(t,\mathbf {v},\mathbf {w})$ and $\tilde{\mathbf {w}}={\overline{\varvec{\phi }}}(t,\mathbf {v},\mathbf {w})$. To see that this condition is applicable, first note that $(t,\mathbf {v},\mathbf {w})\in B_{\eta _{C4}}(({\hat{t}},\hat{\mathbf {v}},\hat{\mathbf {w}}))$ because $\eta \le \eta _{C4}$. Moreover, in light of (50) and (51), we are guaranteed that $(t,\tilde{\mathbf {v}},\tilde{\mathbf {w}})\in B_{\eta _{C4}}(({\hat{t}},\hat{\mathbf {v}},\hat{\mathbf {w}}))\cap D_{\varPi }$. Thus, (46) gives

$$\begin{aligned}&d_H(\varPi _i^L(t,\mathbf {v},\mathbf {w}),\varPi _i^L(t,{\underline{\varvec{\phi }}}(t,\mathbf {v},\mathbf {w}),{\overline{\varvec{\phi }}}(t,\mathbf {v},\mathbf {w})))\nonumber \\&\quad \le \alpha (t)\max \left( \Vert \mathbf {v}-{\underline{\varvec{\phi }}}(t,\mathbf {v},\mathbf {w})\Vert ,\Vert \mathbf {w}-{\overline{\varvec{\phi }}}(t,\mathbf {v},\mathbf {w})\Vert \right) , \nonumber \\&\quad = \alpha (t)\max (\Vert \max (\mathbf {v}-\mathbf {\varvec{\phi }}(t),\mathbf {0})\Vert ,\Vert \max (\mathbf {\varvec{\phi }}(t)-\mathbf {w},\mathbf {0})\Vert ). \end{aligned}$$

(52)

Next, we apply Condition 1 of Hypothesis 1 to the point $(t,{\underline{\varvec{\phi }}}(t,\mathbf {v},\mathbf {w}),{\overline{\varvec{\phi }}}(t,\mathbf {v},\mathbf {w}))\in D_{\varPi }$. This is possible because ${\underline{\varvec{\phi }}}(t,\mathbf {v},\mathbf {w})\le \varvec{\phi }(t)\le {\overline{\varvec{\phi }}}(t,\mathbf {v},\mathbf {w})$ and $\phi _i(t)=\min (\phi _i(t),v_i)={\underline{\phi }}_i(t,\mathbf {v},\mathbf {w})$. Thus, Condition 1 of Hypothesis 1 ensures that ${\dot{\phi }}_i(t)\in \varPi _i^L(t,{\underline{\varvec{\phi }}}(t,\mathbf {v},\mathbf {w}),{\overline{\varvec{\phi }}}(t,\mathbf {v},\mathbf {w}))$. Then, by the definition of the Hausdorff metric, (52) implies that $\exists \sigma _i\in \varPi _i^L(t,\mathbf {v},\mathbf {w})$ satisfying (48). This proves Condition 1 of Hypothesis 2, and Condition 2 follows from an analogous argument. $\square$

To prove Theorem 1, we will apply Theorem 3 with the following definitions:

$$\begin{aligned} D_{\varPi }&\equiv \left\{ \begin{array}{rl} t &{}\in I \\ \mathbf {v} &{}\in {\mathbb {R}}^{n_x} \\ \mathbf {w} &{}\in {\mathbb {R}}^{n_x} \end{array}: \begin{array}{l} \mathbf {v}\le \mathbf {w} \\ (t,U,{\mathscr {B}}_i^{L/U}([\mathbf {v},\mathbf {w}])) \in D_{{\mathscr {R}}} \\ \forall i\in \{1,\dots ,n_x\} \end{array} \right\} \end{aligned}$$

(53)

$$\begin{aligned} \varPi _i^{L}(t,\mathbf {v},\mathbf {w})&\equiv {\mathscr {R}}_i[t,U,{\mathscr {B}}_i^L([\mathbf {v},\mathbf {w}])] \end{aligned}$$

(54)

$$\begin{aligned} \varPi _i^{U}(t,\mathbf {v},\mathbf {w})&\equiv {\mathscr {R}}_i[t,U,{\mathscr {B}}_i^U([\mathbf {v},\mathbf {w}])] \end{aligned}$$

(55)

Lemma 2

Let $(\mathbf {x}_0,\mathbf {u},\mathbf {x})\in X_0\times {\mathscr {U}}\times \mathscr {AC}(I,{\mathbb {R}}^{n_x})$ be any solution of (2). Under Assumption 1, definitions (53)–(55) satisfy Hypothesis 1 with $\varvec{\phi }\equiv \mathbf {x}$.

Proof

To verify Condition 1 of Hypothesis 1, choose any $(t,\mathbf {v},\mathbf {w})\in D_{\varPi }$ such that $\mathbf {v}\le \varvec{\phi }(t)\le \mathbf {w}$ and $\phi _i(t)=v_i$ for some $i\in \{1,\dots ,n_x\}$. These conditions imply that $\mathbf {x}(t)=\varvec{\phi }(t)\in {\mathscr {B}}_i^L([\mathbf {v},\mathbf {w}])$. Moreover, by the definition of $D_{\varPi }$, $(t,\mathbf {v},\mathbf {w})\in D_{\varPi }$ implies that $(t,U,{\mathscr {B}}_i^L([\mathbf {v},\mathbf {w}]))\in D_{{\mathscr {R}}}$. Since $(\mathbf {x}_0,\mathbf {u},\mathbf {x})$ is a solution of (2), Condition 1 of Assumption 1 implies that

$$\begin{aligned} \dot{\mathbf {x}}(t)\in {\mathscr {R}}\left[ t,U,{\mathscr {B}}_i^L([\mathbf {v},\mathbf {w}])\right] . \end{aligned}$$

(56)

By (54), it follows that ${\dot{\phi }}_i(t)={\dot{x}}_i(t)\in \varPi ^L_i(t,\mathbf {v},\mathbf {w})$. This proves Condition 1 of Hypothesis 1, and Condition 2 follows from an analogous argument.

To verify Condition 3 of Hypothesis 1, choose any $({\hat{t}},\hat{\mathbf {v}},\hat{\mathbf {w}})\in D_{\varPi }\cap A$. By (53), $({\hat{t}},U,{\mathscr {B}}_i^{L/U}([\hat{\mathbf {v}},\hat{\mathbf {w}}]))\in D_{{\mathscr {R}}}$ for all $i\in \{1,\ldots ,n_x\}$. By Condition 2 of Assumption 1, $D_{{\mathscr {R}}}$ is open with respect to t and Z. Thus, there must exist $\eta >0$ such that

$$\begin{aligned} (t,Z)\in B_{\eta }({\hat{t}})\times B_{\eta }({\mathscr {B}}_i^{L/U}([\hat{\mathbf {v}},\hat{\mathbf {w}}])) \quad \implies \quad (t,U,Z)\in D_{{\mathscr {R}}}. \end{aligned}$$

(57)

Moreover, by the definition of ${\mathscr {B}}_i^{L/U}$, it follows that

$$\begin{aligned} (t,Z)\in B_{\eta }({\hat{t}})\times B_{\eta }([\hat{\mathbf {v}},\hat{\mathbf {w}}])&\quad \implies \quad (t,{\mathscr {B}}_i^{L/U}(Z))\in B_{\eta }({\hat{t}})\times B_{\eta }({\mathscr {B}}_i^{L/U}([\hat{\mathbf {v}},\hat{\mathbf {w}}])), \nonumber \\&\quad \implies \quad (t,U,{\mathscr {B}}_i^{L/U}(Z))\in D_{{\mathscr {R}}}. \end{aligned}$$

(58)

We claim that Condition 3 of Hypothesis 1 holds with this $\eta$. To see this, choose any point $(t,\mathbf {v},\mathbf {w})$ in $B_{\eta }(({\hat{t}},\hat{\mathbf {v}},\hat{\mathbf {w}}))\cap A$. It suffices to show that $(t,\mathbf {v},\mathbf {w})\in D_{\varPi }$. Since $(t,\mathbf {v},\mathbf {w})\in A$, we have $t\in I$ and $\mathbf {v}\le \mathbf {w}$. Moreover, since $(t,\mathbf {v},\mathbf {w})\in B_{\eta }(({\hat{t}},\hat{\mathbf {v}},\hat{\mathbf {w}}))$, it follows from the definition of $d_H$ that $d_H([\mathbf {v},\mathbf {w}],[\hat{\mathbf {v}},\hat{\mathbf {w}}])\le \eta$. Finally, since $|t-{\hat{t}}|\le \eta$ as well, (58) ensures that $(t,U,{\mathscr {B}}_i^{L/U}([\mathbf {v},\mathbf {w}]))\in D_{{\mathscr {R}}}$. Thus, by (53), $(t,\mathbf {v},\mathbf {w}) \in D_{\varPi }$, as desired.

To verify Condition 4 of Hypothesis 1, choose any $({\hat{t}},\hat{\mathbf {v}},\hat{\mathbf {w}})\in D_{\varPi }$. By (53), $({\hat{t}},U,{\mathscr {B}}_i^{L/U}([\hat{\mathbf {v}},\hat{\mathbf {w}}]))\in D_{{\mathscr {R}}}$ for all $i\in \{1,\ldots ,n_x\}$. Thus, by Condition 3 of Assumption 1, there exists $\eta ,L>0$ such that

$$\begin{aligned} d_H({\mathscr {R}}(t,U,Z),{\mathscr {R}}(t,U,{\tilde{Z}}))&\le Ld_H(Z,{\tilde{Z}}), \end{aligned}$$

(59)

for every $t\in B_{\eta }({\hat{t}})$ and $Z,{\tilde{Z}}\in B_{\eta }({\mathscr {B}}_i^{L/U}([\hat{\mathbf {v}},\hat{\mathbf {w}}]))$. We claim that Condition 4 of Hypothesis 1 holds with this choice of $\eta$ and $\alpha =L$. To see this, choose any $(t,\mathbf {v},\mathbf {w}),(t,\tilde{\mathbf {v}},\tilde{\mathbf {w}})\in B_{\eta }(({\hat{t}},\hat{\mathbf {v}},\hat{\mathbf {w}}))\cap D_{\varPi }$. It suffices to show that

$$\begin{aligned} d_H(\varPi _i^L(t,\mathbf {v},\mathbf {w}),\varPi _i^L(t,\tilde{\mathbf {v}},\tilde{\mathbf {w}}))&\le L\max \left( \Vert \mathbf {v}-\tilde{\mathbf {v}}\Vert ,\Vert \mathbf {w}-\tilde{\mathbf {w}}\Vert \right) , \end{aligned}$$

(60)

$$\begin{aligned} d_H(\varPi _i^U(t,\mathbf {v},\mathbf {w}),\varPi _i^U(t,\tilde{\mathbf {v}},\tilde{\mathbf {w}}))&\le L\max \left( \Vert \mathbf {v}-\tilde{\mathbf {v}}\Vert ,\Vert \mathbf {w}-\tilde{\mathbf {w}}\Vert \right) . \end{aligned}$$

(61)

By (54) and the definition of the Hausdorff metric $d_{H}$,

$$\begin{aligned} d_H(\varPi _i^L(t,\mathbf {v},\mathbf {w}),\varPi _i^L(t,\tilde{\mathbf {v}},\tilde{\mathbf {w}}))&\le d_H({\mathscr {R}}(t,U,{\mathscr {B}}_i^{L}([\mathbf {v},\mathbf {w}])),{\mathscr {R}}(t,U,{\mathscr {B}}_i^{L}([\tilde{\mathbf {v}},\tilde{\mathbf {w}}]))). \end{aligned}$$

(62)

But, as argued above, the fact that $(t,\mathbf {v},\mathbf {w})$ and $(t,\tilde{\mathbf {v}},\tilde{\mathbf {w}})$ are elements of $B_{\eta }(({\hat{t}},\hat{\mathbf {v}},\hat{\mathbf {w}}))$ implies that ${\mathscr {B}}_i^{L}([\mathbf {v},\mathbf {w}])$ and ${\mathscr {B}}_i^{L}([\tilde{\mathbf {v}},\tilde{\mathbf {w}}])$ are elements of $B_{\eta }({\mathscr {B}}_i^{L/U}([\hat{\mathbf {v}},\hat{\mathbf {w}}]))$. Then, using (59), we have

$$\begin{aligned} d_H(\varPi _i^L(t,\mathbf {v},\mathbf {w}),\varPi _i^L(t,\tilde{\mathbf {v}},\tilde{\mathbf {w}}))&\le Ld_H({\mathscr {B}}_i^{L}([\mathbf {v},\mathbf {w}]),{\mathscr {B}}_i^{L}([\tilde{\mathbf {v}},\tilde{\mathbf {w}}])), \end{aligned}$$

(63)

$$\begin{aligned}&\le L\max \left( \Vert \mathbf {v}-\tilde{\mathbf {v}}\Vert ,\Vert \mathbf {w}-\tilde{\mathbf {w}}\Vert \right) , \end{aligned}$$

(64)

as desired. The proof of (61) is analogous. $\square$

We now prove Theorem 1. Choose any $\mathbf {x}^L,\mathbf {x}^U\in \mathscr {AC}(I,{\mathbb {R}}^{n_x})$ and suppose that Conditions 1–3 of Theorem 1 hold. Moreover, let $(\mathbf {x}_0,\mathbf {u},\mathbf {x})\in X_0\times {\mathscr {U}}\times \mathscr {AC}(I,{\mathbb {R}}^{n_x})$ be any solution of (2). We show that the hypotheses of Theorem 3 are satisfied with $\mathbf {v}=\mathbf {x}^L$, $\mathbf {w}=\mathbf {x}^U$, $\varvec{\phi }=\mathbf {x}$, and definitions (53)–(55). As a consequence, $\mathbf {x}(t)\in [\mathbf {x}^L(t),\mathbf {x}^U(t)]$, $\forall t\in I$, as desired.

With the definition of $D_{\varPi }$ in (53), Condition 1 of Theorem 3 follows directly from Condition 1 of Theorem 1. Condition 2 of Theorem 3 also follows directly from Condition 2 of Theorem 1 since $\varvec{\phi }(t_0)=\mathbf {x}(t_0)=\mathbf {x}_0\in X_0\subset [\mathbf {x}^L(t_0),\mathbf {x}^U(t_0)]=[\mathbf {v}(t_0),\mathbf {w}(t_0)]$. Finally, Condition 3 of Theorem 3 follows from Condition 3 of Theorem 1. To see this, choose any $\sigma _i\in \varPi _i^L(t,\mathbf {v}(t),\mathbf {w}(t))=\varPi _i^L(t,\mathbf {x}^L(t),\mathbf {x}^U(t))$. By (54),

$$\begin{aligned} \sigma _i\in {\mathscr {R}}_i[t,U,{\mathscr {B}}_i^{L}([\mathbf {x}^L(t),\mathbf {x}^U(t)])]. \end{aligned}$$

(65)

Therefore, by Condition 3(a) of Theorem 1, we must have ${\dot{v}}_i(t)\le \sigma _i$. This proves Condition 3(a) of Theorem 3. Condition 3(b) is proved analogously. Since all of the hypotheses of Theorem 3 are met, we conclude that

$$\begin{aligned} \mathbf {x}(t)=\varvec{\phi }(t)\in [\mathbf {v}(t),\mathbf {w}(t)]=[\mathbf {x}^L(t),\mathbf {x}^U(t)], \quad \forall t\in I. \end{aligned}$$

(66)

This completes the proof of Theorem 1. $\square$