Optimization-Constrained Differential Equations with Active Set Changes

Abstract

Foundational theory is established for nonlinear differential equations with embedded nonlinear optimization problems exhibiting active set changes. Existence, uniqueness, and continuation of solutions are shown, followed by lexicographically smooth (implying Lipschitzian) parametric dependence. The sensitivity theory found here accurately characterizes sensitivity jumps resulting from active set changes via an auxiliary nonsmooth sensitivity system obtained by lexicographic directional differentiation. The results in this article hold under easily verifiable regularity conditions (linear independence of constraints and strong second-order sufficiency), which are shown to imply generalized differentiation index one of a nonsmooth differential-algebraic equation system obtained by replacing the optimization problem with its optimality conditions and recasting the complementarity conditions as nonsmooth algebraic equations. The theory in this article is computationally relevant, allowing for implementation of dynamic optimization strategies (i.e., open-loop optimal control), and recovers (and rigorously formalizes) classical results in the absence of active set changes. Along the way, contributions are made to the theory of piecewise differentiable functions.

Appendix

Proof of Proposition 2.1

Without loss of generality, let $\{{\mathbf {f}}_{(1)},\ldots ,{\mathbf {f}}_{(k)}\}$ be a set of essentially active $C^1$ selection functions of ${\mathbf {f}}$ at ${\mathbf {x}}$. $\varLambda {\mathbf {f}}({\mathbf {x}})$ is nonempty since ${\mathbf {J}}{\mathbf {f}}_{(i)}({\mathbf {x}}) \in \varLambda {\mathbf {f}}({\mathbf {x}})$ for each $i\in \{1,\ldots ,k\}$. $\varLambda {\mathbf {f}}({\mathbf {x}})$ is a finite set, with $|\varLambda {\mathbf {f}}({\mathbf {x}})| \le n k$, and therefore compact. We proceed in showing upper semicontinuity as follows: choose any $\varepsilon >0$. Since $\partial _{\mathrm{B}}{\mathbf {f}}$ is upper semicontinuous [28], there exists $\delta >0$ such that $\partial _{\mathrm{B}} {\mathbf {f}}({\mathbf {y}}) \subset B_{\varepsilon ^*}(\partial _{\mathrm{B}}{\mathbf {f}}({\mathbf {x}}))$ for all ${\mathbf {y}}\in B_{\delta }({\mathbf {x}})$, where $\varepsilon ^*=\varepsilon /n$ and $B_{\varepsilon ^*}(\partial _{\mathrm{B}}{\mathbf {f}}({\mathbf {x}}))=\left\{ {\mathbf {F}}+ \varepsilon ^* {\mathbf {Y}}: {\mathbf {F}}\in \partial _{\mathrm{B}}{\mathbf {f}}({\mathbf {x}}), \Vert {\mathbf {Y}}\Vert <1\right\}$. Choose any point ${\mathbf {x}}^{\delta } \in B_{\delta }({\mathbf {x}})$. Enumerate the B-subdifferential of ${\mathbf {f}}$ at ${\mathbf {x}}$ and ${\mathbf {x}}^{\delta }$ by, respectively, $\{{\mathbf {F}}_{(1)},\ldots ,{\mathbf {F}}_{(k)}\}=\partial _{\mathrm{B}}{\mathbf {f}}({\mathbf {x}})$ and $\{{\mathbf {F}}^{\delta }_{(1)},\ldots ,{\mathbf {F}}^{\delta }_{(q)}\} =\partial _{\mathrm{B}}{\mathbf {f}}({\mathbf {x}}^{\delta })$.

Let $\pmb {\varGamma }:\partial _{\mathrm{B}} {\mathbf {f}}({\mathbf {x}}^{\delta }) \rightarrow \partial _{\mathrm{B}} {\mathbf {f}}({\mathbf {x}})$ be defined as follows: given ${\mathbf {F}}^{\delta } \in \partial _{\mathrm{B}} {\mathbf {f}}({\mathbf {x}}^{\delta })$, let $\pmb {\varGamma }({\mathbf {F}}^{\delta })={\mathbf {F}}\in \partial _{\mathrm{B}} {\mathbf {f}}({\mathbf {x}})$ be such that ${\mathbf {F}}^{\delta }={\mathbf {F}}+\varepsilon ^* {\mathbf {Y}}$ for some $\Vert {\mathbf {Y}}\Vert <1$. For any $i\in \{1,\ldots ,n\}$, let the mapping $\varTheta _i:\varLambda {\mathbf {f}}({\mathbf {x}}^{\delta }) \rightarrow \partial _{\mathrm{B}}{\mathbf {f}}({\mathbf {x}}^{\delta })$ be defined as follows: given ${\mathbf {F}}^{\varLambda } \in \varLambda {\mathbf {f}}({\mathbf {x}}^{\delta })$, let $\varTheta _i({\mathbf {F}}^{\varLambda })={\mathbf {F}}^{\delta } \in \partial _{\mathrm{B}}{\mathbf {f}}({\mathbf {x}}^{\delta })$ be such that $\text {row}_i({\mathbf {F}}^{\varLambda })=\text {row}_i({\mathbf {F}}^{\delta })$.

Choose any $\bar{{\mathbf {F}}}^{\delta } \in \varLambda {\mathbf {f}}({\mathbf {x}}^{\delta })$ and let

$$\begin{aligned}\bar{{\mathbf {F}}}= \begin{bmatrix} \text {row}_1(\pmb {\varGamma }(\varTheta _1(\bar{{\mathbf {F}}}^{\delta }))) \\ \text {row}_2(\pmb {\varGamma }(\varTheta _2(\bar{{\mathbf {F}}}^{\delta }))) \\ \vdots \\ \text {row}_n(\pmb {\varGamma }(\varTheta _n(\bar{{\mathbf {F}}}^{\delta }))) \end{bmatrix}. \end{aligned}$$

For any $i\in \{1,\ldots ,n\}$, $\pmb {\varGamma }(\varTheta _i(\bar{{\mathbf {F}}}^{\delta }))={\mathbf {F}}_{(j)} \in \partial _{\mathrm{B}} {\mathbf {f}}({\mathbf {x}})$ s.t. ${\mathbf {F}}^{\delta }_{(l)}={\mathbf {F}}_{(j)}+\varepsilon ^* {\mathbf {Y}}_{(i)}$ for some ${\mathbf {F}}^{\delta }_{(l)} \in \partial _{\mathrm{B}}{\mathbf {f}}({\mathbf {x}}^{\delta })$ satisfying $\text {row}_i(\bar{{\mathbf {F}}}^{\delta })=\text {row}_i({\mathbf {F}}^{\delta }_{(l)})$ and $\Vert {\mathbf {Y}}_{(i)}\Vert <1$. Thus, $\text {row}_i(\bar{{\mathbf {F}}})=\text {row}_i(\pmb {\varGamma }(\varTheta _i(\bar{{\mathbf {F}}}^{\delta }))))=\text {row}_i({\mathbf {F}}_{(j)})$, implying that $\bar{{\mathbf {F}}} \in \varLambda {\mathbf {f}}({\mathbf {x}})$. Moreover, $\text {row}_i(\bar{{\mathbf {F}}}^{\delta }-\bar{{\mathbf {F}}}) =\text {row}_i(\bar{{\mathbf {F}}}^{\delta })-\text {row}_i(\pmb {\varGamma }(\varTheta _i(\bar{{\mathbf {F}}}^{\delta })))$, from which it follows that $\text {row}_i(\bar{{\mathbf {F}}}^{\delta }-\bar{{\mathbf {F}}})=\text {row}_i(\bar{{\mathbf {F}}}^{\delta })-\text {row}_i(\pmb {\varGamma }({\mathbf {F}}_{(l)}^{\delta })) =\text {row}_i(\bar{{\mathbf {F}}}^{\delta })-\text {row}_i({\mathbf {F}}_{(j)})$ where

$$\begin{aligned} \text {row}_i({\mathbf {F}}_{(j)}) =\text {row}_i({\mathbf {F}}_{(l)}^{\delta } - \varepsilon ^* {\mathbf {Y}}_{(l)}) =\text {row}_i(\bar{{\mathbf {F}}}^{\delta }) - \varepsilon ^* \text {row}_i({\mathbf {Y}}_{(i)}). \end{aligned}$$

From this, it follows that $\text {row}_i(\bar{{\mathbf {F}}}^{\delta }-\bar{{\mathbf {F}}}) =\text {row}_i(\bar{{\mathbf {F}}}^{\delta })-\text {row}_i({\mathbf {F}}_{(j)})$, and therefore

$$\begin{aligned} \text {row}_i(\bar{{\mathbf {F}}}^{\delta }-\bar{{\mathbf {F}}}) =\text {row}_i(\bar{{\mathbf {F}}}^{\delta })-(\text {row}_i(\bar{{\mathbf {F}}}^{\delta }) - \varepsilon ^* \text {row}_i({\mathbf {Y}}_{(l)}))=\varepsilon ^* \text {row}_i({\mathbf {Y}}_{(i)}). \end{aligned}$$

The above result holds for any $i\in \{1,\ldots ,n\}$, implying that $\bar{{\mathbf {F}}}^{\delta }-\bar{{\mathbf {F}}}= \varepsilon ^* \bar{{\mathbf {Y}}}$ where

$$\begin{aligned} \bar{{\mathbf {Y}}}= \begin{bmatrix} \text {row}_1({\mathbf {Y}}_{(1)}) \\ \text {row}_2({\mathbf {Y}}_{(2)}) \\ \vdots \\ \text {row}_n({\mathbf {Y}}_{(n)}) \\ \end{bmatrix}. \end{aligned}$$

Note that $\Vert {\mathbf {Y}}_{(i)}\Vert <1$ for each i, from which it follows that $\Vert {\mathbf {Y}}_{(i)}\Vert _{\infty } < \sqrt{n}$. Thus,

$$\begin{aligned} \Vert \bar{{\mathbf {Y}}}\Vert \le \sqrt{n} \Vert \bar{{\mathbf {Y}}}\Vert _{\infty } \le \sqrt{n} \max \{\Vert {\mathbf {Y}}_{(i)}\Vert _{\infty }\} < n. \end{aligned}$$

Hence, $\bar{{\mathbf {F}}}^{\delta }=\bar{{\mathbf {F}}}+\varepsilon \left( \frac{1}{n} \bar{{\mathbf {Y}}}\right)$ where $\Vert \frac{1}{n} \bar{{\mathbf {Y}}}\Vert <1$. Thus, $\bar{{\mathbf {F}}}^{\delta } \in B_{\varepsilon }(\varLambda {\mathbf {f}}({\mathbf {x}}))$, from which upper semicontinuity follows.

Choose $n^* \in {\mathbb {N}}$ such that $B_{1/n^*}(\varOmega ) \subset X$. If ${\mathbf {f}}$ is not CCO on a neighborhood of $\varOmega$, then for any $n \ge n^*$, there must exist ${\mathbf {x}}_{n} \in B_{1/n}(\varOmega ) {\setminus } \varOmega$ such that ${\mathbf {f}}$ is not CCO at ${\mathbf {x}}_{n}$. Let ${\mathbf {x}}^* \in \varOmega$ be an accumulation point of the sequence $\{{\mathbf {x}}_{n}\}$. Then, since ${\mathbf {f}}$ is CCO on $\varOmega$, ${{\,\mathrm{sign}\,}}(\det ({\mathbf {F}}))=1$ for all ${\mathbf {F}}\in \varLambda {\mathbf {f}}({\mathbf {x}}^*)$, without loss of generality. Then, by the arguments in the proof of [32, Lemma 7.5.2], upper semicontinuity of $\varLambda {\mathbf {f}}$ at ${\mathbf {x}}^*$, along with continuity of the determinant, imply the existence of $\rho >0$ such that ${{\,\mathrm{sign}\,}}(\det ({\mathbf {F}}^{\rho }))=1$ for all ${\mathbf {F}}^{\rho } \in \varLambda {\mathbf {f}}({\mathbf {x}}^{\rho })$ and all ${\mathbf {x}}^{\rho } \in B_{\rho }({\mathbf {x}}^*) \subset X$. However, for some $\widetilde{n} \ge n^*$, ${\mathbf {x}}_{\widetilde{n}} \in B_{\rho }({\mathbf {x}}^*)$, a contradiction. $\square$

Proof of Proposition 2.2

The function ${\mathbf {f}}$ is $PC^1$ at $({\mathbf {x}}^*,{\mathbf {y}}^*)$ by construction, with

$$\begin{aligned} \partial _{\mathrm{B}} {\mathbf {f}}({\mathbf {x}}^*,{\mathbf {y}}^*) = \left\{ \left[ \begin{array}{c|c} {\mathbf {I}}_{n} &{} {\mathbf {0}}\\ \hline {\mathbf {J}}_{{\mathbf {x}}}{\mathbf {g}}_{(i)}({\mathbf {x}}^*,{\mathbf {y}}^*) &{} {\mathbf {J}}_{{\mathbf {y}}}{\mathbf {g}}_{(i)}({\mathbf {x}}^*,{\mathbf {y}}^*) \end{array}\right] : {\mathbf {g}}_{(i)} \in \mathscr {F}_{{\mathbf {g}},({\mathbf {x}}^*,{\mathbf {y}}^*)} \right\} \end{aligned}$$

where $\mathscr {F}_{{\mathbf {g}},({\mathbf {x}}^*,{\mathbf {y}}^*)}=\{{\mathbf {g}}_{(i)} \}$ is a set of essentially active $C^1$ selection functions of ${\mathbf {g}}$ at $({\mathbf {x}}^*,{\mathbf {y}}^*)$. The combinatorial Jacobian of ${\mathbf {f}}$ at $({\mathbf {x}}^*,{\mathbf {y}}^*)$ is

$$\begin{aligned} \varLambda {\mathbf {f}}({\mathbf {x}}^*,{\mathbf {y}}^*) = \left\{ \left[ \begin{array}{c|c} {\mathbf {I}}_{n} &{} {\mathbf {0}}\\ \hline {\mathbf {X}}^*_{(j)} &{} {\mathbf {Y}}^*_{(j)} \end{array}\right] : j=1,2,\ldots ,n_s \right\} \end{aligned}$$

where $n_s \le 2^{|\mathscr {F}_{{\mathbf {g}},({\mathbf {x}}^*,{\mathbf {y}}^*)}|}$, and for any $j \in \{1,\ldots ,n_{s}\}$, $k \in \{1,\ldots ,m\}$,

$$\begin{aligned} \text {row}_k[{\mathbf {X}}^*_{(j)} \quad {\mathbf {Y}}^*_{(j)}]=\text {row}_k[{\mathbf {J}}_{\mathbf {x}}{\mathbf {g}}_{(i)}({\mathbf {x}}^*,{\mathbf {y}}^*) \quad {\mathbf {J}}_{\mathbf {y}}{\mathbf {g}}_{(i)}({\mathbf {x}}^*,{\mathbf {y}}^*)] \end{aligned}$$

for some ${\mathbf {g}}_{(i)} \in \mathscr {F}_{{\mathbf {g}},({\mathbf {x}}^*,{\mathbf {y}}^*)}$ by definition of $\varLambda {\mathbf {f}}$. That is, ${\mathbf {f}}$ is CCO at $({\mathbf {x}}^*,{\mathbf {y}}^*)$ if and only if ${\mathbf {g}}$ is CCO w.r.t. ${\mathbf {y}}$ at $({\mathbf {x}}^*,{\mathbf {y}}^*)$, as required. $\square$

Proof of Theorem 2.1

First we demonstrate ${\mathbf {g}}$ is CCO w.r.t. ${\mathbf {y}}$ on a neighborhood of $\varOmega$: Define the mapping ${\mathbf {f}}: W \rightarrow {\mathbb {R}}^{n} \times {\mathbb {R}}^m:({\mathbf {x}},{\mathbf {y}}) \mapsto ({\mathbf {x}},{\mathbf {g}}({\mathbf {x}},{\mathbf {y}}))$. Then ${\mathbf {f}}$ is CCO at $({\mathbf {x}}^*,{\mathbf {y}}^*)$ if and only if ${\mathbf {g}}$ is CCO w.r.t. ${\mathbf {y}}$ at $({\mathbf {x}}^*,{\mathbf {y}}^*)$ by Proposition 2.2, from which it follows that ${\mathbf {f}}$ is CCO on $\varOmega$. Consequently, ${\mathbf {f}}$ is CCO on $B_{\gamma }(\varOmega ) \subset W$ for some $\gamma >0$ by Proposition 2.1, from which it follows that ${\mathbf {g}}$ is CCO w.r.t. ${\mathbf {y}}$ on $B_{\gamma }(\varOmega ) \subset W$, implying (b) holds.

Next, we argue (a) holds: Since ${\mathbf {g}}$ is CCO with respect to ${\mathbf {y}}$ and ${\mathbf {g}}({\mathbf {x}},{\mathbf {y}})={\mathbf {0}}$ for each $({\mathbf {x}},{\mathbf {y}}) \in \varOmega$, we may proceed as in the proof of [20, Theorem 3.5], but instead apply the $PC^1$ local implicit function theorem [31, Theorem 3.4] to each point in $\varOmega$ to furnish the family $\mathscr {F}_{{\mathbf {r}}}=\{{\mathbf {r}}_{\mathbf {x}}:{\mathbf {x}}\in \pi _x \varOmega \}$ of local $PC^1$ implicit functions, with corresponding collection of domains $\{N_{{\mathbf {x}}} \subset \pi _{\mathbf {x}}W: {\mathbf {x}}\in \pi _x \varOmega \}$ that are neighborhoods of ${\mathbf {x}}$, where $\pi _x W=\{{\mathbf {x}}:\exists ({\mathbf {x}},{\mathbf {y}}) \in W\}$. Then, continuing as in [20, Theorem 3.5], a $PC^1$ extended implicit function ${\mathbf {r}}:B_{\delta }(\pi _x \varOmega ) \subset \pi _x W \rightarrow {\mathbb {R}}^m$ can be constructed, for some $\delta >0$, using finitely many of these $PC^1$ local implicit functions, say $\{{\mathbf {r}}_{{\mathbf {x}}_{(1)}},{\mathbf {r}}_{{\mathbf {x}}_{(2)}},\ldots ,{\mathbf {r}}_{{\mathbf {x}}_{(q)}}\}$, such that for each ${\mathbf {x}}\in B_{\delta }(\pi _x \varOmega )$, $({\mathbf {x}},{\mathbf {r}}({\mathbf {x}}))$ is the unique vector in $B_{\xi }(\varOmega )$ satisfying ${\mathbf {g}}({\mathbf {x}},{\mathbf {r}}({\mathbf {x}}))={\mathbf {0}}$, for some $\xi >0$. Moreover, the function ${\mathbf {r}}$ is $PC^1$ on its domain by construction; for any ${\mathbf {x}}\in B_{\delta }(\pi _x \varOmega )$, ${\mathbf {r}}({\mathbf {x}}) \in \{{\mathbf {r}}_{{\mathbf {x}}_{(1)}}({\mathbf {x}}),{\mathbf {r}}_{{\mathbf {x}}_{(2)}}({\mathbf {x}}),\ldots ,{\mathbf {r}}_{{\mathbf {x}}_{(q)}}({\mathbf {x}})\}$. Hence, Conclusion (i) holds with $\rho =\min (\xi ,\gamma )$, shrinking $\delta$ if necessary.

Lastly, since a $PC^1$ function is L-smooth, the arguments from the proof of [21, Theorem 3.2] can be repeated, with the above $PC^1$ extended implicit function result, to show Eq. (5) holds. $\square$

Proof of Corollary 2.1

Conclusion (i) follows since $\{(t,{\mathbf {p}}_0,{\mathbf {z}}(t,{\mathbf {p}}_0)): t \in T, {\mathbf {p}}\in P\} \subset G_{\mathrm{R}}^{P} \subset G_{\mathrm{R}}^\mathrm{L}$. To show (ii), suppose that $(t_0,{\mathbf {p}}_0,{\mathbf {x}}_0,{\mathbf {y}}_0) \in G_{\mathrm{R}}^P \cap G_{\mathrm{C}}^0 \subset G_{\mathrm{R}}^{\mathrm{L}} \cap G_{\mathrm{C}}^0$. Then, since $G_{\mathrm{R}}^P$ is open, there exists $\rho >0$ sufficiently small such that $B_{\rho }(t_0,{\mathbf {p}}_0,{\mathbf {x}}_0,{\mathbf {y}}_0) \subset G_{\mathrm{R}}^P$, and, by continuity of the (local) solution $t \mapsto (t,{\mathbf {p}}_0,{\mathbf {z}}(t,{\mathbf {p}}_0))$, there exists $\alpha \in (0,\varepsilon )$ such that

$$\begin{aligned} \{(t,{\mathbf {p}}_0,{\mathbf {z}}(t,{\mathbf {p}}_0)):t\in [t_0-\alpha ,t_0+\alpha ]\} \subset B_{\rho }(t_0,{\mathbf {p}}_0,{\mathbf {x}}_0,{\mathbf {y}}_0) \subset G_{\text {R}}^{ \mathrm P}. \end{aligned}$$

To prove (iii), note that compactness of $\{(t,{\mathbf {p}}_0,\widetilde{{\mathbf {z}}}(t,{\mathbf {p}}_0)): t\in [t_l,t_u]\} \subset G_{\mathrm{R}}^{ \mathrm P}$ combined with openness of $G_{\mathrm{R}}^{ \mathrm P}$ implies the existence of $\rho >0$ such that $B_{\rho }(\{(t,{\mathbf {p}}_0,\widetilde{{\mathbf {z}}}(t,{\mathbf {p}}_0)): t\in [t_l,t_u]\}) \subset G_{\mathrm{R}}^{ \mathrm P}$. In addition, since $t \mapsto (t,{\mathbf {p}}_0,\widetilde{{\mathbf {z}}}(t,{\mathbf {p}}_0))$ is continuous, there exists $\varepsilon \in (0, \alpha )$ such that

$$\begin{aligned} \{(t,{\mathbf {p}}_0,\widetilde{{\mathbf {z}}}(t,{\mathbf {p}}_0)): t\in [t_l-\varepsilon ,t_u+\varepsilon ]\} \subset B_{\rho }(\{(t,{\mathbf {p}}_0,\widetilde{{\mathbf {z}}}(t,{\mathbf {p}}_0)): t\in [t_l,t_u]\}) \subset G_{\mathrm{R}}^{ \mathrm P}, \end{aligned}$$

implying P-regularity of $\widetilde{{\mathbf {z}}}$ on $[t_l-\alpha ,t_u+\alpha ] \times \{{\mathbf {p}}_0\}$. Lastly, to prove maximal regular continuation, define the set of augmented graphs of regular continuations as

$$\begin{aligned} \varGamma _{\mathrm{ext}}:=\{ \{(t,{\mathbf {p}}_0,\widehat{{\mathbf {z}}}(t,{\mathbf {p}}_0)):t \in \widehat{T}\}: \widehat{{\mathbf {z}}} \text { is a P-regular continuation of } {\mathbf {z}}\text { on } \widehat{T}\}. \end{aligned}$$

$\varGamma _{\mathrm{ext}}$ is nonempty since $\{(t,{\mathbf {p}}_0,\widetilde{{\mathbf {z}}}(t,{\mathbf {p}}_0)):t \in [t_l-\alpha ,t_u+\alpha ]\} \in \varGamma _{\mathrm{ext}}$. Zorn’s Lemma can be applied with the generalized inequality $\varPhi \preceq \varPsi$ for $\varPhi , \varPsi \in \varGamma _{\mathrm{ext}}$ if and only if $\varPhi \subset \varPsi$, which is a partial ordering (see the proof of [20, Theorem 6.4]). Given any nonempty totally ordered subset $\varGamma _{\mathrm{ext}}^*=\{\varOmega _{(i)}:i \in A\} \subset \varGamma _{\mathrm{ext}}$, where $\varOmega _{(i)}=\{(t,{\mathbf {p}}_0,{\mathbf {z}}_{(i)}(t,{\mathbf {p}}_0)):t \in T_{(i)}\}$, the element $\varOmega _u=\{(t,{\mathbf {p}}_0,{\mathbf {z}}_u(t)):t \in T_u\}$ is an upper bound of $\varGamma _{\mathrm{ext}}^*$, where $T_u=\bigcup _{i \in A} T_{(i)}$ and the mapping ${\mathbf {z}}_u:t \mapsto {\mathbf {z}}_{(i)}(\cdot ,{\mathbf {p}}_0), \; t \in T_{(i)}$ is single-valued since for any $i, j \in A$, ${\mathbf {z}}_{(i)}(t,{\mathbf {p}}_0)={\mathbf {z}}_{(j)}(t,{\mathbf {p}}_0)$ for all $t \in T_{(i)} \cap T_{(j)}$. Zorn’s Lemma implies that $\varGamma _{\mathrm{ext}}$ contains maximal elements; that is, there exists $\varOmega _{\mathrm{max}}= \{(t,{\mathbf {p}}_0,{\mathbf {z}}_{\mathrm{max}}(t,{\mathbf {p}}_0)):t \in T_{\mathrm{max}}\} \in \varGamma _{\mathrm{ext}}$, such that $\varOmega \subset \varOmega _{\mathrm{max}}$ for any $\varOmega \in \varGamma _{\mathrm{ext}}$. It is then possible to show that $T_{\mathrm{max}}=(t_L,t_U)$ since otherwise P-regularity allows for application of Theorem 2.2 (ii) to $(t_L,{\mathbf {p}}_0,{\mathbf {z}}_{\mathrm{max}}(t_L,{\mathbf {p}}_0))$ or $(t_U,{\mathbf {p}}_0,{\mathbf {z}}_{\mathrm{max}}(t_U,{\mathbf {p}}_0))$ to continue the solution on a strict superset of $T_{\mathrm{max}}$. $\square$