A Maximum Principle for a Time-Optimal Bilevel Sweeping Control Problem

Abstract

In this article, we investigate a time-optimal state-constrained bilevel optimal control problem whose lower-level dynamics feature a sweeping control process involving a truncated normal cone. By bilevel, it is meant that the optimization of the upper level problem is carried out over the solution set of the lower level problem.This problem instance arises in structured crowd motion control problems in a confined space. We establish the corresponding necessary optimality conditions in the Gamkrelidze’s form. The analysis relies on the smooth approximation of the lower level sweeping control system, thereby dealing with the resulting lack of Lipschitzianity with respect to the state variable inherent to the sweeping process, and on the flattening of the bilevel structure via an exact penalization technique. Necessary conditions of optimality in the Gamkrelidze’s form are applied to the resulting standard approximating penalized state-constrained single-level problem, and the main result of this article is obtained by passing to the limit.

Appendix

We provide here the proofs of the propositions 3–6 stated in Sect. 3.

Proof of Proposition 3

Fix some $k\in {\mathbb {N}}$. From Proposition 2, let $(T_k,y_k)$ be a pair such that ${{\mathcal {F}}}^k_L(T_k,y_k) \ne \emptyset$, with $y_k(0)=y_0$, and such that $\dot{y}_k=v_k\in {{{\mathcal {V}}}}$. Then, there exists a triple $(x_k,u_k,u_{0,k})\in AC([0,T_k];{\mathbb {R}}^n)\,\times \,{{{\mathcal {U}}}}\,\times \,{{{\mathcal {U}}}}_0$, satisfying $(D^k)$. Here, and in what follows, the sets ${{{\mathcal {U}}}}$ and ${{{\mathcal {U}}}}_0$ are defined on the time interval $[0,T_k]$.

The non-emptiness of the set of sequences $\left\{ (T_k,y_k, x_k,v_k, u_k,u_{0,k})\right\} _{k=1}^\infty$ satisfying $1)-3)$ follows from the fact that, for each k, the velocity set of the dynamics of $(D^k)$ contains the one of the dynamics of $(P_L)$ (as observed in the proof of Proposition 2), and, thus, obvious fact that the constant sequence $\{(T,y,x,v,u,u_0)\}$, which is feasible for $(D^k)$, satisfies $(x,u,u_0)\,\in \,{{{\mathcal {F}}}}_L(T,y)$. By inspection of the data of $(P_L)$ and $(P_H)$, we can construct a sequence with the stated properties, and such that, on $[0,\max \{T_k, T_{k+1}\}]$, and for some constant ${\widetilde{M}}\,\in \,{\mathbb {R}}^+\,$, $\varDelta \left( (v_k,u_k,u_{0,k}), (v_{k+1},u_{k+1},u_{0,k+1})\right) \,\le \,{\widetilde{M}} 2^{-k}.$

The boundedness of $f(\cdot ,\cdot )$ (by assumption H1), and of the approximation to the truncated normal cone defined by (18) implies that $\{\dot{x}_k\}$ is a bounded sequence in $L_1$. Let us see now that $\{x_k\}$ is an equicontinous and uniformly bounded sequence. Indeed, take any $\epsilon >0$, and consider $\tau _1$, and $\tau _2$ such that $T_k>\tau _2 > \tau _1$, and $\displaystyle \tau _2-\tau _1\,\le \,\frac{\epsilon }{M_1+M}$. Then,

$$\begin{aligned}&|x_k(\tau _2)\,-\,x_k(\tau _1)| \,\le \,\int _{\tau _1}^{\tau _2}\,\,\,|\dot{x}_k(s)| \mathrm{d}s \\&\quad \le \,\int _{\tau _1}^{\tau _2}\,\,\,\left( |f(x_k(s),u_k(s))| \,+\,|u_{0,k}(s) c(\gamma _k, x_k(s),y_k(s)) (x_k(s)\,-\,y_k(s))|\right) \mathrm{d}s \,\\&\quad \le \,(M_1\,+\,M) (\tau _2\,-\,\tau _1)\,\le \,\epsilon . \end{aligned}$$

Since the $x_k(0)$’s are bounded, the uniform boundedness of $\{x_k\}$ is obtained from the fact that, for all $t\,\in \,[0,{\overline{T}}]$, where $\displaystyle {\overline{T}}= \max _{i \in {\mathbb {N}}}\{T,T_i\}$, we have $|x_k(t)|\,\le \,|x_k(0)|\,+\,|x_k(t)-x_k(0)| \,\le \,|x_k(0)| \,+\,(M_1\,+\,M) t \,\le \,| x_k(0)| \,+\,(M_1\,+\,M) {\overline{T}}.$ Due to the uniform boundedness, and the equicontinuity of $\{x_k\}$, we conclude, from the application of Arzela–Ascoli theorem, the existence of a subsequence (without relabeling) such that $x_k$ converges uniformly to an absolutely continuous function x. From the boundedness of $\{\dot{x}_k\}$ in $L_1$, and, from the Dunford-Pettis theorem, there exists a subsequence (we do not relabel) $\{\dot{x}_k\}$ converging weakly in $L_1$ to some function $\xi \in L_1$. Moreover, since $\{x_k(0)\}$ is a bounded sequence, we arrange by further subsequence extraction that $x_k(0) \rightarrow x(0)$ for some $x(0)\in Q_1+y_0$. Hence, $\dot{x}(t) = \xi (t)$ a.e., and

$$\begin{aligned} x_k(t) = x_k(0) + \int _{0}^{t} \dot{x}_k(s) \mathrm{d}s \rightarrow _k x(t) := x(0) + \int _{0}^{t} \xi (s) \mathrm{d}s \qquad \text{ uniformly }. \end{aligned}$$

Next, we prove that $\dot{x}(t) \in f(x(t),u(t))- N_{Q_1+y(t)}^M(x(t))$ a.e.. From the convexity of $Q_1$, and from the definition of the normal cone to a convex set, this amounts to show the following inequality

$$\begin{aligned} \langle f(x(t),u(t))- \dot{x}(t), {\tilde{x}}- x(t) \rangle \le 0 \quad \forall {\tilde{x}} \in Q_1+y(t). \end{aligned}$$

(30)

For this purpose, let us examine the quantity $\langle f(x_k(t),u_k(t))- \dot{x}_k(t), {\tilde{x}}- x_k(t) \rangle$ for the control $u_k$, and the pair $(T_k,y_k)$, for which we recall that ${\mathcal {F}}_L^k(T_k,y_k)\ne \emptyset$. We consider two cases: ${\tilde{x}} \in \text{ int } (Q_1\,+\,y_k(t))$, and ${\tilde{x}} \in \partial (Q_1\,+\,y_k(t))$.

We start by taking an arbitrary ${\tilde{x}} \in \text{ int } (Q_1\,+\,y_k(t))$. From the a.e. equality

$$\begin{aligned} f(x_k(t),u_k(t))- \dot{x}_k(t) = u_{0,k}(t) c(\gamma _k, x_k(t),y_k(t))( x_k(t)-y_k(t)), \end{aligned}$$

the convexity of $h_L$, and the expression of its gradient, $\nabla _x h_L(x_k(t),y_k(t))= x_k(t)-y_k(t)$, (there is no loss of generality in assuming $x_k(t)\ne y_k(t)$), we conclude that

$$\begin{aligned} \langle f(x_k(t),u_k(t))- \dot{x}_k(t), {\tilde{x}}- x_k(t) \rangle\le & {} u_{0,k}(t) c( \gamma _k,x_k(t),y_k(t))\nonumber \\&(h_L({\tilde{x}},y_k(t))-\, h_L(x_k(t),y_k(t))) . \end{aligned}$$

(31)

Define $Z_k(t,{\tilde{x}}):= u_{0,k}(t) c( \gamma _k,x_k(t),y_k(t))(h_L({\tilde{x}},y_k(t)) - h_L(x_k(t),y_k(t)))$. From (19), we have that, either $\displaystyle \lim _{k\rightarrow \infty }\,Z_k(t,{\tilde{x}}) = 0$, if $x_k(t)\,\in \,\text{ int } (Q_1\,+\,y_k(t))$, or $Z_k(t,{\tilde{x}})\,\le \,0$ if $x_k(t) \,\in \,\partial (Q_1\,+\,y_k(t))$. Let us take the limit in (31). As already proved, the sequence $\{\dot{x}_k\}$ converges weakly in $L_1$ to some $\xi$, such that $\dot{x}(t)=\xi (t)$ a.e.. By Mazur’s theorem, there exists a sequence $\{\chi _n\}$, such that $\displaystyle \chi _n= \sum _{k=n}^{\infty } \beta _k\dot{x}_k$, where $\beta _k \,\ge \,0$ and $\displaystyle \sum _{k=n}^{+\infty } \beta _k=1$, and $\chi _n$ converges in the $L_1$-norm to $\xi$. This also implies that, up to a subsequence (we do not relabel), $\{\chi _n\}$ converges to $\xi$ a.e..

Now, let us take a Lebesgue point $t\,\in \,[0,T_k]$, and some $\tau > 0$ such that $t\,+\,\tau \le T _k$. Then, we may write

$$\begin{aligned}&\frac{1}{\tau } \sum _{k=n}^{\infty } \beta _k \int _{t}^{t+\tau } \,\,\,Z_k(s,{\tilde{x}}) \mathrm{d}s \nonumber \\&\ge \frac{1}{\tau } \sum _{k=n}^{\infty }\beta _k \int _{t}^{t+\tau } \,\,\,\langle f(x_k(s),u_k(s))- \dot{x}_k(s), {\tilde{x}}- x(s) \rangle \mathrm{d}s \end{aligned}$$

(32)

$$\begin{aligned}&\qquad + \frac{1}{\tau } \sum _{k=n}^{\infty }\beta _k \int _{t}^{t+\tau } \,\,\,\langle f(x_k(s),u_k(s))- \dot{x}_k(s), x(s)- x_k(s) \rangle \mathrm{d}s \nonumber \\&\quad \ge \frac{1}{\tau } \sum _{k=n}^{\infty }\beta _k \int _{t}^{t+\tau } \,\,\,\langle f(x_k(s),u_k(s))- \dot{x}_k(s), {\tilde{x}}- x(s) \rangle \mathrm{d}s \nonumber \\&\qquad -\, M\frac{1}{\tau }\sum _{k=n}^{\infty }\beta _k \int _{t}^{t+\tau } \,\,\,| x(s)- x_k(s) | \mathrm{d}s, \end{aligned}$$

(33)

being inequality (33) due to the boundedness of $|f(x_k(s),u_k(s))- \dot{x}_k(s)|$. From the uniform convergence of $x_k$ to x, we have that $\displaystyle M\frac{1}{\tau }\int _{t}^{t+\tau } \,\,\,| x(s)- x_k(s) | \mathrm{d}s \rightarrow 0$. Moreover, we can write

$$\begin{aligned}&\frac{1}{\tau } \sum _{k=n}^{\infty }\beta _k \int _{t}^{t+\tau } \langle f(x_k(s),u_k(s))- \dot{x}_k(s), {\tilde{x}}- x(s) \rangle \mathrm{d}s \\&\quad =\frac{1}{\tau } \int _{t}^{t+\tau } \langle f(x(s),u(s)) -\xi (s), {\tilde{x}}- x(s) \rangle \mathrm{d}s \\&\qquad \qquad \qquad +\, \frac{1}{\tau } \int _{t}^{t+\tau } \,\,\left\langle \sum _{k=n}^{\infty }\beta _k [f(x_k(s),u_k(s))- f(x(s),u_k(s))], {\tilde{x}}\,- \,x(s) \,\right\rangle \mathrm{d}s\\&\qquad \qquad \qquad + \frac{1}{\tau } \int _{t}^{t+\tau }\,\,\left\langle \xi (s)- \sum _{k=n}^{\infty }\beta _k \dot{x}_k(s), {\tilde{x}}\,-\,x(s) \,\right\rangle \mathrm{d}s \\&\qquad \qquad \qquad +\, \frac{1}{\tau } \int _{t}^{t+\tau } \,\,\left\langle \sum _{k=n}^{\infty }\beta _k f(x(s),u_k(s))\,-\,f(x(s),u(s)), {\tilde{x}}\,-\,x(s)\,\right\rangle \mathrm{d}s. \end{aligned}$$

It is easy to check that each one of the last three terms converges to zero. The second term is due to the fact that $\displaystyle \sum _{k=n}^{\infty }\beta _k \dot{x}_k(t) \rightarrow \xi (t)$, the third one follows from the Lipschitz continuity of f w.r.t. x, and the uniform convergence of $x_k$ to x, and the last one is a consequence of Mazur’s theorem, by taking into account that $\{u_k\}$ weakly converges in $L_1$ to u, which entails that, on $[t,t+\tau )$, when $k\rightarrow \infty$, $\displaystyle \sum _{k=n}^{\infty }\beta _k f(x(s),u_k(s))$ converges to f(x(s), u(s)). By passing the relation (32) to the limit in n, we obtain $\displaystyle \frac{1}{\tau } \int _{t}^{t+\tau } \langle f(x(s),u(s)) -\xi (s), {\tilde{x}}- x(s) \rangle \mathrm{d}s \le \lim _{n\rightarrow \infty }\frac{1}{\tau } \sum _{k=n}^{\infty } \beta _k \int _{t}^{t+\tau } Z_k(s,{\tilde{x}}) \mathrm{d}s \le 0.$ Now, by taking the limit $\tau \rightarrow 0$, and, since t is a Lebesgue point, then, for all ${\tilde{x}}\,\in \,\text{ int }(Q_1\,+\,y(t))$, we conclude that

$$\begin{aligned} \langle f(x(t),u(t)) -\dot{x}(t), {\tilde{x}}- x(t) \rangle \le 0. \end{aligned}$$

(34)

Let us consider now, any point ${\tilde{x}} \,\in \,\partial (Q_1\,+\,y)$. Take some $\lambda \,\in \,[0,1)$. Clearly, $h_L(\lambda ({\tilde{x}}-y)+y,y)= \frac{1}{2}(\lambda ^2 - 1)R_1^2 <0.$ We are, therefore, reduced to the previous case replacing ${\tilde{x}}$ in (34) by $y+ \lambda ({\tilde{x}} -y)$. By taking the limit $\lambda \rightarrow 1$, and by using the continuity of $h_L(\cdot ,y)$, we have that, for all t, $h_L(x(t),y(t)) \le 0$. Thus, we conclude that $x(\cdot )$ is a solution to (D).

It remains to show the uniqueness of the solution to (D). This is a consequence of the hypomonotonicity of the truncated normal cone to a convex set (cf. [42]), and of the application of the Gronwall’s inequality. Indeed, let (T, y) be a given parameter, and take a measurable u such that $u(t) \in U$, [0, T]-a.e. By contradiction, let $x_1(\cdot )$, and $x_2(\cdot )$ be distinct solutions to (D) with $x_1(0)=x_2(0)$, and let ${\tilde{v}}(x_1,y)$, and ${\tilde{v}}(x_2,y)$ be the velocity components of $x_1$, and $x_2$, respectively, in $N_{Q_1+y}^M(x_1)$, and $N_{Q_1+y}^M(x_2)$. Thus, we may write $\langle \dot{x}_1- \dot{x}_2, x_1-x_2 \rangle \,= \,\langle f(x_1,u) - {\tilde{v}}(x_1,y)- f( x_2,u) + {\tilde{v}}(x_2,y), x_1-x_2 \rangle$. The Lipschitz continuity of $f(\cdot ,u)$ (assumption H1) implies that

$$\begin{aligned} \langle \dot{x}_1- \dot{x}_2, x_1-x_2 \rangle \le K_f |x_1-x_2|^2 - \langle {\tilde{v}}(x_1,y) - {\tilde{v}}(x_2,y), x_1-x_2 \rangle . \end{aligned}$$

(35)

From the hypomonotonicity of the truncated normal cone, we may write the inequality

$$\begin{aligned} \langle {\tilde{v}}(x_1,y) - {\tilde{v}}(x_2,y), x_1-x_2 \rangle \ge - |x_1-x_2|^2. \end{aligned}$$

(36)

By replacing (36) in (35), we obtain $\displaystyle \frac{1}{2} \frac{\mathrm{d}}{\mathrm{d}t} (|x_1-x_2|^2) \le (K_f+1) |x_1-x_2|^2$. By applying Gronwall’s lemma, and using the fact that $x_1(0)=x_2(0)$, we obtain $x_1(\cdot )=x_2(\cdot )$, thus confirming the uniqueness of the solution to (D). $\square$

Proof of Proposition 4

Here, the index k in the specification of the multipliers, the state, and control variables is omitted for the sake of simplicity. Let (T, y) be a parameter inherited from $(P_H)$ such that ${\mathcal {F}}^k_L(T,y) \,\ne \,\emptyset$, and denote by $(x,u_L)$ a feasible control process to $(P_L^k(T,y))$ whose existence is asserted by Proposition 2. Then, $\varphi ^k(\omega ,v)$ is finite where $(\omega , v)$ corresponds to (T, y). Standard compactness arguments (see [47, Lemma 2.4]) allow us to conclude that, for a sequence $\{(T_j,y_j)\}$ converging in $L_2$ to (T, y), and a sequence of $\{(x_j,u_j)\}$ with $(x_j,u_j)$ feasible for $(P^k_L(T_j,y_j))$, there is a subsequence $\{x_j\}$ converging uniformly to x for which $\exists \, u_L$ feasible, and such that $(x,u_L)$ is feasible for $(P^k_L(T,y))$, and $\displaystyle J_L(x(0),u_L; T,y)\,\le \,\lim \inf _j J_L(x_j(0),u_j; T_j,y_j)$. This, with the boundedness of $\varphi ^k$ in its domain, entails its lower semicontinuity.

Moreover, let (T, y) be such that $\partial ^P\varphi ^k (\omega ,v) \ne \emptyset$. Assume that such $(x,u_L)$ is an optimal solution to $(P_L^k)$, i.e. $(x,u_L) \in \varPsi ^k(\omega ,v)$, being the latter the set of optimal solutions to $(P_L^k)$, and that $\partial ^P \varphi ^k(\omega ,v)\,\ne \,\emptyset$. Remark that, by [47, Lemma 2.4] applied to $\varphi ^k$, this property holds a.e.. Let $(\zeta _1,\zeta _2)\,\in \,\partial ^P \varphi ^k(\omega ,v)$. Here, $(\omega ,v)$ are the controls associated with (T, y) as defined in (21). By definition of the proximal subdifferential in the Hilbert space $L_2$, for some ${\bar{c}}>0$, and for all $(\omega ',v')$ sufficiently near $(\omega ,v)$ in the $L_2$-norm, i.e., $\exists \,\varepsilon \,>\,0$ such that $\Vert (\omega ',v')-(\omega ,v)\Vert _2^2 <\varepsilon$, we have

$$\begin{aligned}&\int _0^{T^*}\,\langle \zeta _1, \omega '-\omega \rangle \mathrm{d}t \\&\quad + \int _0^{T^*}\,\langle \zeta _2, v'-v)\rangle \mathrm{d}t \le \varphi ^k(\omega ',v') - \varphi ^k(\omega ,v) +\, {\bar{c}}\Vert \omega -\omega '\Vert _{2}^2 + {\bar{c}} \Vert v-v'\Vert _{2}^2. \end{aligned}$$

Here, $\Vert \cdot \Vert _2$ represents the $L_2$-norm. Equivalently,

$$\begin{aligned} \varphi ^k(\omega ',v')- & {} \int _0^{T^*}\,\langle (\zeta _1(t), \zeta _2(t)),(\omega '(t),v'(t))\rangle \mathrm{d}t + {\bar{c}}\Vert \omega -\omega '\Vert _{2}^2 + {\bar{c}} \Vert v-v'\Vert _{2}^2 \\&\qquad \qquad \ge \int _{0}^{T^*}\,|u_L(t)|^2 w(t)\mathrm{d}t - \int _0^{T^*}\,\langle (\zeta _1(t),\zeta _2(t)),(\omega (t), v(t))\rangle \mathrm{d}t. \end{aligned}$$

Denote by $(x',u_L')$ an admissible solution to the approximate lower level problem corresponding to $(\omega ',v')$ (or equivalently $(T',y')$). Then,

$$\begin{aligned}&\int _{0}^{T^*} \,|u_L'(t)|^2\omega '(t) \mathrm{d}t - \int _0^{T^*}\,\langle (\zeta _1(t), \zeta _2(t)),(\omega '(t),v'(t)) \rangle \mathrm{d}t \\&\qquad \qquad \qquad +\, {\bar{c}}\Vert \omega -\omega '\Vert _{2}^2 + {\bar{c}} \Vert v-v'\Vert _{2}^2 \\&\qquad \qquad \quad \ge \int _{0}^{T^*} \,|u_L(t)|^2 \omega (t)\mathrm{d}t- \int _0^{T^*}\,\langle (\zeta _1(t),\zeta _2(t)),(\omega (t),v(t)) \rangle \mathrm{d}t. \end{aligned}$$

This means that $(x,u,u_0;\omega ,v)$ is an optimal solution to

$$\begin{aligned} (P_{aux}) \text{ Minimize }&\int _{0}^{T^*}\,\,\left[ |u_L'(t)|^2\omega '(t) - \langle (\zeta _1(t),\zeta _2(t)),(\omega '(t),v'(t))\rangle \right] \mathrm{d}t+ {\bar{c}}\Vert \omega -\omega '\Vert _2^2 + {\bar{c}} \Vert v-v'\Vert _{2}^2, \\ \qquad \qquad \text{ subject } \text{ to }&\dot{x}'= \left( \,f(x',u') -u_0' \min \left\{ \frac{M}{r_1}, \gamma _k e^{\gamma _k h_L(x',y')}\right\} (x'-y')\,\right) \omega ',\;\; \\&\dot{y}'=v'\omega ', \; \;\dot{t} = \omega '\quad [0,T^*]-\text{ a.e. }, \\&x'(0) \in Q_1 + y_0', \;\; y'(0)=y_0', \;\; y'(T^*) \in {\bar{E}}, \;\; t(0)=0, \\&v' \in {\mathcal {V}}, \;\; u'\in {\mathcal {U}}, \;\, u_0'\in {\mathcal {U}}_0, \;\; \omega ' \in L_2([0,T^*],{\mathbb {R}}^+), \text{ and } h_L(x',y') \le 0, \; \\&h_H(y') \le 0\;\; \text{ on } [0,T^*]. \end{aligned}$$

By considering ${\overline{H}}_L^k$ is as defined in (23), the Hamilton-Pontryagin function for $(P_{aux})$ in the Gamkrelidze’s form is

$$\begin{aligned} H_L^k(y', x', v',u',{\bar{p}},{\bar{\mu }}, {\bar{\lambda }},\omega '; v,u,w):= & {} \left( {\overline{H}}_L^k+{\bar{\lambda }} \zeta _1\right) \omega '\,+ \,{\bar{\lambda }}\langle \zeta _2,v\rangle \,\\&-\,\,{\bar{\lambda }} c\left( |\omega '\,-\,\omega |^2\,+\,|u'\,-\,u|^2\,+\,|v'\,-\,v|^2\right) \,. \end{aligned}$$

The application of the necessary conditions of optimality to the reference control process to $(P_{aux})$ in the Gamkrelidze’s form of [3] with nonsmooth data [35], asserts the existence of a multiplier comprising an adjoint arc ${\bar{p}}\,=\,(p_H,p_L)\,\in \,AC([0,T^*];{\mathbb {R}}^{2n})$, nonincreasing scalar function ${\bar{\mu }}\,=\,(\mu _H,\mu _L) \,\in \,NBV([0,T^*];{\mathbb {R}}^2)$, with $\mu _H$ constant on $\{t\,\in \,[0,T^*]\,:\,|y-z|\,>\,R_1\; \forall \, z\,\in \,\partial Q\}$, and $\mu _L$ constant on $\{t\,\in \,[0,T^*]\,:\,|y-x|\,<\,R_1\}$, and a scalar ${\bar{\lambda }}\,\ge \,0$, not all zero, satisfying:

• i) Adjoint and primal system: $\displaystyle (- \dot{{\bar{p}}}, \dot{y},\dot{x}) \in \partial ^C_{(y',x',{\bar{p}})} H_L^k(y, x, v,u,{\bar{p}},{\bar{\mu }}, {\bar{\lambda }},\omega ) \quad [0, T^*]-\text{ a.e. }$,

• ii) Transversality conditions: $p_H(0)\,\in \,{\mathbb {R}}^n$, $p_H(T)\,\in \,- N_{{\bar{E}}}(y(T))\,+\,\mu _H(T) (y(T)\,-\,q_0)\,- \,\mu _L (T) (x(T)\,-\,y(T))$, $p_L(0)\,\in \,N_{Q_1}(x(0)\,-\,y_0)$, and $p_L(T))\,=\,(\mu _L(0) (x(0)\,- \,y_0),\mu _L(T) (x(T)\,- \,y(T)))$, and

• iii) Maximum condition on $(\omega ',v')$: $\displaystyle \max _{\omega ',v'} \Big \{{\overline{H}}_L^k(v')\omega '\,+\,{\bar{\lambda }} \zeta _1\omega '\,+\,{\bar{\lambda }} \langle \zeta _2,v' \rangle \,- \,{\bar{\lambda }} {\bar{c}} |\omega \,-\,\omega '|^2\,-\,{\bar{\lambda }} {\bar{c}} |v\,-\,v'|^2\Big \}\,=\,({\overline{H}}_L^k (v)\,+ \,{\bar{\lambda }} \zeta _1)\omega \,+\,{\bar{\lambda }} \langle \zeta _2,v\rangle .$

By applying the Lagrange multiplier rule, the maximum condition with respect to $\omega '$, and $v'$, yields

$$\begin{aligned} -{\bar{\lambda }} (\zeta _1,\zeta _2) \in \{{\overline{H}}_L^k(v)\} \times \nabla _{v} {\overline{H}}_L^k(v)\omega + \{0\} \times N_{{\mathcal {V}}}(v). \end{aligned}$$

(37)

Clearly, this inclusion corresponds to the last inclusion specifying the generalized gradient of $\varphi ^k$ of Proposition 4.

These conclusions are extended to pairs $(\omega ,v)$ for which $\partial ^P \,\varphi ^k(\omega ,v)$ might be an empty set. Take any $(\zeta _1,\zeta _2)\,\in \,\partial \varphi ^k(\omega ,v)$, where $\dot{t}\,=\,\omega$ and $\dot{y}\,=\,v$ a.e., such that $v\,\in \,{\mathcal {V}}$, and $\omega \,\in \,L_2([0,T^*],{\mathbb {R}}^+)$, and satisfying the corresponding endpoint and state constraints (i.e., $y(0)\,=\,y_0$, $y(T^*)\,\in \,{\bar{E}}$, $t(0)\,=\,0$, and $h_H(y)\,\le \,0$). From the definition of limiting subdifferential, there exists a sequence $\{(\omega _i,v_i)\}$ converging in $L_2$ to $(\omega ,v)$ with $\varphi ^k(\omega _i,v_i)$ converging to $\varphi ^k(\omega ,v)$ by continuity of $\varphi ^k$, and $\{(\zeta _1^{i},\zeta _2^{i})\}$ satisfying $(\zeta _1^{i},\zeta _2^{i})\,\in \,\partial ^P\,\varphi ^k(\omega _i,v_i)$, such that, for a fixed k, and, for all i, $(\zeta _1^{i},\zeta _2^{i})$ converges weakly in $L_2$ to $(\zeta _1,\zeta _2)$.

By compactness arguments, the sequence $\{(\omega _i,v_i)\}$ with $(\omega _i,v_i)\,{\in }\,L_2([0,T^*],{\mathbb {R}}^+) \,\times \,{\mathcal {V}}$ can be chosen so that $\dot{t}_i\,=\,\omega _i$, and $\dot{y}_i\,=\,v_i$ a.e., and the corresponding endpoint and state constraints are satisfied. The analysis above implies that, for each one of such $(\omega _i,v_i)$’s, there exists $(p_H^{i},p_L^{i},\mu _H^{i},\mu _L^{i},{\bar{\lambda }}^{i})$ associated with the pair $(x_i,u_i)$ such that conditions (i)-(iii) above hold, and that $(x_i,u_i)$ is an optimal solution to $(P_L^k)$, i.e., $(x_i,u_i) \,\in \,\varPsi ^k(\omega _i,v_i)$.

The sequences $\{p_H^{i}\}, \{p_L^i \}$ are bounded, hence $\{\dot{p}_H^{i}\}, \{\dot{p}_L^i\}$ are uniformly integrably bounded (as follows from the adjoint system). Then, there exist subsequences (we do not relabel) $\{p_H^{i}\}, \{p_L^i\}$, uniformly converging to some absolutely continuous arcs respectively, $p_H$, and $p_L$. Moreover, we can extract (without relabeling) subsequences $\{{\bar{\lambda }}^{i}\}$, $\{\mu _H^{i}\}$, and $\{\mu _L^{i}\}$ converging, respectively, to some ${\bar{\lambda }} \,\ge \,0$, and to some nonincreasing functions of bounded variation $\mu _H$, and $\mu _L$ such that $\mu _H$ is constant on $\{t\,\in \,[0,T^*]\,:\,|y- z|\,>\,R_1\, \forall \, z\,\in \,\partial Q\}$, and $\mu _L$ is constant on $\{t\,\in \,[0,T^*]\,: |y- x|\,<\,R_1\}$.

Therefore, from the compactness of the feasible set of $(P_L^k)$, we have that $x_i\rightarrow x$ uniformly, and $u_i\rightarrow u$ a.e., where $(x,u_L)$ is a feasible process to $(P_L^k)$ associated with (T, y) (or equivalently $(\omega ,v)$). Moreover, $(x,u_L)\,\in \,\varPsi ^k(\omega ,v)$, since the graph of $\varPsi ^k(\cdot ,\cdot )$ is closed (as a result of the closure of the graph of ${\mathcal {F}}^k(T,y)$, and $(x_i,u_i) \,\in \,\varPsi ^k(\omega _i,v_i)$). Then, by passing to the limit $i\,\rightarrow \,\infty$ in the necessary conditions of optimality and by compactness arguments, the statement of Proposition 4 regarding the expression of $\partial \varphi ^k(\omega ,v)$ holds.

Now, it remains to show that $\varphi ^k$ is Lipschitz continuous in its domain. By [19, Theorem 3.6], $\varphi ^k$ is Lipschitz continuous near (w, v) with rank $\mathbf{L}$ if all the proximal subgradients in $\partial ^P\varphi ^k(w',v')$ for all $(w',v')$ of the domain of $\varphi ^k$ near (w, v) are bounded in $L_2$ by the constant $\mathbf{L}$. This can be easily concluded from the necessary conditions obtained above, and the boundedness of the adjoint variable with simple arguments. $\square$

Proof of Proposition 5

Let ${\bar{\theta }}= (v,u,u_0,\omega )\in \varTheta := {{{\mathcal {V}}}}\times {{{\mathcal {U}}}}\times {{{\mathcal {U}}}}_0\times L_2([0,T^*];{\mathbb {R}}^+)$. Under assumptions H1-H6, Proposition 1 entails the existence of a solution to $(P_F)$ denoted by $( y^*, x^*, z^*, t^*, v^*, u^*, u_0^*, \omega ^*)$.

Proposition 3 applied to $(D^k)$ cast in the new time parametrization (i.e., on the fixed time interval $[0,T^*]$), yields the existence of a sequence $\{(x_k(0),{\bar{\theta }}_k)\}$, with $(x_k,u_k,u_{0,k},\omega _k)\,\in \,{{{\mathcal {F}}}}_L^k(t_k(T^*),y_k)$, such that ${\bar{\theta }}_k \rightarrow {\bar{\theta }}^*$, a.e. on $[0, T^*]$, and $\{x_k\}$ is such that $x_k \rightarrow x^*$ uniformly, being $x^*$ the unique solution to $(P_L)$ with data $(x^*(0),y^*,t^*,z^*,{\bar{\theta }}^*)$.

By observing that the construction of the approximating problem $({\widetilde{P}}_F^k)$ involves only the approximation to $(P_L)$ (i.e., $(P_H)$ remains intact), it is clear that Proposition 3 can be readily applied to $(P_F)$ in the reparameterized time by choosing a control process so that $z^k(T^*)\,- \,\varphi ^{k}(\omega _k,v_k)\le 0$. This is always possible due to the fact that ${{\mathcal {F}}}_L^k(t_k(T^*),y_k)$, with, $\displaystyle t_k(T^*)= \int _0^{T^*}\,\,\omega _k(s)\mathrm{d}s$, is compact and, thus, the existence of solution to $(P_L^k)$ is guaranteed.

Since, it is straightforward to conclude that the cost functional of $(P_F^k)$ is lower semi-continuous, and that ${\mathbb {R}}^n\,\times \,\varTheta$ with $|\cdot |\,+\,\varDelta (\cdot ,\cdot )$ is a complete metric space, we just need to show that there is a sequence of positive numbers $\{\varepsilon _k\}$ satisfying $\varepsilon _k \rightarrow 0$ as $k \rightarrow \infty$ so that $(y^*,x^*,z^*,t^*,{\bar{\theta }}^*)$ is a $\varepsilon _k^2$-minimizer to $(P_F^k)$. Let $\varepsilon _k^2 := |x^*(0)\,-\,x_k(0)| \,+\,\varDelta ({\bar{\theta }}^*,{\bar{\theta }}_k)$. From Proposition 3, it is clear that the sequence $\{\varepsilon _k\}$ satisfies the needed requirements. Thus, Proposition 5 follows immediately (for k sufficiently large, $\varepsilon _k < 1$), from Ekeland’s Variational Principle. $\square$

Proof of Proposition 6

Let $(T^*,y^*,x^*,u^*,z^*,u^*_0)$ be a minimizer to $(P_F)$. Then, the partial calmness, cf. Definition 1, implies that $\exists \rho \,\ge \,0$, such that, for any feasible control process $(y,x,z,t,{\bar{\theta }})$ of $(P_F)$, we have

$$\begin{aligned} J_H(t^*(T^*),y^*;x^*(0),u_L^*)\le & {} J_H(t(T^*),y;x(0),u_L)\,\nonumber \\&\quad +\,\rho (z(t(T^*))\,-\,\varphi (\omega ,v)). \end{aligned}$$

(38)

Consider now $({\widetilde{P}}_F^k)$, with $\chi _k=(y_k,x_k,z_k,t_k)$, and ${\bar{\theta }}_k = (v_k,u_k,u_{0,k},\omega _k)$ as its state and control variables, respectively. We preserve the notation in the limit by dropping the subscript k and refer to the solution by adding superscript “*” to either these variables or their components. Let us assume that $({\widetilde{P}}_F^k)$ is not partially calm. Then, for its minimizer (see Proposition 5), $(\chi ^*_k,{\bar{\theta }}^*_k)$, there exists a nonnegative sequence $\{\beta _k\}$ converging to some $\beta \,\ge \,0$, and a control process $(\chi _k,{\bar{\theta }}_k)$ feasible to $({\widetilde{P}}_F^{k})$, such that $|x^*(0)\,- \,x_k(0)|\,+\,\varDelta ({\bar{\theta }}^*,{\bar{\theta }}_k) < \varepsilon _k$, and satisfying

$$\begin{aligned}&t^*_k(T^*) > t_k(T^*) + \varepsilon _k\left( |x_k^*(0) \,-\,x_k(0)| \,+ \,\varDelta ({\bar{\theta }}_k^*,{\bar{\theta }}_k)\right) \,\nonumber \\&+ \beta _k\left( z_k(t_k(T^*))\,-\,\varphi (\omega _k,v_k)\right) . \end{aligned}$$

(39)

Moreover, from Proposition 5, we have that the sequence of approximating control processes $\{(\chi ^*_k,{\bar{\theta }}_k^*)\}$ satisfies

$$\begin{aligned}&\chi _k^* \rightarrow \chi ^* \text { uniformly, and } \end{aligned}$$

(40)

$$\begin{aligned}&{\bar{\theta }}_k^* \rightarrow {\bar{\theta }}^* \text { a.e.} \end{aligned}$$

(41)

Since, Proposition 5 holds for all $(\chi _k,{\bar{\theta }}_k)$ feasible for $({\widetilde{P}}_F^k)$ such that $|x^*(0)\,- \,x_k(0)| + \varDelta ({\bar{\theta }}^*,{\bar{\theta }}_k) < \varepsilon _k,$ we have that (41) asserted for $\{{\bar{\theta }}_k^*\}$ also holds for $\{{\bar{\theta }}_k\}$. Furthermore, since $x_k(0)\rightarrow x^*(0)$, and, in the light of the convergence $[0,T^*]$-a.e. of $\{{\bar{\theta }}_k\}$ to ${\bar{\theta }}^*$, straightforward arguments lead us to conclude that (40) also holds for $\{\chi _k\}$.

Now, since $|x_k^*(0)- x_k(0)|\,\le \,|x_k^*(0)- x^*(0)|+|x^*(0)-x_k(0)|$, $\varDelta ({\bar{\theta }}^*_k,{\bar{\theta }}_k)< \varDelta ({\bar{\theta }}_k^*,{\bar{\theta }}^*) \,+ \,\varDelta ({\bar{\theta }}^*,{\bar{\theta }}_k)$, and that, by definition $z_k(t_k(T^*))\,\ge \,\varphi (\omega _k,v_k)$, we conclude, by passing to the limit, while choosing $\beta \,\ge \,\rho$ without any loss of generality, that $\displaystyle \lim _{k\rightarrow \infty } t_k^*(T^*)\,> \,\lim _{k\rightarrow \infty } t_k(T^*)$. This is a contradiction since $t_k^*(T^*)\,\le \,t_k(T^*)$ for all k. Thus, $({\widetilde{P}}_F^k)$ has to be partially calm. $\square$