2.1. Assumptions and main results

First, let us give the necessary conditions to guarantee the existence and uniqueness of the solution u to the HJB equation (2.9) on the space $\textrm{C}^{0,1}\big(\overline{\mathcal{O}}\big)\cap \textrm{W}^{2,\infty}_{\textrm{loc}}(\mathcal{O})$ :

Let $\ell$ be in $\mathbb{I}$ . Then the following hold:

(H4) Let $\mathcal{S}(d)$ be the set of $d\times d$ symmetric matrices. The coefficients of the differential part of $ \mathcal{L}_{\ell}$ , $a_{\ell}=(a_{\ell\,ij})_{d\times d}\,:\,\overline{\mathcal{O}}\longrightarrow\mathcal{S}(d)$ , $b_{\ell}=(b_{\ell\,1},\dots,b_{\ell\,d})\,:\,\overline{\mathcal{O}}\longrightarrow\mathbb{R}^{d}$ , and $c_{\ell}\,:\,\overline{\mathcal{O}}\longrightarrow\mathbb{R}$ , are such that $a_{\ell\,ij},b_{\ell\,i},c_{\ell}\in \textrm{C}^{2,\alpha^{\prime}}\big(\overline{\mathcal{O}}\big)$ , $c_{\ell}>0$ on $\mathcal{O}$ , and $||a_{\ell\,ij}||_{\textrm{C}^{2,\alpha^{\prime}}\big(\overline{\mathcal{O}}\big)},$ $||b_{\ell\,i}||_{\textrm{C}^{2,\alpha^{\prime}}\big(\overline{\mathcal{O}}\big)},||c_{\ell}||_{\textrm{C}^{2,\alpha^{\prime}}\big(\overline{\mathcal{O}}\big)}$ are bounded by some finite positive constant $\Lambda$ . We assume that there exists a real number $\theta>0$ such that

(2.11) \begin{equation} \langle a_{\ell}(x) \zeta,\zeta\rangle\geq \theta|\zeta|^{2},\ \text{for all}\ x\in\overline{\mathcal{O}},\ \zeta\in \mathbb{R}^{d}. \end{equation}

Under Assumptions (H1)–(H4), the first main goal obtained in this document is as follows.

Remark 2.4. Previous to this work, the HJB equation (2.9) was studied in 1983 by Lenhart and Belbas [Reference Lenhart and Belbas13], in the absence of $\big|\textrm{D}^{1}u_{\ell}\big|-g_{\ell}$ . They proved that the unique solution to their HJB equation belongs to $\textrm{W}^{2,\infty}\big(\overline{\mathcal{O}}\big)$ . This HJB equation is related to optimal switching stochastic control problems. Afterwards, Yamada [Reference Yamada18] analyzed the HJB equation for (2.8) when $\vartheta_{\ell,\kappa}=0$ and $g_{\ell}=g$ for $\ell,\kappa\in\mathbb{I}$ . This case is an example of a system with loops of zero cost whose HJB equation has the form

(2.12) \begin{equation} \max\Big\{\max_{\ell\in\mathbb{I}}\{[c_{\ell}-\mathcal{L}_{\ell}]\tilde{u}-h_{\ell}\},\big|\textrm{D}^{1}\tilde{u}\big|-g\Big\}=0 \ \text{in}\ \mathcal{O},\quad\text{s.t.}\ \tilde{u}=0\ \text{on}\ \partial\mathcal{O}. \end{equation}

Yamada showed that there exists a solution $\tilde{u}$ to (2.12) in $\textrm{C}^{0,1}\big(\overline{\mathcal{O}}\big)\cap \textrm{W}^{2,\infty}_{\textrm{loc}}(\mathcal{O})$ that does not depend on the elements of $\mathbb{I}$ , and assuming $g>0$ , he guaranteed that $\tilde{u}$ belongs to $\text{C}^{1}(\mathcal{O})\cap \textrm{C}\big(\overline{\mathcal{O}}\big)$ and is the unique viscosity solution to (2.12). Comparing those papers with ours, we see that the results presented here are more general than those of [Reference Lenhart and Belbas13], and under the assumption of the absence of loops of zero cost in the system, we guarantee the unique solution u to (2.9) in the a.e. sense.

In addition to the statement in (H2), we need to assume that the domain set is convex, which will permit us to verify that the value function V and the solution u to (2.9) agree on $\mathcal{O}$ .

Under Assumptions (H1) and (H3)–(H5), the second main goal obtained in this document is as follows.

To finalize, let us comment on the assumptions mentioned above. Under (H1)–(H4) and using the Schaefer fixed point theorem (see, e.g., [Reference Evans8, Theorem 4, p. 539]), for each $\varepsilon,\delta\in(0,1)$ fixed, we guarantee the existence and uniqueness of the classical nonnegative solution $u^{\varepsilon,\delta}$ to the NPDS (3.1); see Proposition 3.1 and the appendix for the proof. Also, those assumptions are required to show some a priori estimates of $u^{\varepsilon,\delta}$ , which are independent of $\varepsilon,\delta$ ; see Lemma 3.1. Since (H1) holds, $c_{\ell}>0$ on $\overline{\mathcal{O}}$ , and using Lemma 3.1, we verify that $u^{\varepsilon}$ is the unique nonnegative solution to the HJB equation (2.20); see Subsection 3.1. Once again making use of (H1) and $c_{\ell}>0$ on $\overline{\mathcal{O}}$ , we prove that the solution u to the HJB equation (2.9) is unique; see Subsection 3.2. Finally, Assumption (H5) helps to check that the value function V, given in (2.18), and the solution u to the HJB equation (2.9) agree on $\overline{\mathcal{O}}$ .

2.2. $\varepsilon$ -PACS control problem

To prove the theorems above (Theorems 2.1 and 2.2), we first study an $\varepsilon$ -PACS control problem that is closely related to the value function problem seen previously. At the same time, the solution to the HJB equation related to this stochastic control problem (see Proposition 2.1) helps to guarantee the existence and regularity of the solution to the system of variational inequalities (2.9). The verification lemma for this part, which is divided into two lemmas (Lemmas 4.1 and 4.2), will be presented below to provide the proofs of Theorem 2.1 and Proposition 2.1.

Define the penalized controls set $\mathcal{U}^{\varepsilon}$ in the following way:

(2.13)

with $\varepsilon\in(0,1)$ fixed, where C is some fixed positive constant independent of $\varepsilon$ . For each $\big(\tilde{x},\tilde{\ell}\big)\in\overline{\mathcal{O}}\times\mathbb{I}$ and $(\xi,\varsigma)\in\mathcal{U}^{\varepsilon}\times\mathcal{S}$ , the process evolves as

(2.14)

(2.15)

where $\tilde{\tau}_{i}=\tau_{i}\wedge\tau$ and . Before introducing the corresponding functional cost $\mathcal{V}_{\xi,\varsigma}$ of $(\xi,\varsigma)\in\mathcal{U}^{\varepsilon}\times\mathcal{S}$ , let us define the penalization function $\psi_{\varepsilon}$ . Consider $\varphi$ as a function from $\mathbb{R}$ to itself that is in $\textrm{C}^{\infty}(\mathbb{R})$ and satisfies

(2.16) \begin{align}&\varphi(t)=0,\ t\leq0,\quad\varphi(t)>0,\ t>0,\\&\varphi(t)=t-1,\ t\geq2,\quad\varphi^{\prime}(t)\geq0,\ \varphi^{\prime\prime}(t)\geq0.\end{align}

Then, $\psi_{\varepsilon}$ is taken as $\psi_{\varepsilon}(t)=\varphi(t/\varepsilon)$ , for each $\varepsilon\in(0,1)$ . Also, for each $(x,\ell)\in\overline{\mathcal{O}}\times\mathbb{I}$ fixed, we define the Legendre transform of $H_{\ell}^{\varepsilon}(\gamma,x)\,:\!=\, H^{\varepsilon}(\gamma,x,\ell)\,:\!=\,\psi_{\varepsilon}\big(|\gamma|^{2}-g_{\ell}(x)^{2}\big)$ by

\begin{equation*}l_{\ell}^{\varepsilon}(y,x)\,:\!=\, l^{\varepsilon}(y,x,\ell)\,:\!=\, \sup_{\gamma\in\mathbb{R}^{d}}\left\{\langle \gamma,y \rangle-H_{\ell}^{\varepsilon}(\gamma,x)\right\},\quad \text{for}\ y\in\mathbb{R}^{d}.\end{equation*}

Notice that, for each $(x,\ell)\in\overline{\mathcal{O}}\times\mathbb{I}$ fixed, $H_{\ell}^{\varepsilon}(\gamma,x)$ is a $\textrm{C}^{2}$ and convex function with respect to the variable $\gamma\in\mathbb{R}^{d}$ , since $\psi_{\varepsilon}\in \textrm{C}^{\infty}(\mathbb{R})$ is convex. The penalized cost of $(\xi,\varsigma)\in\mathcal{U}^{\varepsilon}\times\mathcal{S}$ is defined by

(2.17)

The value function for this problem is given by

(2.18) \begin{equation}V_{\tilde{\ell}}^{\varepsilon}(\tilde{x})\,:\!=\,\inf_{(\xi,\varsigma)\in\mathcal{U}^{\varepsilon}}\mathcal{V}_{\xi,\varsigma}\big(\tilde{x},\tilde{\ell}\big),\end{equation}

whose corresponding HJB equation is

(2.19) \begin{align}\max\bigg\{[c_{\ell}-\mathcal{L}_{\ell}] u_{\ell}^{\varepsilon}+ \sup_{y\in\mathbb{R}^{d}}\Big\{\Big\langle \textrm{D}^{1}u_{\ell}^{\varepsilon},y\Big\rangle-l_{\ell}^{\varepsilon}(y,\cdot)\Big\}-h_{\ell},u_{\ell}^{\varepsilon}-\mathcal{M}_{\ell}u^{\varepsilon}\bigg\}= 0 ,\ \text{in}\ \mathcal{O},&\\\text{s.t.}\ \ u_{\ell}^{\varepsilon}=0,\ \text{on}\ \partial\mathcal{O}, &\end{align}

where $\mathcal{L}_{\ell}, \mathcal{M}_{\ell}$ are as in (2.3), (2.10), respectively. Observe that (2.19) can be rewritten as

(2.20) \begin{align}\max\left\{[c_{\ell}-\mathcal{L}_{\ell}] u_{\ell}^{\varepsilon}+\psi_{\varepsilon}\Big(\Big|\textrm{D}^{1}u_{\ell}^{\varepsilon}\Big|^{2}- g_{\ell}^{2}\Big)-h_{\ell},u_{\ell}^{\varepsilon}-\mathcal{M}_{\ell}u^{\varepsilon}\right\}= 0 ,\ \text{in}\ \mathcal{O},&\\\text{s.t.}\ \ u_{\ell}^{\varepsilon}=0,\ \text{on}\ \partial\mathcal{O},&\end{align}

because of $H^{\varepsilon}_{\ell}(\gamma,x)=\sup_{y\in\mathbb{R}^{d}}\big\{\langle \gamma,y\rangle-l^{\varepsilon}_{\ell}(y,x)\big\}$ . Under Assumptions (H1)–(H4), the following result is obtained, whose proof is given in the next section; see Subsection 3.1.

3.1. Proof of Proposition 2.1

Let $\varepsilon\in(0,1)$ and $\ell\in\mathbb{I}$ be fixed. Since $u^{\varepsilon,\delta}_{\ell}, \ \partial_{ij}u^{\varepsilon,\delta}_{\ell}$ are bounded, uniformly in $\delta$ , on the spaces $\Big(\text{C}^{1}\big(\overline{\mathcal{O}}\big),||{\cdot}||_{\text{C}^{1}(\mathcal{O})}\Big)$ and $\Big(\text{C}\big(\overline{B}_{ r}\big),||{\cdot}||_{\text{C}(B_{ r})}\Big)$ , respectively, where $B_{r}\subset\mathcal{O}$ (see Lemma 3.1), and using the Arzelà–Ascoli compactness criterion (see [Reference Evans8, p. 718]) and the fact that for each $p\in(1,\infty)$ , $\Big(\textrm{L}^{p}\big(B_{\beta r}\big),||{\cdot}||_{\textrm{L}^{p}(B_{r})}\Big)$ , with $B_{r}\subset\mathcal{O}$ , is a reflexive space (see [Reference Adams and Fournier1, Theorem 2.46, p. 49]), it can be proven that there exist a subsequence $\Big\{u^{\varepsilon,\delta_{\hat{n}}}_{\ell}\Big\}_{\hat{n}\geq1}$ of $\big\{u^{\varepsilon,\delta}_{\ell}\big\}_{\delta\in(0,1)}$ and a function $u^{\varepsilon}_{\ell}$ in $\textrm{C}^{0,1}\big(\overline{\mathcal{O}}\big)\cap \textrm{W}^{2,\infty}_{\textrm{loc}}(\mathcal{O})$ such that

(3.5) \begin{align}&\text{$u^{\varepsilon,\delta_{\hat{n}}}_{\ell}\underset{\delta_{\hat{n}}\rightarrow0}{\longrightarrow}u^{\varepsilon}_{\ell}$ in $\textrm{C}\big(\overline{\mathcal{O}}\big)$, $\partial_{i}u^{\varepsilon,\delta_{\hat{n}}}_{\ell}\underset{\delta_{\hat{n}}\rightarrow0}{\longrightarrow}\partial_{i} u^{\varepsilon}_{\ell}$ $\text{in}\ \text{C}_{\textrm{loc}}(\mathcal{O})$,}\\ &\text{$\partial_{ij}u^{\varepsilon,\delta_{\hat{n}}}_{\ell}\underset{\delta_{\hat{n}}\rightarrow0}{\longrightarrow}\partial_{ij}u^{\varepsilon}_{\ell}$, weakly $ \textrm{L}^{p}_{\textrm{loc}}(\mathcal{O})$, for each $p\in(1,\infty)$.}\end{align}

We proceed to proving Proposition 2.1.

Proof of Proposition 2.1. Existence. Taking $\kappa\in\mathbb{I}\setminus\{\ell\}$ , and using (2.3), (3.1), and Lemma 3.1, we have that $\psi_{\delta}\Big(u^{\varepsilon,\delta}_{\ell}-u^{\varepsilon,\delta}_{\kappa} -\vartheta_{\ell,\kappa}\Big)$ is locally bounded, uniformly in $\delta$ . From here and (3.5), we obtain that $u^{\varepsilon}_{\ell}-u^{\varepsilon}_{\kappa} -\vartheta_{\ell,\kappa}\leq0$ in $\mathcal{O}$ . Then,

(3.6) \begin{equation} u^{\varepsilon}_{\ell}-\mathcal{M}_{\ell}u^{\varepsilon}\leq0,\quad \text{in}\ \mathcal{O}. \end{equation}

Note that the previous inequality is true on the boundary set $\partial\mathcal{O}$ , since $u^{\varepsilon,\delta}_{\ell}=u^{\varepsilon,\delta}_{\kappa}=0$ on $\partial\mathcal{O}$ and $\vartheta_{\ell,\kappa}\geq0$ . Recall that the operator $\mathcal{M}_{\ell}$ is defined in (2.10). On the other hand, since $u^{\varepsilon,\delta_{\hat{n}}}_{\ell}$ is the unique solution to (3.1), when $\delta=\delta_{\hat{n}}$ , it follows that

(3.7) \begin{equation} \int_{B_{r}}\bigg\{[c_{\ell}-\mathcal{L}_{\ell}] u_{\ell}^{\varepsilon,\delta_{\hat{n}}}+ \psi_{\varepsilon}\Big(\Big|\textrm{D}^{1} u_{\ell}^{\varepsilon,\delta_{\hat{n}}}\Big|^{2}- g_{\ell}^{2}\Big)\bigg\}\varpi\textrm{d} x\leq \int_{B_{r}}h_{\ell}\varpi\textrm{d} x ,\quad\text{for}\ \varpi\in\mathcal{B}(B_{r}), \end{equation}

where

(3.8) \begin{equation} \mathcal{B}(A)\,:\!=\,\big\{\varpi\in \text{C}^{\infty}_{\text{c}}(A)\,:\, \varpi\geq0\ \text{and}\ \text{supp}[\varpi]\subset A\subset\mathcal{O} \big\}. \end{equation}

By (3.5) and letting $\delta_{\hat{n}}\rightarrow0$ in (3.7), we obtain that

(3.9) \begin{equation} [c_{\ell}-\mathcal{L}_{\ell}] u_{\ell}^{\varepsilon}+ \psi_{\varepsilon}\Big(\Big|\textrm{D}^{1} u_{\ell}^{\varepsilon}\Big|^{2}- g_{\ell}^{2}\Big)\leq h_{\ell}\quad \text{a.e.}\ \text{in} \, \mathcal{O}. \end{equation}

From (3.6) and (3.9), $\max\Big\{[c_{\ell}-\mathcal{L}_{\ell}] u_{\ell}^{\varepsilon}+\psi_{\varepsilon}\Big(\Big|\textrm{D}^{1} u_{\ell}^{\varepsilon}\Big|^{2}- g_{\ell}^{2}\Big)-h_{\ell},u^{\varepsilon}_{\ell}-\mathcal{M}_{\ell}u^{\varepsilon}\Big\}\leq 0$ a.e. in $\mathcal{O}$ . We shall prove that if

(3.10) \begin{equation} u^{\varepsilon}_{\ell}(x^{*})-\mathcal{M}_{\ell}u^{\varepsilon}(x^{*})<0,\quad \text{for some}\ x^{*}\in\mathcal{O}, \end{equation}

then there exists a neighborhood $\mathcal{N}_{x^{*}}\subset\mathcal{O}$ of $x^{*}$ such that

(3.11) \begin{equation} [c_{\ell}-\mathcal{L}_{\ell}] u_{\ell}^{\varepsilon}+\psi_{\varepsilon}\Big(\Big|\textrm{D}^{1} u_{\ell}^{\varepsilon}\Big|^{2}- g_{\ell}^{2}\Big)= h_{\ell},\quad \text{a.e.}\ \text{in} \, \mathcal{N}_{x^{*}}. \end{equation}

Assume (3.10) holds. Then, taking $\kappa\in\mathbb{I}\setminus\{\ell\}$ , we see that $u^{\varepsilon}_{\ell}-u^{\varepsilon}_{\kappa} -\vartheta_{\ell,\kappa}\leq u^{\varepsilon}_{\ell}-\mathcal{M}_{\ell}u^{\varepsilon}<0$ at $x^{*}$ . Since $u^{\varepsilon}_{\ell}-u^{\varepsilon}_{\kappa} $ is a continuous function, there exists a ball $B_{r_{\kappa}}\subset\mathcal{O}$ such that $x^{*}\in B_{r_{\kappa}}$ and $u^{\varepsilon}_{\ell}-u^{\varepsilon}_{\kappa} -\vartheta_{\ell,\kappa}<0$ in $B_{r_{\kappa}}$ . From here, and defining $\mathcal{N}_{x^{*}}$ as $\bigcap_{\kappa\in\mathbb{I}\setminus\{\ell\}}B_{r_{\kappa}}$ , we have that $\mathcal{N}_{x^{*}}\subset\mathcal{O}$ is a neighborhood of $x^{*}$ and

(3.12) \begin{equation} u^{\varepsilon}_{\ell}-u^{\varepsilon}_{\kappa} -\vartheta_{\ell,\kappa}<0,\quad \text{ in}\ \mathcal{N}_{x^{*}},\ \text{for}\ \kappa\in\mathbb{I}\setminus\{\ell\}. \end{equation}

Meanwhile, observe that

(3.13) \begin{equation} \Big|\Big|u^{\varepsilon,\delta_{\hat{n}}}_{\ell}-u^{\varepsilon,\delta_{\hat{n}}}_{\kappa} -\big(u^{\varepsilon}_{\ell}-u^{\varepsilon}_{\kappa} \big)\Big|\Big|_{\text{C}(\mathcal{O})}\underset{\delta_{\hat{n}}\rightarrow0}{\longrightarrow}0,\quad\text{for}\ \kappa\in\mathbb{I}\setminus\{\ell\}, \end{equation}

since (3.5) holds. Then, by (3.12)–(3.13), we have that for each $\kappa\in\mathbb{I}\setminus\{\ell\}$ , there exists a $\delta^{(\kappa)}\in(0,1)$ such that if $\delta_{\hat{n}}\leq\delta^{(\kappa)}$ , then $u^{\varepsilon,\delta_{\hat{n}}}_{\ell}-u^{\varepsilon,\delta_{\hat{n}}}_{\kappa} -\vartheta_{\ell,\kappa}<0$ in $\mathcal{N}_{x^{*}}$ . Taking $\delta^{\prime}\,:\!=\,\min_{\kappa\in\mathbb{I}\setminus\{\ell\}}\{\delta^{(\kappa)}\}$ , it follows that $u^{\varepsilon,\delta_{\hat{n}}}_{\ell}-u^{\varepsilon,\delta_{\hat{n}}}_{\kappa} -\vartheta_{\ell,\kappa}<0$ in $\mathcal{N}_{x^{*}}$ , for all $\delta_{\hat{n}}\leq\delta^{\prime}$ and $\kappa\in\mathbb{I}\setminus\{\ell\}$ . From here and since for each $\delta_{\hat{n}}\leq\delta^{\prime}$ , $u^{\varepsilon,\delta_{\hat{n}}}_{\ell}$ is the unique solution to (3.1), when $\delta=\delta_{\hat{n}}$ , this implies that

\begin{equation*} \int_{\mathcal{N}_{x^{*}}}\Big\{[c_{\ell}-\mathcal{L}_{\ell}] u_{\ell}^{\varepsilon,\delta_{\hat{n}}}+ \psi_{\varepsilon}\Big(\Big|\textrm{D}^{1} u_{\ell}^{\varepsilon,\delta_{\hat{n}}}\Big|^{2}- g_{\ell}^{2}\Big)\Big\}\varpi\textrm{d} x= \int_{\mathcal{N}_{x^{*}}}h_{\ell}\varpi\textrm{d} x, \quad\text{for}\ \varpi\in\mathcal{B}(\mathcal{N}_{x^{*}}). \end{equation*}

Therefore, (3.11) holds. Hence, we get that for each $\varepsilon\in(0,1)$ , $u^{\varepsilon}=\big(u^{\varepsilon}_{1},\dots,u^{\varepsilon}_{m}\big)$ is a solution to the HJB equation (2.19).

Proof of Proposition 2.1. Uniqueness. Let $\varepsilon\in(0,1)$ be fixed. Suppose that $u^{\varepsilon}=\big(u^{\varepsilon}_{1},\dots,u^{\varepsilon}_{m}\big)$ and $v^{\varepsilon}=\big(v^{\varepsilon}_{1},\dots,v^{\varepsilon}_{m}\big)$ are two solutions to the HJB equation (2.20) whose components belong to $\textrm{C}^{0,1}\big(\overline{\mathcal{O}}\big)\cap \textrm{W}^{2,\infty}_{\textrm{loc}}(\mathcal{O})$ . Take $(x_{0},\ell_{0})\in\overline{\mathcal{O}}\times\mathbb{I}$ such that

(3.14) \begin{equation} u^{\varepsilon}_{\ell_{0}}(x_{0})-v^{\varepsilon}_{\ell_{0}}(x_{0})=\sup_{(x,\ell)\in\overline{\mathcal{O}}\times\mathbb{I}}\big\{u^{\varepsilon}_{\ell}(x)-v_{\ell}^{\varepsilon}(x)\big\}. \end{equation}

Notice that by (3.14), we only need to verify that

(3.15) \begin{equation} u^{\varepsilon}_{\ell_{0}}(x_{0})-v^{\varepsilon}_{\ell_{0}}(x_{0})\leq0, \end{equation}

which is trivially true if $x_{0}\in\partial\mathcal{O}$ , since $u^{\varepsilon}_{\ell_{0}}-v^{\varepsilon}_{\ell_{0}}=0$ on $\partial\mathcal{O}$ . Let us assume $x_{0}\in\mathcal{O}$ . We shall verify (3.15) by contradiction. Suppose that $u^{\varepsilon}_{\ell_{0}}-v^{\varepsilon}_{\ell_{0}}>0$ at $x_{0}$ . Then, by continuity of $u^{\varepsilon}_{\ell_{0}}-v^{\varepsilon}_{\ell_{0}}$ , there exists a ball $B_{r_{1}}(x_{0})\subset\mathcal{O}$ such that

(3.16) \begin{equation} c_{\ell_{0}}\Big[u^{\varepsilon}_{\ell_{0}}-v^{\varepsilon}_{\ell_{0}}\Big]\geq\min_{x\in B_{r_{1}}(x_{0})}\Big\{c_{\ell_{0}}(x)\Big[u^{\varepsilon}_{\ell_{0}}(x)-v^{\varepsilon}_{\ell_{0}}(x)\Big]\Big\}>0,\quad \text{in}\ B_{r_{1}}(x_{0}). \end{equation}

The last inequality is true because of $c_{\ell_{0}}>0$ in $\mathcal{O}$ . Taking $\ell_{1}\in\mathbb{I}$ such that

(3.17) \begin{equation} \mathcal{M}_{\ell_{0}}v^{\varepsilon}(x_{0})=v^{\varepsilon}_{\ell_{1}}(x_{0} )+\vartheta_{\ell_{0},\ell_{1}}, \end{equation}

by (2.20) and (3.14), we get that $v^{\varepsilon}_{\ell_{0}}-\big(v^{\varepsilon}_{\ell_{1}}+\vartheta_{\ell_{0},\ell_{1}}\big) =v^{\varepsilon}_{\ell_{0}}-\mathcal{M}_{\ell_{0}}v^{\varepsilon}\leq u^{\varepsilon}_{\ell_{0}}-\mathcal{M}_{\ell_{0}}u^{\varepsilon}\leq 0$ at $x_{0}$ . If $v^{\varepsilon}_{\ell_{0}}(x_{0})-\mathcal{M}_{\ell_{0}}v^{\varepsilon}(x_{0})<0$ , there exists a ball $B_{r_{2}}(x_{0})\subset\mathcal{O}$ such that $v^{\varepsilon}_{\ell_{0}}-\mathcal{M}_{\ell_{0}}v^{\varepsilon}<0$ in $B_{r_{2}}(x_{0})$ . Moreover, from (2.20),

(3.18) \begin{align} \big[c_{\ell_{0}}-\mathcal{L}_{\ell_{0}}\big] v_{\ell_{0}}^{\varepsilon}+\psi_{\varepsilon}\Big(\big|\textrm{D}^{1} v_{\ell_{0}}^{\varepsilon}\big|^{2}- g_{\ell_{0}}^{2}\Big)-h_{\ell_{0}}&=0,\\ \big[c_{\ell_{0}}-\mathcal{L}_{\ell_{0}}\big] u_{\ell_{0}}^{\varepsilon}+\psi_{\varepsilon}\Big(\big|\textrm{D}^{1} u_{\ell_{0}}^{\varepsilon}\big|^{2}- g_{\ell_{0}}^{2}\Big)-h_{\ell_{0}}&\leq0, \quad\text{in}\ B_{r_{2}}(x_{0}). \end{align}

Notice that $\psi_{\varepsilon}\Big(\Big|\textrm{D}^{1} u_{\ell_{0}}^{\varepsilon}\Big|^{2}- g_{\ell_{0}}^{2}\Big)-\psi_{\varepsilon}\Big(\Big|\textrm{D}^{1} v_{\ell_{0}}^{\varepsilon}\Big|^{2}- g_{\ell_{0}}^{2}\Big)$ is a continuous function in $\mathcal{O}$ , since $\partial_{i}u^{\varepsilon}_{\ell_{0}},\partial_{i}v^{\varepsilon}_{\ell_{0}}\in \text{C}^{0}(\mathcal{O})$ , which satisfies $\psi_{\varepsilon}\Big(\Big|\textrm{D}^{1} u_{\ell_{0}}^{\varepsilon}\Big|^{2}- g_{\ell_{0}}^{2}\Big)-\psi_{\varepsilon}\Big(\Big|\textrm{D}^{1} v_{\ell_{0}}^{\varepsilon}\Big|^{2}- g_{\ell_{0}}^{2}\Big)=0$ at $x_{0}$ , since $x_{0}$ is the point where $u^{\varepsilon}_{\ell_{0}}-v^{\varepsilon}_{\ell_{0}}$ attains its maximum. Meanwhile, by Bony’s maximum principle (see [Reference Lions14]), it is known that for every $r\leq r_{3}$ , with $r_{3}>0$ small enough,

(3.19) \begin{equation} \textrm{tr}\big[a_{\ell_{0}}\textrm{D}^{2}\big[u^{\varepsilon}_{\ell_{0}}-v^{\varepsilon}_{\ell_{0}}\big]\big]\leq0, \quad\text{a.e.}\ \text{in}\ B_{r}(x_{0}). \end{equation}

So, from (3.16), (3.18), and (3.19), we have that for every $r\leq \hat{r}\,:\!=\,\min\{r_{1},r_{2},r_{3}\}$ ,

\begin{align*} 0&\geq \textrm{tr}\big[a_{\ell_{0}}\textrm{D}^{2}\big[u^{\varepsilon}_{\ell_{0}}-v^{\varepsilon}_{\ell_{0}}\big]\big]\notag\\ &\geq c_{\ell_{0}}\big[u^{\varepsilon}_{\ell_{0}}-v^{\varepsilon}_{\ell_{0}}\big]+\big\langle b_{\ell_{0}},\textrm{D}^{1}\big[u^{\varepsilon}_{\ell_{0}}-v^{\varepsilon}_{\ell_{0}}\big]\big\rangle+\psi_{\varepsilon}\Big(\Big|\textrm{D}^{1} u_{\ell_{0}}^{\varepsilon}\Big|^{2}- g_{\ell_{0}}^{2}\Big)-\psi_{\varepsilon}\Big(\Big|\textrm{D}^{1} v_{\ell_{0}}^{\varepsilon}\Big|^{2}- g_{\ell_{0}}^{2}\Big)\notag\\ &\geq \min_{x\in B_{r_{1}}(x_{0})}\Big\{c_{\ell_{0}}(x)\Big[u^{\varepsilon}_{\ell_{0}}(x)-v^{\varepsilon}_{\ell_{0}}(x)\Big]\Big\} +\big\langle b_{\ell_{0}},\textrm{D}^{1}\big[u^{\varepsilon}_{\ell_{0}}-v^{\varepsilon}_{\ell_{0}}\big]\big\rangle\notag\\ &\quad+\psi_{\varepsilon}\Big(\Big|\textrm{D}^{1} u_{\ell_{0}}^{\varepsilon}\Big|^{2}- g_{\ell_{0}}^{2}\Big)-\psi_{\varepsilon}\Big(\Big|\textrm{D}^{1} v_{\ell_{0}}^{\varepsilon}\Big|^{2}- g_{\ell_{0}}^{2}\Big),\quad\text{a.e.}\ \text{in} \, B_{r}(x_{0}). \end{align*}

Then,

(3.20) \begin{align}\lim_{r\rightarrow0}\bigg\{\mathop{\textrm{inf} \,\textrm{ess}}\limits_{B_{r}(x_{0})}\bigg[\psi_{\varepsilon}\Big(\Big|\textrm{D}^{1} u_{\ell_{0}}^{\varepsilon}\Big|^{2}- g_{\ell_{0}}^{2}\Big) & -\psi_{\varepsilon}\Big(\Big|\textrm{D}^{1} v_{\ell_{0}}^{\varepsilon}\Big|^{2}- g_{\ell_{0}}^{2}\Big)\bigg]\bigg\}\nonumber\\& <-\min_{x\in B_{r_{1}}(x_{0})}\Big\{c_{\ell_{0}}(x)\Big[u^{\varepsilon}_{\ell_{0}}(x)-v^{\varepsilon}_{\ell_{0}}(x)\Big]\Big\}<0.\end{align}

This means $\psi_{\varepsilon}\Big(\Big|\textrm{D}^{1} u_{\ell_{0}}^{\varepsilon}\Big|^{2}- g_{\ell_{0}}^{2}\Big)-\psi_{\varepsilon}\Big(\Big|\textrm{D}^{1} v_{\ell_{0}}^{\varepsilon}\Big|^{2}- g_{\ell_{0}}^{2}\Big)$ is not continuous at $x_{0}$ which is a contradiction. Thus,

(3.21) \begin{equation} 0=v^{\varepsilon}_{\ell_{0}}-\big(v^{\varepsilon}_{\ell_{1}}+\vartheta_{\ell_{0},\ell_{1}}\big) =v^{\varepsilon}_{\ell_{0}}-\mathcal{M}_{\ell_{0}}v^{\varepsilon}\leq u^{\varepsilon}_{\ell_{0}}-\mathcal{M}_{\ell_{0}}u^{\varepsilon}\leq 0\quad\text{at}\ x_{0}. \end{equation}

This implies that

(3.22) \begin{align} u^{\varepsilon}_{\ell_{1}}({x_{0} }) & -v^{\varepsilon}_{\ell_{1}}({x_{0} })\geq u^{\varepsilon}_{\ell_{0}}(x_{0})-v^{\varepsilon}_{\ell_{0}}(x_{0})>0,\\v^{\varepsilon}_{\ell_{0}}(x_{0}) & =v^{\varepsilon}_{\ell_{1}}({x_{0} })+\vartheta_{\ell_{0},\ell_{1}}.\nonumber \end{align}

By (3.14) and (3.22), we have that $u^{\varepsilon}_{\ell_{1}}-v^{\varepsilon}_{\ell_{1}}$ attains its maximum at $ x_{0} \in\mathcal{O}$ , where its value agrees with $u^{\varepsilon}_{\ell_{0}}(x_{0})-v^{\varepsilon}_{\ell_{0}}(x_{0})$ . Then, replacing $u^{\varepsilon}_{\ell_{0}}-v^{\varepsilon}_{\ell_{0}}$ by $u^{\varepsilon}_{\ell_{1}}-v^{\varepsilon}_{\ell_{1}}$ above and repeating the same arguments seen in (3.17)–(3.21), we get that there is a regime $\ell_{2}\in\mathbb{I}$ such that

\begin{align*} &u^{\varepsilon}_{\ell_{2}}(x_{0})-v^{\varepsilon}_{\ell_{2}}(x_{0})=u^{\varepsilon}_{\ell_{1}}(x_{0})-v^{\varepsilon}_{\ell_{1}}(x_{0})=u^{\varepsilon}_{\ell_{0}}(x_{0})-v^{\varepsilon}_{\ell_{0}}(x_{0})>0,\notag\\ &v^{\varepsilon}_{\ell_{1}}(x_{0})=v^{\varepsilon}_{\ell_{2}}(x_{0})+\vartheta_{\ell_{1},\ell_{2}}. \end{align*}

Recursively, we obtain a sequence of regimes $\{\ell_{i}\}_{i\geq0}$ such that

(3.23) \begin{align} &u^{\varepsilon}_{\ell_{i}}(x_{0})-v^{\varepsilon}_{\ell_{i}}(x_{0})=u^{\varepsilon}_{\ell_{i-1}}(x_{0})-v^{\varepsilon}_{\ell_{i-1}}(x_{0})=\cdots=u^{\varepsilon}_{\ell_{0}}(x_{0})-v^{\varepsilon}_{\ell_{0}}(x_{0})>0,\notag\\ &v^{\varepsilon}_{\ell_{i}}(x_{0})=v^{\varepsilon}_{\ell_{i+1}}(x_{0})+\vartheta_{\ell_{i},\ell_{i+1}}. \end{align}

Since $\mathbb{I}$ is finite, there is a regime $\ell^{\prime}$ that will appear infinitely often in $\{\ell_{i}\}_{i\geq0}$ . Let $\ell_{n}=\ell^{\prime}$ , for some $n>1$ . After $\hat{n}$ steps, the regime $\ell^{\prime}$ reappears, i.e. $\ell_{n+\hat{n}}=\ell^{\prime}$ . Then, by (3.23), we get

(3.24) \begin{equation} v^{\varepsilon}_{\ell^{\prime}}(x_{0})=v^{\varepsilon}_{\ell^{\prime}}(x_{0})+\vartheta_{\ell^{\prime},\ell_{n+1}}+\vartheta_{\ell_{n+1},\ell_{n+2}}+\cdots+\vartheta_{\ell_{n+\hat{n}-1},\ell^{\prime}}. \end{equation}

Notice that (3.24) contradicts the assumption that there is no loop of zero cost (see Equation (2.7)). From here we conclude that (3.15) must hold. Taking $v^{\varepsilon}-u^{\varepsilon}$ and proceeding in the same way as before, it follows that for each $\ell\in\mathbb{I}$ , $v^{\varepsilon}_{\ell}-u^{\varepsilon}_{\ell}\leq 0$ in $ \mathcal{O} $ , and hence we conclude that the solution $u^{\varepsilon}$ to the HJB equation (2.20) is unique.

3.2. Proof of Theorem 2.1

In view of Lemma 3.1 and by (3.5), the following inequalities hold for each $\ell\in\mathbb{I}$ :

(3.25) \begin{equation}0\leq u^{\varepsilon}_{\ell}\leq C_{1}\quad\text{and}\quad \big|\textrm{D}^{1}u^{\varepsilon}_{\ell}\big|\leq C_{4},\quad\text{in}\ \overline{\mathcal{O}},\end{equation}

for some positive constant $C_{4}=C_{4}(d,\Lambda,\alpha^{\prime})$ . The constant $C_{1}$ is as in (3.2). Moreover, for each $B_{\beta r}\subset\mathcal{O}$ , there exists a positive constant $C_{5}=C_{5}(d,\Lambda,\alpha^{\prime})$ such that

(3.26) \begin{equation}\big|\big|\textrm{D}^{2}u^{\varepsilon}_{\ell}\big|\big|_{L^{p}\big(B_{\beta r}\big)}\leq C_{5},\quad \text{for each}\ p\in(1,\infty).\end{equation}

Then, from (3.25)–(3.26) and using again the Arzelà–Ascoli compactness criterion and the fact that $\Big(\text{L}^{p}\big(B_{\beta r}\big),||{\cdot}||_{L^{p}\big(B_{\beta r}\big)}\Big)$ is a reflexive space, we have that there exist a subsequence $\big\{u^{\varepsilon_{n}}_{\ell}\big\}_{n\geq1}$ of $\big\{u^{\varepsilon}_{\ell}\big\}_{\varepsilon\in(0,1)}$ and a $u_{\ell}$ in $\textrm{C}^{0,1}\big(\overline{\mathcal{O}}\big)\cap \textrm{W}^{2,\infty}_{\textrm{loc}}(\mathcal{O})$ such that

(3.27) \begin{align}&\text{$u^{\varepsilon_{n}}_{\ell}\underset{\varepsilon_{n}\rightarrow0}{\longrightarrow}u_{\ell}$ in $\textrm{C}\big(\overline{\mathcal{O}}\big)$, $\partial_{i}u^{\varepsilon_{n}}_{\ell}\underset{\varepsilon_{n}\rightarrow0}{\longrightarrow}\partial_{i} u_{\ell}$ $\text{in}\ C_{\textrm{loc}}(\mathcal{O})$,}\\ &\text{$\partial_{ij}u^{\varepsilon_{n}}_{\ell}\underset{\varepsilon_{n}\rightarrow0}{\longrightarrow}\partial_{ij}u_{\ell}$, weakly $ \text{L}^{p}_{\textrm{loc}}(\mathcal{O})$, for each $p\in(1,\infty)$.}\end{align}

We proceed to proving Theorem 2.1.

Proof of Theorem 2.1. Existence. Now, let $\ell\in\mathbb{I}$ be fixed. Since $u^{\varepsilon_{n}}_{\ell}$ is the unique strong solution to the HJB equation (2.20) when $\varepsilon=\varepsilon_{n}$ , which belongs to $\text{C}^{0,1}\big(\overline{\mathcal{O}}\big)$ , it follows that for each $\kappa\in\mathbb{I}\setminus\{\ell\}$ , $u^{\varepsilon_{n}}_{\ell}-\big(u^{\varepsilon_{n}}_{\kappa} +\vartheta_{\ell,\kappa}\big)\leq u^{\varepsilon_{n}}_{\ell}-\mathcal{M}_{\ell}u^{\varepsilon_{n}}\leq0$ in $\mathcal{O}$ . From here and (3.27), we have that $u_{\ell}-u_{\kappa} -\vartheta_{\ell,\kappa}\leq0$ in $\mathcal{O}$ . Then, $u_{\ell}-\mathcal{M}_{\ell}u\leq0$ , in $\mathcal{O}$ . Also, we know that $[c_{\ell}-\mathcal{L}_{\ell}] u_{\ell}^{\varepsilon_{n}}+ \psi_{\varepsilon_{n}}\Big(\big|\textrm{D}^{1} u_{\ell}^{\varepsilon_{n}}\big|^{2}- g_{\ell}^{2}\Big)\leq h_{\ell}$ a.e. in $\mathcal{O}$ . Thus,

(3.28) \begin{align} &0\leq\psi_{\varepsilon_{n}}\Big(\big|\textrm{D}^{1} u_{\ell}^{\varepsilon_{n}}\big|^{2}- g_{\ell}^{2}\Big)\leq h_{\ell}-[c_{\ell}-\mathcal{L}_{\ell}] u_{\ell}^{\varepsilon_{n}},\quad \text{a.e.}\ \text{in}\ \mathcal{O}. \end{align}

Consequently, by (H3), (3.25), (3.26), and (3.28), there exists a positive constant $C_{6}=C_{6}(d,\Lambda,\alpha^{\prime})$ such that

\begin{equation*}0\leq\int_{B_{r}}\psi_{\varepsilon_{n}}\Big(\big|\textrm{D}^{1} u_{\ell}^{\varepsilon_{n}}\big|^{2}- g_{\ell}^{2}\Big)\varpi\textrm{d} x\leq\int_{B_{r}} \big\{h_{\ell}-[c_{\ell}-\mathcal{L}_{\ell}] u_{\ell}^{\varepsilon_{n}}\big\}\varpi\textrm{d} x\leq C_{6}\end{equation*}

for each $\varpi\in\mathcal{B}(B_{r})$ , with $\mathcal{B}({\cdot})$ as in (3.8). Thus, using definition of $\psi_{\varepsilon}$ (see (2.16)), and since $\big|\textrm{D}^{1}u_{\ell}^{\varepsilon_{n}}\big|^{2}-g^{2}_{\ell}$ is continuous in $\mathcal{O}$ , we have that for each $B_{r}\subset \mathcal{O}$ , there exists $\varepsilon^{\prime}\in(0,1)$ small enough so that for all $\varepsilon_{n}\leq \varepsilon^{\prime}$ , $\big|\textrm{D}^{1}u^{\varepsilon_{n}}_{\ell}\big|-g_{\ell}\leq0$ in $B_{r}$ . Then, since (3.27) holds, it follows that $|\textrm{D}^{1} u_{\ell}|\leq g_{\ell}$ in $\mathcal{O}$ . From (3.28), we get $\int_{B_{r}}\big\{[c_{\ell}-\mathcal{L}_{\ell}] u_{\ell}^{\varepsilon_{n}}-h_{\ell}\big\}\varpi\textrm{d} x\leq 0$ , for each $\varpi\in \mathcal{B}(B_{r})$ . From here and (3.27), we obtain that $[c_{\ell}-\mathcal{L}_{\ell}] u_{\ell}-h_{\ell}\leq 0$ a.e. in $\mathcal{O}$ . Therefore, by what we saw previously,

(3.29) \begin{equation} \max\Big\{[c_{\ell}-\mathcal{L}_{\ell}] u_{\ell}-h_{\ell},\big|\textrm{D}^{1} u_{\ell}\big|- g_{\ell},u_{\ell}-\mathcal{M}_{\ell}u\Big\}\leq 0 ,\quad \text{a.e.}\ \text{in}\ \mathcal{O}. \end{equation}

Without loss of generality we assume that $ u_{\ell}(x^{*})-\mathcal{M}_{\ell}u(x^{*})<0$ , for some $x^{*}\in\mathcal{O}$ . Otherwise, equality holds in (3.29). Then, for each $\kappa\in\mathbb{I}$ such that $\kappa\neq\ell$ , $u_{\ell}-(u_{\kappa} +\vartheta_{\ell,\kappa})\leq u_{\ell}-\mathcal{M}_{\ell}u<0$ at $x^{*}$ . There exists a ball $B_{r_{1}}(x^{*})\subset\mathcal{O}$ such that

(3.30) \begin{equation} u_{\ell}-(u_{\kappa} +\vartheta_{\ell,\kappa})\leq u_{\ell}-\mathcal{M}_{\ell}u<0,\quad\text{in}\ B_{r_{1}}(x^{*}), \end{equation}

by the continuity of $u_{\ell}-u_{\kappa}$ in $\overline{\mathcal{O}}$ . Now, consider that $\big|\textrm{D}^{1}u_{\ell}\big|-g_{\ell}<0$ for some $x^{*}_{1}\in B_{r_{1}}(x^{*})$ . Otherwise, equality again holds in (3.29). By continuity of $\big|\textrm{D}^{1}u_{\ell}\big|-g_{\ell}$ , we have that for some $B_{r_{2}}(x^{*}_{1})\subset\mathcal{O}$ , $\big|\textrm{D}^{1}u_{\ell}\big|-g_{\ell}<0$ in $B_{r_{2}}(x^{*}_{1})$ . From here, using (3.27), (3.30) and taking $\mathcal{N}\,:\!=\, B_{r_{1}}(x^{*})\cap B_{r_{2}}\big(x^{*}_{1}\big)$ , it can be verified that there exists an $\varepsilon^{\prime}\in(0,1)$ small enough so that for each $\varepsilon_{n}\leq\varepsilon^{\prime}$ , $|\textrm{D}^{1}u^{\varepsilon_{n}}_{\ell}|-g_{\ell}<0$ and $ u^{\varepsilon_{n}}_{\ell}-\mathcal{M}_{\ell}u^{\varepsilon_{n}}<0$ in $\mathcal{N}$ . Thus, $[c_{\ell}-\mathcal{L}_{\ell}] u_{\ell}^{\varepsilon_{n}}= h_{\ell}$ a.e. in $\mathcal{N}$ , since $u^{\varepsilon_{n}}$ is the unique solution to the HJB equation (2.20), when $\varepsilon=\varepsilon_{n}$ . Then, $\int_{\mathcal{N}}\big\{[c_{\ell}-\mathcal{L}_{\ell}] u_{\ell}^{\varepsilon_{n}}-h_{\ell}\big\}\varpi\textrm{d} x= 0$ , for each $\varpi\in\mathcal{B}(\mathcal{N})$ . Hence, letting $\varepsilon_{n}\rightarrow0$ and again using (3.27), we get that $u=(u_{1},\dots,u_{m})$ is a solution to the HJB equation (2.9).

Proof of Theorem 2.1. Uniqueness. Suppose that

\begin{equation*}u=(u_{1},\dots,u_{m})\quad \text{and}\quad v=(v_{1},\dots,v_{m})\end{equation*}

are two solutions to the HJB equation (2.9) whose components belong to $\textrm{C}^{0,1}\big(\overline{\mathcal{O}}\big)\cap \textrm{W}^{2,\infty}_{\textrm{loc}}(\mathcal{O})$ . Take $(x_{0},\ell_{0})\in\overline{\mathcal{O}}\times\mathbb{I}$ such that

(3.31) \begin{equation} u_{\ell_{0}}(x_{0})-v_{\ell_{0}}(x_{0})=\sup_{(x,\ell)\in\overline{\mathcal{O}}\times\mathbb{I}}\{u_{\ell}(x)-v_{\ell}(x)\}. \end{equation}

As before (see Subsection 3.1), we only need to verify that

(3.32) \begin{equation} u_{\ell_{0}}-v_{\ell_{0}}\leq0,\quad \text{at}\ x_{0}\in\mathcal{O}. \end{equation}

Assume that $u_{\ell_{0}}-v_{\ell_{0}}>0$ at $x_{0}$ . Then there exists a ball $B_{r_{1}}(x_{0})\subset\mathcal{O}$ such that

\begin{equation*}c_{\ell_{0}}[u_{\ell_{0}}-v_{\ell_{0}}]\geq\min_{x\in B_{r_{1}}(x_{0})}\big\{c_{\ell_{0}}(x)\big[u_{\ell_{0}}(x)-v_{\ell_{0}}(x)\big]\big\}>0\end{equation*}

in $B_{r_{1}}(x_{0}),$ by the continuity of $u_{\ell_{0}}-v_{\ell_{0}}$ in $\overline{\mathcal{O}}$ and the fact that $c_{\ell_{0}}>0$ in $\mathcal{O}$ . Meanwhile, from (3.31), $v_{\ell_{0}}-\mathcal{M}_{\ell_{0}}v\leq u_{\ell_{0}}-\mathcal{M}_{\ell_{0}}u\leq 0$ at $x_{0}$ . If $v_{\ell_{0}}-\mathcal{M}_{\ell_{0}}v<0$ at $x_{0}$ , there exists a ball $B_{r_{2}}(x_{0})\subset\mathcal{O}$ such that $v_{\ell_{0}}-\mathcal{M}_{\ell_{0}}v<0$ in $B_{r_{2}}(x_{0})$ . Now, consider the auxiliary function $f_{\varrho}\,:\!=\, u_{\ell_{0}}-v_{\ell_{0}}-\varrho u_{\ell_{0}}$ , with $\varrho\in(0,1)$ . Notice that $f_{\varrho}=0$ on $\partial\mathcal{O}$ , for $\varrho\in(0,1)$ , and

(3.33) \begin{equation} f_{\varrho}\uparrow u_{\ell_{0}}-v_{\ell_{0}}\ \text{uniformly in}\ \mathcal{O},\ \text{when}\ \varrho\downarrow 0. \end{equation}

In addition, there is a $\varrho^{\prime}\in(0,1)$ small enough so that $\sup_{x\in B_{r_{2}}(x_{0})}\{f_{\varrho}(x)\}>0$ for all $\varrho\in(0,\varrho^{\prime})$ , since $u_{\ell_{0}}-v_{\ell_{0}}>0$ at $x_{0}$ . By (3.31) and (3.33), there exists $\hat{\varrho}\in(0,\varrho^{\prime})$ small enough so that $f_{\hat{\varrho}}$ has a local maximum at $x_{\hat{\varrho}}\in B_{r_{1}}(x_{0})\cap B_{r_{2}}(x_{0})$ . It follows that

\begin{equation*}\big|\textrm{D}^{1}v_{\ell_{0}}\big(x_{\hat{\varrho}}\big)\big|=[1-\hat{\varrho}]\big|\textrm{D}^{1}u_{\ell_{0}}\big(x_{\hat{\varrho}}\big)\big|<\big|\textrm{D}^{1}u_{\ell_{0}}\big(x_{\hat{\varrho}}\big)\big|\leq g\big(x_{\hat{\varrho}}\big).\end{equation*}

Thus, there exists a ball $B_{r_{3}}\big(x_{\hat{\varrho}}\big)\subset B_{r_{1}}(x_{0})\cap B_{r_{2}}(x_{0})$ such that $[c_{\ell_{0}}-\mathcal{L}_{\ell_{0}}] v_{\ell_{0}}-h_{\ell_{0}}=0$ and $[c_{\ell_{0}}-\mathcal{L}_{\ell_{0}}] u_{\ell_{0}}-h_{\ell_{0}}\leq0$ in $B_{r_{3}}\big(x_{\hat{{\varrho}}}\big)$ . Then, by Bony’s maximum principle, we have that

$$\begin{equation*}0\geq\lim_{r\rightarrow0}\bigg\{\mathop{\textrm{inf} \, \textrm{ess}}\limits_{B_{r}\big(x_{\hat{\varrho}}\big)} \textrm{tr}\Big[a_{\ell_{0}}\textrm{D}^{2}f_{\hat{\varrho}}\Big]\bigg\}\geq c_{\ell_{0}}f_{\hat{\varrho}}+\hat{\varrho} h_{\ell_{0}}\end{equation*}$$

at $x_{\hat{\varrho}},$ which is a contradiction since $\hat{\varrho} h_{\ell}\geq0$ , $f_{\hat{\varrho}}>0$ , and $c_{\ell_{0}}>0$ at $x_{\hat{\varrho}}$ . We conclude that $0=v_{\ell_{0}}-\mathcal{M}_{\ell_{0}}v\leq u_{\ell_{0}}-\mathcal{M}_{\ell_{0}}u\leq 0$ at $x_{0}$ . Using the same arguments seen in the proof of uniqueness of the solution to the HJB equation (2.9) (see Subsection 3.1), it can be verified that there is a contradiction with the assumption that there is no loop of zero cost (see Equation (2.7)). From here we conclude that ((3.32)) must hold. Taking $v-u$ and proceeding in the same way as before, we see that u is the unique solution to the HJB equation (2.9).

4.1. $\varepsilon$ -PACS optimal control problem

Before presenting the second part of the verification lemma and proving it, we first construct the control $\big(\xi^{\varepsilon,*},\varsigma^{\varepsilon,*}\big)$ which turns out to be the optimal strategy for the $\varepsilon$ -PACS control problem. To that end, let us introduce the switching regions.

For any $\ell\in\mathbb{I}$ , let $\mathcal{S}^{\varepsilon}_{\ell}$ be the set defined by

\begin{equation*}\mathcal{S}^{\varepsilon}_{\ell}=\Big\{x\in\mathcal{O}\,:\,u^{\varepsilon}_{\ell}(x)-\mathcal{M}_{\ell}u^{\varepsilon}(x)=0\Big\}.\end{equation*}

Notice that $\mathcal{S}^{\varepsilon}_{\ell}$ is a closed subset of $\mathcal{O}$ and corresponds to the region where it is optimal to switch regimes. The complement $\mathcal{C}^{\varepsilon}_{\ell}$ of $\mathcal{S}^{\varepsilon}_{\ell}$ in $\mathcal{O}$ , where it is optimal to stay in the regime $\ell$ , is the so-called continuation region

\begin{equation*}\mathcal{C}^{\varepsilon}_{\ell}=\Big\{x\in\mathcal{O}\,:\,u^{\varepsilon}_{\ell}(x)-\mathcal{M}_{\ell}u^{\varepsilon}(x)<0\Big\}.\end{equation*}

Remark 4.1. Observe that $u^{\varepsilon}_{\ell}\in W^{2,\infty}_{\textrm{loc}}\big(\mathcal{C}^{\varepsilon}_{\ell}\big)=C_{\textrm{loc}}^{1,1}\big(\mathcal{C}^{\varepsilon}_{\ell}\big)$ . This implies that

\begin{equation*}h_{\ell}-\psi_{\varepsilon}\Big(\Big|\textrm{D}^{1}u^{\varepsilon}_{\ell}\Big|^{2}-g_{\ell}^{2}\Big)\in \textrm{C}^{0,\alpha^{\prime}}_{\textrm{loc}}\big(\mathcal{C}_{\ell}^{\varepsilon}\big),\end{equation*}

since $h_{\ell}\in \textrm{C}^{2,\alpha^{\prime}}\big(\overline{\mathcal{O}}\big))$ and $\psi_{\varepsilon}\Big(\big|\textrm{D}^{1}u^{\varepsilon}_{\ell}\big|^{2}-g_{\ell}^{2}\Big)\in \text{C}^{0,\alpha^{\prime}}_{\textrm{loc}}\big(\mathcal{C}_{\ell}^{\varepsilon}\big)$ . From here and using Theorem 9.19 of [Reference Gilbarg and Trudinger10], we have that $u^{\varepsilon}_{\ell}\in \text{C}_{\textrm{loc}}^{2,\alpha^{\prime}}\big(\mathcal{C}^{\varepsilon}_{\ell}\big)$ .

Also, the set $\mathcal{S}^{\varepsilon}_{\ell}$ satisfies the following property.

Lemma 4.2. Let $\ell$ be in $\mathbb{I}$ . Then

\begin{equation*}\mathcal{S}^{\varepsilon}_{\ell}=\widetilde{\mathcal{S}}^{\varepsilon}_{\ell}\,:\!=\,\bigcup_{\kappa\in\mathbb{I}\setminus\{\ell \}}\mathcal{S}^{\varepsilon}_{\ell,\kappa},\end{equation*}

where $\mathcal{S}^{\varepsilon}_{\ell,\kappa}\,:\!=\,\big\{x\in\mathcal{C}^{\varepsilon}_{\kappa}\,:\,u^{\varepsilon}_{\ell}(x)=u^{\varepsilon}_{\kappa}(x )+\vartheta_{\ell,\kappa} \big\}$ .

Proof. We obtain trivially that $\widetilde{\mathcal{S}}^{\varepsilon}_{\ell}\subset\mathcal{S}^{\varepsilon}_{\ell}$ , since $u^{\varepsilon}_{\ell}-u^{\varepsilon}_{\kappa}-\vartheta_{\ell,\kappa}\leq u^{\varepsilon}_{\ell}-\mathcal{M}_{\ell}u^{\varepsilon}\leq0$ on $\mathcal{O}$ for $\kappa\in\mathbb{I}\setminus\{\ell\}$ . If $x\in\mathcal{S}^{\varepsilon}_{\ell}$ , there is an $\ell_{1}\neq\ell$ where $u^{\varepsilon}_{\ell}(x)=u^{\varepsilon}_{\ell_{1}}(x)+\vartheta_{\ell,\ell_{1}}$ . Notice that x must belong to either $\mathcal{C}^{\varepsilon}_{\ell_{1}}$ or $\mathcal{S}^{\varepsilon}_{\ell_{1}}$ . If $x\in\mathcal{C}^{\varepsilon}_{\ell_{1}}$ , then $x\in\mathcal{S}^{\varepsilon}_{\ell,\ell_{1}}\subset\widetilde{\mathcal{S}}^{\varepsilon}_{\ell} $ . Otherwise, there is an $\ell_{2}\neq\ell_{1}$ such that $u^{\varepsilon}_{\ell_{1}}\!(x)=u^{\varepsilon}_{\ell_{2}}(x)+\vartheta_{\ell_{1},\ell_{2}}$ . This implies

\begin{equation*}u^{\varepsilon}_{\ell}(x)=u^{\varepsilon}_{\ell_{2}}(x)+\vartheta_{\ell,\ell_{1}}+\vartheta_{\ell_{1},\ell_{2}}\geq u^{\varepsilon}_{\ell_{2}}(x)+\vartheta_{\ell,\ell_{2}},\end{equation*}

since (2.6) holds. Then, $u^{\varepsilon}_{\ell}(x)= u^{\varepsilon}_{\ell_{2}}(x)+\vartheta_{\ell,\ell_{2}}$ . Again x must belong to either $\mathcal{C}^{\varepsilon}_{\ell_{2}}$ or $\mathcal{S}^{\varepsilon}_{\ell_{2}}$ . If $x\in\mathcal{C}^{\varepsilon}_{\ell_{2}}$ , we have $x\in\mathcal{S}^{\varepsilon}_{\ell,\ell_{2}}\subset\widetilde{\mathcal{S}}^{\varepsilon}_{\ell} $ . Otherwise, by the same argument as before, and since the number of regimes is finite, it must be that there is some $\ell_{i}\neq\ell$ such that $x\in\mathcal{C}^{\varepsilon}_{\ell_{i}}$ and $u^{\varepsilon}_{\ell}(x)= u^{\varepsilon}_{\ell_{i}}(x)+\vartheta_{\ell,\ell_{i}}$ . Therefore $x\in\mathcal{S}^{\varepsilon}_{\ell,\ell_{i}}\subset\widetilde{\mathcal{S}}^{\varepsilon}_{\ell} $ .

Now we construct the optimal control $\big(\xi^{\varepsilon,*},\varsigma^{\varepsilon,*}\big)$ to the problem (2.18). Let $\big(\tilde{x},\tilde{\ell}\big)$ be in $\overline{\mathcal{O}}\times\mathbb{I}$ . The dynamics of the process and $\big(\xi^{\varepsilon,*},\varsigma^{\varepsilon,*}\big)$ is given recursively in the following way:

1. (i) Define $\tau^{*}_{0}=0$ and $\ell^{*}_{0-}=\tilde{\ell}$ . If $\tilde{x}\notin\mathcal{C}^{\varepsilon}_{\tilde{\ell}}$ , take $\tau^{*}_{1}\,:\!=\,0$ and pass to Item (ii) because of Lemma 4.2. Otherwise, the process $X^{\varepsilon,*}$ evolves as

   (4.6)

   
   with $X^{\varepsilon,*}_{0}=\tilde{x}$ , ,
   (4.7)

   
   The control process $\xi^{\varepsilon,*}=\big(\unicode{x1D55F}^{\varepsilon,*},\zeta^{\varepsilon,*}\big)$ is defined by
   (4.8)

   
   where $\gamma_{0}\in\mathbb{R}^{d}$ is a fixed unit vector, and , with and
   (4.9)

   

2. (ii) Recursively, letting $i\geq1$ and defining

   (4.10)

   
   if $\tau^{*}_{i}<\tau^{*}$ , the process $X^{\varepsilon,*}$ evolves as
   (4.11)

   
   where
   (4.12)

   
   with $\gamma_{0}\in\mathbb{R}^{d}$ being a fixed unit vector, and , with and
   (4.13)

   

Remark 4.3. Suppose that $\tau^{*}_{i}<\tau^{*}$ for some $i>0$ . We notice that for ,

, , and, by (2.16) and (3.25), this yields that . Also we see that if , by Lemma 4.2.

Remark 4.5. Taking

we see that it is a càdlàg process.

Proof. Take , with $\mathcal{O}_{q}\,:\!=\,\{x\in\mathcal{O}\,:\,\textrm{dist}(x,\partial\mathcal{O} )>1/q\}$ and q a positive integer large enough. By Remarks 4.1 and 4.3, it is known that $u^{\varepsilon}_{\ell}$ is a $\text{C}^{2}$ function on $\mathcal{C}^{\varepsilon}_{\ell}$ and that if . Then, using integration by parts and Itô’s formula in on the interval $\big[\hat{\tau}^{*,q}_{i},\hat{\tau}^{*,q}_{i+1}\big)$ , and taking expected value on it, we obtain a similar expression to that of (4.1). Now, considering that , which was defined in (4.2), is a square-integrable martingale, and since

\begin{equation*}\big[c_{\ell^{*}_{i}}-\mathcal{L}_{\ell^{*}_{i}}\big] u_{\ell^{*}_{i}}^{\varepsilon}=h_{\ell^{*}_{i}}-\psi_{\varepsilon}\Big(\Big|\textrm{D}^{1} u_{\ell^{*}_{i}}^{\varepsilon}\Big|^{2}- g_{\ell^{*}_{i}}^{2}\Big)\end{equation*}

on $\mathcal{C}^{\varepsilon}_{\ell^{*}_{i}}$ and the supremum of $l^{\varepsilon}_{\ell}(\eta,x)$ is attained if $\gamma$ is related to $\eta$ by $\eta=2\psi^{\prime}_{\varepsilon}\big(|\gamma|^{2}-g_{\ell}(x)^{2}\big)\gamma$ , i.e.,

\begin{equation*}l^{\varepsilon}_{\ell}\Big(2\psi^{\prime}_{\varepsilon}\Big(|\gamma|^{2}- g_{\ell}(x)^{2}\Big)\gamma,x\Big)=2\psi^{\prime}_{\varepsilon}\Big(|\gamma|^{2}- g_{\ell}(x)^{2}\Big)|\gamma|^{2}-\psi_{\varepsilon}\Big(|\gamma|^{2}- g_{\ell}(x)^{2}\Big),\end{equation*}

it can be checked that

(4.14)

Notice that $\hat{\tau}^ {*,q}_{i}\uparrow\tilde{\tau}^{*}_{i}$ as $q\rightarrow\infty$ , $\mathbb{P}_{\tilde{x}}$ -a.s. Consequently, letting first $q\rightarrow\infty$ and then in (4.14), we see that

(4.15)

for $i\geq0$ , with $\mathcal{D}\big[\tau^{*}_{i+1},\ell^{*}_{i},\ell^{*}_{i+1},X^{\varepsilon,*}, u^{\varepsilon}\big]$ as in (4.4). By (4.7) and (4.10),

\begin{equation*}\mathcal{D}\big[\tau^{*}_{i+1},\ell^{*}_{i},\ell^{*}_{i+1},X^{\varepsilon,*}, u^{\varepsilon}\big]\unicode{x1D7D9}_{\big\{\tau^{*}_{i+1}<\tau^{*}\big\}}=0.\end{equation*}

Therefore, from here and (4.15), we obtain the desired result that was given in the lemma above.

4.2. Proof of Theorem 2.2

To conclude this section, we present the proof of Theorem 2.2, which will be proven under Assumptions (H1) and (H3)–(H5). Recall that $\mathcal{U}$ and $\mathcal{S}$ are the families of controls $\xi=(\unicode{x1D55F},\zeta)$ and $\varsigma=(\tau_{i},\ell_{i})_{i\geq0}$ that satisfy (2.1) and (2.2), respectively. The functional cost $V_{\xi,\varsigma}$ is given by (1.2) for $(\xi,\varsigma)\in\mathcal{U}\times\mathcal{S}$ .

Proof of Theorem 2.2. Let $\{u^{\varepsilon_{n}}\}_{n\geq1}$ be the sequence of unique strong solutions to the HJB equation (2.20), when $\varepsilon=\varepsilon_{n}$ , which satisfy (3.27). From Lemma 4.3, we know that

\begin{equation*}u^{\varepsilon_{n}}_{\tilde{\ell}}(\tilde{x})=\mathcal{V}_{\xi^{\varepsilon_{n},*},\varsigma^{\varepsilon_{n},*}}\big(\tilde{x},\tilde{\ell}\big)= V^{\varepsilon_{n}}\big(\tilde{x},\tilde{\ell}\big)\quad\text{for $\big(\tilde{x},\tilde{\ell}\big)\in\overline{\mathcal{O}}\times\mathbb{I}$},\end{equation*}

with $(\xi^{\varepsilon_{n},*},\varsigma^{\varepsilon_{n},*})$ as in (4.7)–(4.10) and (4.12)–(4.13), when $\varepsilon=\varepsilon_{n}$ . Notice that

\begin{equation*}l_{\varepsilon_{n}}(x,\beta\gamma)\geq\big\langle\beta\gamma, g_{\ell^{*}_{i}} (x)\gamma\big\rangle-\psi_{\varepsilon_{n}}\big(| g_{\ell^{*}_{i}}(x)\gamma|^{2}- g_{\ell^{*}_{i}}(x)^{2}\big)=\beta g _{\ell^{*}_{i}} (x),\end{equation*}

with $\beta\in\mathbb{R}$ and $\gamma\in\mathbb{R}^{d}$ a unit vector. Then, from here and considering $(X^{\varepsilon_{n},*},I^{\varepsilon_{n},*})$ governed by (4.6) and (4.11), it follows that

(4.16)

Notice that $V_{\xi^{\varepsilon_{n},*},\varsigma^{\varepsilon_{n},*}}$ is the cost function given in (1.2) corresponding to the control $(\xi^{\varepsilon_{n},*},\varsigma^{\varepsilon_{n},*})$ where the second term in the right-hand side of (2.5) is zero, since $\zeta^{\varepsilon_{n},*}$ has the continuous part only. Letting $\varepsilon_{n}\rightarrow0$ in (4.16), we have $u_{\tilde{\ell}}(\tilde{x})\geq V_{\tilde{\ell}}(\tilde{x})$ for each $\big(\tilde{x},\tilde{\ell}\big)\in\overline{\mathcal{O}}\times\mathbb{I}$ .

Let $\big\{u^{\varepsilon_{n},\delta_{\hat{n}}}\big\}_{n,\hat{n}\geq1}$ be the sequence of unique solutions to the NPDS (3.1), when $\varepsilon=\varepsilon_{n}$ and $\delta=\delta_{\hat{n}}$ , which satisfy (3.5) and (3.27). Let us assume (X, I) evolves as in (1.1) with initial state $\big(\tilde{x},\tilde{\ell}\big)\in\overline{\mathcal{O}}\times\mathbb{I}$ and the control process $(\xi,\varsigma)$ belongs to $\mathcal{U}\times\mathcal{S}$ . Take , with $i\geq1$ , $\mathcal{O}_{q}\,:\!=\,\{x\in\mathcal{O}\,:\,dist(x,\partial\mathcal{O} )>1/q\}$ , and q a positive integer large enough. Using integration by parts and Itô’s formula in on $ \big[\hat{\tau}^{q}_{i},\hat{\tau}^{q}_{i+1}\big)$ , $i\geq0$ , we get that

(4.17)

where is as in (4.2) and

Since for , $i\geq0$ , and $\mathcal{O}$ is a convex set, by the mean value theorem, we have

Taking the expected value in (4.17), and since is a square-integrable martingale and $\big[c_{\ell_{i}}-\mathcal{L}_{\ell_{i}}\big]u^{\varepsilon_{n},\delta_{\hat{n}}}_{\ell_{i}}\leq h_{\ell_{i}}$ , we obtain

(4.18)

Notice that for ,

with $\lambda\in[0,1]$ . Then, letting $\varepsilon_{n},\delta_{\hat{n}}\rightarrow0$ in (4.18), by the dominated convergence theorem and using $\big|\textrm{D}^{1} u_{\ell_{i}}\big|\leq g_{\ell_{i}}$ , we have

(4.19)

with as in (2.5). Again, letting $q\rightarrow\infty$ and in (4.19), we have

(4.20)

for $i\geq0$ , with $\tilde{\tau}_{i}=\tau_{i}\wedge\tau$ and $\mathcal{D}[\tau_{i+1},\ell_{i},\ell_{i+1},X, u ]$ as in (4.4). Noticing that

\begin{equation*}\mathcal{D}[\tau_{i+1},\ell_{i},\ell_{i+1},X, u ]\unicode{x1D7D9}_{\{\tau_{i+1}<\tau\}}\leq0\end{equation*}

and using (1.2), (4.20), we obtain $u_{\tilde{\ell}}(\tilde{x})\leq V_{\xi,\varsigma}\big(\tilde{x},\tilde{\ell}\big)$ for each $\big(\tilde{x},\tilde{\ell}\big)\in\overline{\mathcal{O}}\times\mathbb{I}$ . Since the previous property is true for each control $(\xi,\varsigma)\in\mathcal{U}\times\mathcal{S}$ , we conclude that $u_{\tilde{\ell}}(\tilde{x})\leq V_{\tilde{\ell}}(\tilde{x})$ for each $\big(\tilde{x},\tilde{\ell}\big)\in\overline{\mathcal{O}}\times\mathbb{I}$ .

4.3. On $\varepsilon$ -PACS optimal controls

Previously, we checked that the value function $V_{ \ell}$ , defined in (2.8), satisfies the HJB equation (2.9). This means that for each $\ell\in\mathbb{I}$ fixed, the domain set $\mathcal{O}$ is divided into three parts. Consider $\mathcal{C}_{\ell}\,:\!=\,\mathcal{O}\setminus(\mathcal{N}_{\ell}\cup \mathcal{S}_{\ell})$ where

\begin{equation*}\mathcal{N}_{\ell}\,:\!=\,\big\{x\in\mathcal{O}\,:\,|\textrm{D}^{1}V_{\ell}|= g_{\ell}\big\}\quad \text{and} \quad \mathcal{S}_{\ell}=\{x\in\mathcal{O}\,:\,V_{\ell}=\mathcal{M}_{\ell}V\}.\end{equation*}

The open set $\mathcal{C}_{\ell}$ is where $V_{\ell}$ satisfies the elliptic partial differential equation $[c_{\ell}-\mathcal{L}_{\ell}] V_{\ell}=h_{\ell}$ , which suggests that the ‘optimal control’ $(\xi^{*},\varsigma^{*})$ corresponding to this problem will not be exercised when the process $X^{\xi^{*},\varsigma^{*}}$ is in $\mathcal{C}_{\ell}$ . Otherwise, either $\xi^{*}$ or $\varsigma^{*}$ will be exercised on $X^{\xi^{*},\varsigma^{*}}$ in the following way:

1. (i) if $X^{\xi^{*},\varsigma^{*}}\in\mathcal{N}_{\ell}\setminus\mathcal{S}_{\ell}$ , the singular control $\xi^{*}$ will act on $X^{\xi^{*},\varsigma^{*}}$ in such a way that $ X^{\xi^{*},\varsigma^{*}} $ will be pushed back to some point $y \in\partial\mathcal{C}_{\ell}$ ;

2. (ii) if $X^{\xi^{*},\varsigma^{*}}\in \mathcal{S}_{\ell}$ , the switching control $\varsigma^{*}$ will be executed in such a way that the process $ X^{\xi^{*},\varsigma^{*}} $ will switch to some regime $\kappa\neq\ell$ at time $\tau_{\kappa}\leq\tau$ .

The construction of an optimal strategy for the problem (2.8) still remains an open problem of interest. A way to carry out this construction is to verify first that $\partial\mathcal{C}_{\ell}$ is at least of class $\text{C}^{1}$ , which is not easy to do.

In the literature, we can see that the existence of an optimal dividend/switching strategy, which is an example of a mixed singular/switching control problem, has been solved when the payoff expected value is given by (1.4) and the cash reserve process switches between m-regimes governed by different Poisson processes with drifts; see [Reference Azcue and Muler2], in which the authors proved that their solution is stationary with a band structure.

Another way to address the problem (2.8) is by means of $\varepsilon$ -PACS optimal controls, which have been constructed in (4.6)–(4.13).

By Lemma 4.3 and the proof of Theorem 2.2, it is known that for each $\ell\in\mathbb{I}$ , $V^{\varepsilon_{n}}_{\ell}\rightarrow V_{\ell}$ as $\varepsilon_{n}\downarrow0$ , and $V_{\ell}\leq V_{\xi^{\varepsilon_{n},*},\varsigma^{\varepsilon_{n},*}}(\cdot, \ell)\leq V^{\varepsilon_{n}}_{\ell}\ \text{on} \overline{\mathcal{O}}$ , with $\big(\xi^{\varepsilon_{n},*},\varsigma^{\varepsilon_{n},*}\big)$ as in (4.6)–(4.13), and $V_{\xi^{\varepsilon_{n},*},\varsigma^{\varepsilon_{n},*}}(\cdot,\ell),V_{\ell}, V^{\varepsilon_{n}}_{\ell}$ given by (1.2), (2.8), and (2.18), respectively.

Taking $\varepsilon_{n}$ small enough and assuming that the process is on the regime $\ell^{*}_{i}$ at time , we have that and the control $\big(\xi^{\varepsilon_{n},*},\varsigma^{\varepsilon_{n},*}\big)$ will be exercised as follows:

1. (i) if the controlled process $X^{ \varepsilon_{n},*}$ satisfies

   
   then and will stay in $\mathcal{C}_{\ell_{i}^{*}}$ ;

2. (ii) if , the process will be crossing $\partial\mathcal{C}_{\ell_{i}^{*}}$ persistently;

3. (iii) otherwise, will exercise a force

   
   and in the direction
   
   at in such a way that it will be pushed back to $\partial\mathcal{C}_{\ell_{i}^{*}}$ .

4. (iv) At time , $X^{\varepsilon_{n},*}_{\tau_{i+1}^{*}}$ will be in $\mathcal{S}^{\varepsilon_{n}}_{\ell_{i}^{*}}$ for the first time and will switch to the regime $\ell_{i+1}^{*}$ .

This procedure will be repeated until the time $\tau^{*}$ , which represents the first exit time of the process $X^{\varepsilon_{n},*}$ from the set $\mathcal{O}$ .

5.1. Construction of the solution to the HJB equation (5.1)

Considering $\ell\in\mathbb{I}$ fixed, let us first construct the solution to the following system of variational inequalities:

(5.2) \begin{align}&\max\Big\{\big[c_{{\ell}}-\mathcal{L}_{{\ell}}\big]\tilde{u}_{{\ell}}- h_{{\ell}},\big|\tilde{u}^{\prime}_{{\ell}}\big|- g_{{\ell}}\Big\}= 0\ \text{in}\ (0,l),\quad\text{s.t.}\ \tilde{u}_{{\ell}}(0)=\tilde{u}_{{\ell}}(l)=0.\end{align}

It is well known that the general solution to $[c_{\ell} -\mathcal{L}_{\ell}] \tilde{u}_{\ell}=h_\ell$ on $\big[x^{*}_{\ell},l\big]$ , with $x^{*}_{\ell}>0$ , has the form

\begin{equation*}\tilde{u}_{\ell}(x)=A_{\ell}x^{r_{1,\ell}}+B_{\ell}x^{r_{2,\ell}}+\overline{K}_{\ell}x^{\gamma_{\ell}},\end{equation*}

where

\begin{equation*}\overline{K}_{\ell}=\frac{2 K_{\ell}}{\sigma^{2}(r_{1,\ell}-\gamma_{\ell})(\gamma_{\ell}-r_{2,\ell})}\end{equation*}

and $ r_{1,\ell},r_{2,\ell}$ denote the negative and positive solutions to

(5.3) \begin{equation}r^{2}-\bigg(\dfrac{2b_{\ell}}{\sigma^{2}_{\ell}}+1\bigg)r-\dfrac{2c_{\ell}}{\sigma^{2}_{\ell}}=0.\end{equation}

Taking

(5.4) \begin{equation}\tilde{u}_{\ell}(x)\,:\!=\,\begin{cases}g_{\ell}x&\text{if}\ 0\leq x<x^{*}_{\ell},\\[3pt]A_{\ell}x^{r_{1,\ell}}+B_{\ell}x^{r_{2,\ell}}+\overline{K}_{\ell}x^{\gamma_{\ell}} &\text{if}\ x^{*}_{\ell}\leq x\leq l,\end{cases}\end{equation}

we see easily that

\begin{equation*}\big|\tilde{u}^{\prime}_\ell\big|-g_{\ell}=0\ \text{on}\ \big[0,x^{*}_{\ell}\big)\quad\text{and}\quad [c_{\ell}-\mathcal{L}_{\ell}] \tilde{u}_{\ell}-h_{\ell}=0\ \text{on}\ \big[x^{*}_{\ell},l\big],\end{equation*}

and if $c_{\ell}+b_{\ell}\leq0$ , then

\begin{equation*}[c_{\ell}-\mathcal{L}_{\ell}] \tilde{u}_{\ell}(x)-h_{\ell}(x)=g_{\ell}(c_{\ell}+b_{\ell})x-K_{\ell}x^{\gamma_{\ell}}<0\quad \text{for}\ x\in\big(0,x^{*}_{\ell}\big).\end{equation*}

In order for $\tilde{u}$ to satisfy (5.2), we shall consider parameters $b_{\ell}$ , $\sigma_{\ell}$ , $c_{\ell}$ , and $g_{\ell}$ such that

(5.5) \begin{equation}\big|u^{\prime}_{\ell}\big|-g_{\ell}\leq0\quad\text{on}\ \big(x^{*}_{\ell},l\big).\end{equation}

Additionally, they must satisfy

(5.6) \begin{equation}x^{*}_{\ell}<\bigg(\dfrac{K_{\ell}}{g_{\ell}(c_{\ell}+b_{\ell})}\bigg)^{\frac{1}{1-\gamma_{\ell}}},\quad\text{if}\ c_{\ell}+b_{\ell}>0,\end{equation}

since if (5.6) holds and $x\in\big(0,x^{*}_{\ell}\big)$ , then

(5.7) \begin{align}[c_{\ell}-\mathcal{L}_{\ell}] \tilde{u}_{\ell}(x)-h_{\ell}(x)&=x^{\gamma_{\ell}}\big(g_{\ell}(c_{\ell}+b_{\ell})x^{1-\gamma_{\ell}}-K_{\ell}\big)\notag\\&<x^{\gamma_{\ell}}\big(g_{\ell}(c_{\ell}+b_{\ell})\big(x^{*}_{\ell}\big)^{1-\gamma_{\ell}}-K_{\ell}\big)<0.\end{align}

In order for $\tilde{u}_{\ell}$ to belong to $\text{C}^{1}([0,l])$ , by smooth fit, the parameters $A_{\ell}$ , $B_{\ell}$ , and $x^{*}_{\ell}$ must satisfy the following system of equations:

(5.8) \begin{align}&A_{\ell}l^{r_{1,\ell}}+B_{\ell}l^{r_{2,\ell}}+\overline{K}_{\ell}l^{\gamma_{\ell}}=0, \\&A_{\ell}+B_{\ell}\big(x^{*}_{\ell}\big)^{r_{2,\ell}-r_{1,\ell}}+\overline{K}\big(x^{*}_{\ell}\big)^{\gamma_{\ell}-r_{1,\ell}}-g_{\ell}\big(x^{*}_{\ell}\big)^{1-r_{1,\ell}}=0,\\&r_{1,\ell}A_{\ell}+r_{2,\ell}B_{\ell}\big(x^{*}_{\ell}\big)^{r_{2,\ell}-r_{1,\ell}}+\gamma_{\ell}\overline{K}_{\ell} \big(x^{*}_{\ell}\big)^{\gamma_{\ell}-r_{1,\ell}}-g_{\ell}\big(x^{*}_{\ell}\big)^{1-r_{1,\ell}}=0,\quad\text{s.t.}\ (5.6).\end{align}

By what we have seen previously, we have the next result.

The uniqueness of $\tilde{u}_{\ell}$ is guaranteed by [Reference Evans7, Theorem 1.1].

Numerical examples. Table 1 contains the values that are considered to find numerically the solution $\tilde{u}_{\ell}$ to the variational inequalities (5.2), when $\ell=1,2,3$ and $l=3$ ; see the left panel of Figure 1.

Table 1.

Parameter values for the equations (5.3) and (5.8).

Figure 1.

The left and right panels show plots of $\tilde{u}_{\ell}$ for the parameters given in Tables 2 and 4, respectively.

Table 2 presents the numerical solutions to the system of equations (5.3) and (5.8) for the parameters given in Table 1. Observe that when $\ell=2,3$ , the condition (5.6) is satisfied. The left panel of Figure 3 shows that (5.5) and (5.7) are satisfied for the parameters seen in Table 1. From here observe that $\tilde{u}$ is the numerical solution to the variational inequality (5.2) for the parameters given in Table 1.

Table 2.

Approximate solution to the system of equations (5.3) and (5.8) with the parameters given in Table 1.

Table 3.

Parameter values for the equations (5.3) and (5.8).

Let us now give two examples where the conditions (5.5) and (5.6) are not satisfied. Consider the following parameters:

Table 4 presents the numerical solutions to the system of equations (5.3) and (5.8). The left panel of Figure 2 shows that $\tilde{u}_{1}$ does not satisfy the condition (5.5). Notice that $x^{*}_{2}\approx 0.4411 $ does not satisfy the condition (5.6), leaving a discontinuity of $\tilde{u}_{2}$ at $x^{*}_{2}$ ; see the right panel of Figure 1. Additionally, in the right panel of Figure 2, we see that $\tilde{u}_{2}$ violates (5.7).

Table 4.

Approximate solution to the system of equations (5.3) and (5.8) for the parameters given in Table 3.

Figure 2.

The left and right panels show plots of $[c_{\ell}-\mathcal{L}_{\ell}]\tilde{u}_{\ell}-h_{\ell}$ and $\big|\tilde{u}^{\prime}_{\ell}\big|-g_{\ell}$ on $\big(0,x^{*}_{\ell}\big)$ and $\big(x^{*}_{\ell}, l\big)$ for the parameters given in Table 4.

5.1.1. Construction of the solution to (5.1)

In this part, we consider the vector function $\tilde{u}=\big(\tilde{u}_{1},\dots,\tilde{u}_{m}\big)$ such that its elements are given as (5.4) and satisfy the hypothesis of Proposition 5.1 in such a way that $\tilde{u}_{\ell}$ belongs to $\textrm{C}^{1}([0,l])\cap \textrm{C}^{2}\big([0,l]\setminus\{x^{*}_{\ell}\}\big)$ and is the unique solution of (5.2) for each $\ell\in\mathbb{I}$ . Assume that the entrances of $\tilde{u}$ are different.

We will study the parameters $\vartheta_{\ell,\kappa}\geq0$ where $\tilde{u}$ satisfies the HJB equation (5.1). For that, define first the sets $\mathcal{D}_{\ell,\kappa}$ and $\mathcal{U}_{\ell,\kappa}$ , with $\ell\neq\kappa$ , by

\begin{align*}\mathcal{D}_{\ell,\kappa}&=\big\{x\in(0,l)\,:\, u^{\prime}_{\ell}(x)= u^{\prime}_{\kappa}(x)\big\},\\{\mathcal{U}}_{\ell,\kappa}&=\big\{x\in\mathcal{D}_{\ell,\kappa}\,:\, 0\leq\big[\tilde{u}_{\ell}-\tilde{u}_{\kappa}\big](y)\leq\big[\tilde{u}_{\ell}-\tilde{u}_{\kappa}\big](x)\ \text{for any}\ y\in\mathcal{D}_{\ell,\kappa} \big\}.\end{align*}

Now let us take $\hat{\vartheta}_{\ell,\kappa}$ , with $\ell,\kappa\in\mathbb{I}$ and $\ell\neq\kappa$ , in the following way:

(5.9) \begin{equation}\hat{\vartheta}_{\ell,\kappa}\,:\!=\,\big[\tilde{u}_{\ell}-\tilde{u}_{\kappa}\big]\big(\bar{x}_{\ell,\kappa}\big),\quad\text{for some $\bar{x}_{\ell,\kappa}\in \mathcal{U}_{\ell,\kappa}$ fixed.}\end{equation}

If $ \mathcal{U}_{\ell,\kappa}=\emptyset$ , we set $\hat{\vartheta}_{\ell,\kappa}=0$ . We have the following result.

Lemma 5.1. If $\tilde{u}_{\ell}<\tilde{u}_{\kappa}+\hat{\vartheta}_{\ell,\kappa}$ on $(0,l)\setminus \mathcal{U}_{\ell,\kappa}$ , then $\tilde{u}_{\ell}$ is a solution to the following system of variational inequalities:

(5.10) \begin{align} \max\!\big\{[c_{{\ell}}-\mathcal{L}_{{\ell}}]\tilde{u}_{{\ell}}- h_{{\ell}},\big|\tilde{u}^{\prime}_{{\ell}}\big|- g_{{\ell}},\tilde{u}_{\ell}-\big(\tilde{u}_{\kappa}+\hat{\vartheta}_{\ell,\kappa}\big)\big\}= 0 \ \text{in}\ (0,l),&\\\text{s.t.}\ \tilde{u}_{{\ell}}(0)=\tilde{u}_{{\ell}}(l)=0.& \end{align}

Remark 5.1. Notice that if $\vartheta_{\ell,\kappa}>\hat{\vartheta}_{\ell,\kappa}$ , $\tilde{u}_{\ell}$ satisfies (5.10) with $\hat{\vartheta}_{\ell,\kappa}$ replaced by $\vartheta_{\ell,\kappa}$ , but there would be no opportunity to switch from the state $\ell$ to $\kappa$ since $\tilde{u}_{\ell}<\tilde{u}_{\kappa}+ {\vartheta}_{\ell,\kappa}$ on (0,l).

To conclude this part, let us suppose that for each $\ell\in\mathbb{I}$ fixed, $\tilde{u}_{\ell}$ satisfies the following condition:

(5.11) \begin{equation} \tilde{u}_{\ell}<\tilde{u}_{\kappa}+\hat{\vartheta}_{\ell,\kappa}\quad \text{on}\ (0,l)\setminus \mathcal{U}_{\ell,\kappa}\ \text{for each}\ \kappa\neq\ell. \end{equation}

From here, it is easy to see that

\begin{equation*} \tilde{u}_{{\ell}}-\mathcal{M}_{{\ell}}\tilde{u}=\tilde{u}_{\ell}-\min_{\kappa\neq\ell}\big\{\tilde{u}_{\kappa}+\hat{\vartheta}_{\ell,\kappa}\big\}< 0\quad \text{on}\ (0,l)\setminus\bigcup_{\kappa\neq\ell}\mathcal{U}_{\ell,\kappa}. \end{equation*}

Remark 5.2. Defining $\widetilde{\mathcal{S}}_{\ell}=\{x\in(0,l)\,:\,\tilde{u}_{\ell}=\mathcal{M}_{\ell}\tilde{u}\}$ , we have that

\begin{equation*} \widetilde{\mathcal{S}}_{\ell}=\bigcup_{\kappa\neq\ell}\mathcal{U}_{\ell,\kappa}\quad\text{for}\ \ell\in\mathbb{I}, \end{equation*}

which is the zone where it is optimal to switch from the state $\ell$ to some other state $\kappa$ .

Proposition 5.2. Let $\tilde{u}=\big(\tilde{u}_{1},\dots,\tilde{u}_{m}\big)$ be the vector function whose entrances, given by (5.4), are different and satisfy the hypotheses of Proposition 5.1 and the condition (5.11). If $\hat{\vartheta}_{\ell,\kappa}$ , defined in (5.9), satisfy the conditions (2.6)–(2.7), then $\tilde{u}$ is the unique solution to (5.1).

Remark 5.3. Taking $\widetilde{\mathcal{N}}_{\ell}\,:\!=\,\{x\in(0,l)\,:\,\big|\tilde{u}^{\prime}_{\ell}\big|-g_{\ell}=0\}$ , we have that $\widetilde{\mathcal{C}}_{\ell}\,:\!=\,(0,l)\setminus(\widetilde{\mathcal{N}}_{\ell}\cup\widetilde{\mathcal{S}}_{\ell})$ is the zone where it is not optimal to switch to any other state.

Numerical example. Consider the parameters given in Table 1. The left panel of Table 5 presents the approximate values of $\hat{\vartheta}_{\ell,\kappa}$ which are found using (5.9).

Table 5.

The table on the left presents the approximate values of $\hat{\vartheta}_{\ell,\kappa}$ . The table on the right presents the approximate values of $\hat{\vartheta}_{\ell,\bar{\kappa}}+\hat{\vartheta}_{\bar{\kappa},\kappa}$ , with $\ell\neq\kappa\neq\bar{\kappa}$ .

The right panel of Table 5 presents the approximate values of $\hat{\vartheta}_{\ell,\bar{\kappa}}+\hat{\vartheta}_{\bar{\kappa},\kappa}$ . Note that the inequalities (2.6)–(2.7) are always satisfied, say $\hat{\vartheta}_{1,2}\approx0.1368<\hat{\vartheta}_{1,3}+\hat{\vartheta}_{3,2}\approx0.1444$ . Table 6 presents the approximate values of $\bar{x}_{\ell,\kappa}$ where switching from the state $\ell$ to the state $\kappa$ is recommended.

Table 6.

Approximate values of $\bar{x}_{\ell,\kappa}$ where switching from the state $\ell$ to the state $\kappa$ is recommended.

Finally, Figure 3 shows that the function $\tilde{u}_{\ell}$ indeed satisfies the HJB equation (5.1). The figures in the left column show the plots of $[c_{\ell}-\mathcal{L}_{\ell}]\tilde{u}_{\ell}-h_{\ell}$ and $\big|\tilde{u}^{\prime}_{\ell}\big|-g_{\ell}$ on $\big(0,x^{*}_{\ell}\big)$ and $\big(x^{*}_{\ell}, l\big)$ , respectively. Note that $[c_{\ell}-\mathcal{L}_{\ell}]\tilde{u}_{\ell}-h_{\ell}<0$ on $\big(0,x^{*}_{\ell}\big)$ , and $\big|\tilde{u}^{\prime}_{\ell}\big|-g_{\ell}<0$ on $\big(x^{*}_{\ell}, l\big)$ . The figures in the right column show the plots of $\tilde{u}_{\ell}-\mathcal{M}_{\ell}\tilde{u}_{\ell}$ on (0,l). Also, we observe that

\begin{equation*} \widetilde{\mathcal{N}}_{\ell}=\big(0,{x}^{*}_{\ell}\big)\quad\text{and}\quad\widetilde{\mathcal{S}}_{\ell}=\big\{\bar{x}_{\ell,\kappa},\bar{x}_{\ell,\tilde{\kappa}}\big\},\quad \end{equation*}

for $\ell,\kappa,\tilde{\kappa}\in\mathbb{I}=\{1,2,3\}$ with $\ell\neq\kappa\neq\tilde{\kappa}$ . Suppose, for illustration, that the process has started from the point $\tilde{x}=2$ at the regime 1. Then the optimal strategy requests to switch to the regime 3 at the point ${\bar x}_{1,3}$ . Then, following the figure at the bottom right corner, the optimal strategy requests either to switch to the regime 2 at the point ${\bar x}_{3,2}$ or to the regime 1 at the point ${\bar x}_{3,1}$ , depending on which point is reached first, and so on.

Figure 3.

The figures in the left column show the plots of $[c_{\ell}-\mathcal{L}_{\ell}]\tilde{u}_{\ell}-h_{\ell}$ and $\big|\tilde{u}^{\prime}_{\ell}\big|-g_{\ell}$ on $\big(0,x^{*}_{\ell}\big)$ and $\big(x^{*}_{\ell}, l\big)$ , respectively. The figures in the right column show the plots of $\tilde{u}_{\ell}-\mathcal{M}_{\ell}\tilde{u}_{\ell}$ on (0,l).