Optimal Control for a Parabolic–Hyperbolic Free Boundary Problem Modeling the Growth of Tumor with Drug Application

Abstract

In this paper, we study two optimal control problems for a free boundary problem, which models tumor growth with drug application. This free boundary problem is a multicellular tumor spheroid model and includes five time-dependent partial differential equations. The tumor considered in this model consists of three kinds of cells: proliferative cells, quiescent cells and dead cells. Three different first-order hyperbolic equations are given, which describe the evolution of cells, and other two second-order parabolic equations describe the diffusions of nutrient (e.g., oxygen and glucose) and drug concentrations. Existence and uniqueness of optimal controls are also proved. We use tangent-normal cone techniques to obtain necessary conditions. Then, we employ the Ekeland variational principle to show that there exists unique optimal control for each optimal control problem.

Appendix A

In this section, we provide some essential mathematical concepts, lemmas and theorems, which have been used in the mathematical analysis throughout the paper.

Definition A.1

Let $R(t)~(0\le t\le T)$ be a positive continuous function; then we define

$$\begin{aligned} W_{p}^{2,1 }(Q_T^R ):=\left\{ u\in L^{p} (Q_T^R ) : \partial _x^\alpha \partial _t^k u\in L^{p} (Q_T^R ),~~ |\alpha |+2k\le 2\right\} , \end{aligned}$$

with the norm $||u||_{{W_{p}^{2,1}} (Q_T^R ) }:=\displaystyle {\sum _{|\alpha |+2k\le 2}}||\partial _x^\alpha \partial _t^k u||_{L^{p} (Q_T^R )}$, where $Q_T^R:=\Big \{(x,t)\in \mathbb {R}^{3}\times \mathbb {R}: |x|< R(t) ,0<t<T\Big \}.$

Definition A.2

Let $\Omega \subseteq \mathbb {R}^3$ be an open set and $p>\dfrac{5}{2}$; then we define $D_{p}(\Omega )$ the trace space of $W_{p}^{2,1}(\Omega \times ]0,T[)$ at $t=0$ as follows:

$$\begin{aligned} D_{p}(\Omega ):=\Big \{\varphi :\exists u\in W_{p}^{2,1}(\Omega \times ]0,T[)~s.t.~ u(., 0) = \varphi \Big \}, \end{aligned}$$

equipped with the norm $||\varphi ||_{D_p (\Omega )} := \inf \Big \{T^{-\frac{1}{p}} ||u||_{W_{p}^{2,1}(\Omega \times ]0,T[)} : u\in W_{p}^{2,1} (\Omega \times ]0,T[),~ u(.,0)=\varphi \Big \}.$

Lemma A.1

Assume that $f\left( \rho ,\tau \right) ,~\psi (\rho ,\tau )$ and $\varphi (\tau )$ are bounded continuous functions on $[0, 1] \times [0, T]$ and $\ [0, T]\ (T>0)$, respectively, and let $\overline{c}$ be a constant and ${c}_{ 0}$ be a function on [0, 1] such that $c_0(\left| x\right| )\in D_p(B_1)$ for some $p>5$, where $B_1$ is a unit ball in $\mathbb {R}^3$. Then, the problem

$$\begin{aligned} \dfrac{\partial c }{\partial \tau }= & {} \dfrac{1}{{\rho }^2}\dfrac{\partial }{\partial \rho }\left( {\rho }^2\dfrac{\partial c }{\partial \rho }\right) +\varphi \left( \tau \right) \rho \dfrac{\partial c }{\partial \rho }+\psi \left( \rho ,\tau \right) c+f(\rho ,\tau ) ,\\ 0< & {} \rho<1,~ 0< \tau \le T,\\ \dfrac{\partial c}{\partial \rho }(0,\tau )= & {} 0,~~c \left( 1,\tau \right) =\overline{c},~~ 0<\tau \le T,~c \left( \rho ,0\right) =c_0(\rho ),~~ 0\le \rho \le 1, \end{aligned}$$

has a unique solution such that $c(|x|,\tau )\in W^{2,1}_p(Q^1_T)$, where $Q^1_T:=\Big \{(x,\tau )\in \mathbb {R}^{3}\times \mathbb {R}: |x|< 1,~0<\tau < T\Big \}$. Also there exists a positive constant $\mu$ depending only on p,  T, ${||\psi ||}_{L^{ \infty }}$ and ${||\varphi ||}_{L^{\infty }}$ such that

$$\begin{aligned} {||c (|x|,\tau )||}_{W^{2,1}_p(Q^1_T)}\le \mu \left( \left| \overline{c }\right| +{\left| \left| {c }_0\left( |x|\right) \right| \right| }_{D_p\left( B_1\right) }+||f ||_{L^p}\right) , \end{aligned}$$

(64)

where $\mu$ is bounded for T in any bounded set. Moreover, there exists a positive constant d such that

$$\begin{aligned} {||\partial _x^\beta c \big (|x|,\tau \big ) ||}_{L^{\infty }}\le s_1T^{\frac{1}{2}-\frac{5}{2p}} {||c (|x|,\tau )||}_{W^{2,1}_p\big (Q^1_T\big )}+s_2\delta ^{-1-\frac{5}{p}}||c \big (|x|,\tau \big )||_{L^p}, \end{aligned}$$

(65)

where $|\beta |=1$ and $\delta =\min \{d,\sqrt{T}\}$ and $s_1$, $s_2$ are positive constants, which depend on p. Also we have

$$\begin{aligned} {||c ||}_{L^{\infty }}\le e^{\mu _0T}\Big (\max \{\left| \overline{c }\right| +{\left| \left| {c }_0\right| \right| }_{L^{\infty }}\}+T||f ||_{L^\infty }\Big ), \end{aligned}$$

(66)

where $\mu _0=0$ if $\psi \left( \rho ,\tau \right) \le 0$ and $\mu _0=\displaystyle \max _{\overline{Q^1_{T}}}\psi$ otherwise.

Proof

See [3, 4, 17, 18]. $\square$

Lemma A.2

Let $z\left( \rho ,\tau \right)$ , $h_{ij}(\rho ,\tau )\ (i, j=1,2,3)$ , $g_i(\rho ,\tau )\ (i= 1, 2, 3)$ be bounded continuous functions on $[0, 1]\times [0, T]$, $z\left( \rho ,\tau \right)$ be continuously differentiable with respect to $\rho$ and $z\left( 0,\tau \right) =z\left( 1,\tau \right) =0$. Then, for every ${\alpha }_0,\ {\beta }_0,\ {\gamma }_0\in \ C[0, 1]$, the problem

$$\begin{aligned} \dfrac{\partial \alpha }{\partial \tau }+z\left( \rho ,\tau \right) \dfrac{\partial \alpha }{\partial \rho }= & {} h_{11}\left( \rho ,\tau \right) \alpha +h_{12}\left( \rho ,\tau \right) \beta +h_{13}\left( \rho ,\tau \right) \gamma +g_1(\rho ,\tau ),\\ 0\le & {} \rho \le 1, \ 0<\tau \le T,\\ \dfrac{\partial \beta }{\partial \tau }+z\left( \rho ,\tau \right) \dfrac{\partial \beta }{\partial \rho }= & {} h_{21}\left( \rho ,\tau \right) \alpha +h_{22}\left( \rho ,\tau \right) \beta +h_{23}\left( \rho ,\tau \right) \gamma +g_2(\rho ,\tau ),\\ 0\le & {} \rho \le 1, \ 0<\tau \le T,\\ \dfrac{\partial \gamma }{\partial \tau }+z\left( \rho ,\tau \right) \dfrac{\partial \gamma }{\partial \rho }= & {} h_{31}\left( \rho ,\tau \right) \alpha +h_{32}\left( \rho ,\tau \right) \beta +h_{33}\left( \rho ,\tau \right) \gamma +g_3(\rho ,\tau ),\\ 0\le & {} \rho \le 1, \ 0<\tau \le T,\\ \alpha \left( \rho ,0\right)= & {} {\alpha }_0\left( \rho \right) ,\ \beta \left( \rho ,0\right) ={\beta }_0\left( \rho \right) ,\ \gamma \left( \rho ,0\right) ={\gamma }_0\left( \rho \right) ,~\ 0\le \rho \le 1, \end{aligned}$$

has a unique weak solution, which is continuous with respect to $(\rho ,\tau )$ and

$$\begin{aligned} {\Vert (\alpha ,\beta ,\gamma )\Vert }_{L^{\infty }}\le e^{TA_0\left( T\right) }\left( {\Vert ({\alpha }_0,{\beta }_0,{\gamma }_0)\Vert }_{L^{\infty }}+T{\Vert (g_1,g_2,g_3)\Vert }_{L^{\infty }}\right) , \end{aligned}$$

where $A_0(T)=2{max \{{\Vert h_{ij}\Vert }_{L^{\infty }}i,j=1,2,3\}}.$ If $h_{ij}(\rho ,\tau )\ (i, j=1, 2, 3)$ and $g_{i}(\rho ,\tau )\ (i= 1, 2, 3)$ are continuously differentiable with respect to $\rho$ and ${\alpha }_0,\ {\beta }_0,\ {\gamma }_0\in \ C^1[0, 1]$, then the weak solution of problem is a classical solution. Thus, we have

$$\begin{aligned} \left\| \left( \dfrac{\partial \alpha }{\partial \rho },\dfrac{\partial \beta }{\partial \rho },\dfrac{\partial \gamma }{\partial \rho }\right) \right\| _{L^{\infty }}\le & {} e^{\left( A_0\left( T\right) +A\left( T\right) \right) T}\Big (\left\| ({\alpha }_0',{\beta }_0',\gamma _0')\right\| _{L^{\infty }}\nonumber \\&+TA_1(T)e^{TA\left( T\right) }{\Vert ({\alpha }_0,{\beta }_0,{\gamma }_0)\Vert }_{L^{\infty }}\nonumber \\&+Te^{TA\left( T\right) }{\left\| \left( \dfrac{\partial g_1}{\partial \rho },\dfrac{\partial g_2}{\partial \rho },\dfrac{\partial g_3}{\partial \rho }\right) \right\| }_{L^{\infty }}\Big ), \end{aligned}$$

(67)

where $A\left( T\right) ={\Vert \frac{\partial z}{\partial \rho }\Vert }_{L^{\infty }}, A_1(T)=2{max \Big \{{\Vert \frac{\partial h_{ij}}{\partial \rho } \Vert }_{L^{\infty }},~i,j=1,2,3\Big \}}.$

Proof

See [3, 4]. $\square$

In the following theorem, the existence and uniqueness of the solution of (13)–(26) are proved.

Theorem A.1

Let assumptions A–C and initial condition (26) be satisfied. Then, the problem (13)–(25) has a unique solution for $t\ge 0$. And for every $T>0$, $R\left( t\right) \in C^1\left[ 0,T\right] ,\ C,\ W\in W^{2,1}_p\left( {Q^R_T}\right)$ for $p>5$ and $P,Q,D\in C\left( \overline{Q^R_T}\right)$. Moreover, we have

$$\begin{aligned} P\left( r,t\right) +Q\left( r,t\right) +D\left( r,t\right)= & {} N,~~ 0\le r\le R\left( t\right) ,\ t\ge 0,\\ p,q,d\in & {} C^1\left( [0,1]\times [0,T]\right) . \end{aligned}$$

Proof

Using Lemmas A.1 and A.2, similar to the proof of the main theorem of [3], the desired result can be obtained. $\square$

Definition A.3

[17, 19] Let $0<\alpha <1$ and $\Omega \subset \mathbb {R}^n$ be bounded. Then, $f\in C^{\alpha ,\frac{\alpha }{2}}(\overline{\Omega }\times [0,T])$ iff there exists a positive constant C such that

$$\begin{aligned} |f(x_1,t_1)-f(x_2,t_2)|\le C\Big (|x_1-x_2|^2+|t_1-t_2|\Big )^{\frac{\alpha }{2}},~\forall x_1,x_2\in \overline{\Omega },~\forall t_1,t_2\in [0,T]. \end{aligned}$$

Furthermore, for any nonnegative integer k we define

$$\begin{aligned} C^{2k+\alpha ,k+\frac{\alpha }{2}}\left( \overline{\Omega }\times [0,T]\right):= & {} \left\{ f\in C^{\alpha ,\frac{\alpha }{2}}\left( \overline{\Omega }\times [0,T]\right) : \partial _x^{\beta } \partial _t^{i} f\in C^{\alpha ,\frac{\alpha }{2}}\left( \overline{\Omega }\right. \right. \\&\left. \left. \times [0,T]\right) , |\beta |+2i\le 2k\right\} . \end{aligned}$$

Theorem A.2

Let $\Omega \subset \mathbb {R}^n$ be bounded with boundary $\partial \Omega \in C^\infty$ and $0<\alpha <1$ and

$$\begin{aligned}&\dfrac{\partial c}{\partial t}-\sum _{i=1}^n\sum _{j=1}^n a_{ij}(x,t)D_{ij}c+\sum _{i=1}^n b_i(x,t)D_ic+d(x,t)c=g(x,t),\nonumber \\&(x,t)\in Q_T=\Omega \times ]0,T[,\nonumber \\&c(x,t)=\psi (x,t),~~(x,t)\in \partial Q_T\setminus (\Omega \times \{t=T\}), \end{aligned}$$

(68)

where $\psi \in C^{2+\alpha ,1+\frac{\alpha }{2}}\left( \overline{\Omega }\times [0,T]\right) ,~ a_{ij},b_i,d\in C^{\alpha ,\frac{\alpha }{2}}\left( \overline{\Omega }\times [0,T]\right)$, $a_{ij}=a_{ji}$ and for constants $0<C_1\le C_2$

$$\begin{aligned} C_1|\lambda |^2\le \sum _{i=1}^n\sum _{j=1}^n a_{ij}(x,t)\lambda _i\lambda _j\le C_2|\lambda |^2,~~\forall \lambda \in \mathbb {R}^n,(x,t)\in Q_T. \end{aligned}$$

Also assume that $g\in C^{\alpha ,\frac{\alpha }{2}}\left( \overline{\Omega }\times [0,T]\right)$, then (68) has a unique solution such that

$$\begin{aligned} c,~\dfrac{\partial c}{\partial t},~\dfrac{\partial c}{\partial x_j},~\dfrac{\partial ^2 c}{\partial x_i\partial x_j}\in C^{\alpha ,\frac{\alpha }{2}}\left( \overline{\Omega }\times [0,T]\right) ,~~\forall i,j. \end{aligned}$$

Proof

See [17]. $\square$

Theorem A.3

Let X be a complete metric space and let $f : X \rightarrow ]-\infty , +\infty ]$ be lower semicontinuous and bounded from below and $\not \equiv +\infty$. Let $\epsilon > 0$ and $x_\epsilon \in X$ be such that $f(x_\epsilon )\le \inf \Big \{f(x): x\in X\Big \}+\epsilon .$ Then, there exists $y_\epsilon \in X$ such that

$$\begin{aligned} f(y_\epsilon )\le & {} f(x_\epsilon ),~~~d(x_\epsilon ,y_\epsilon )\le \epsilon ^{\frac{1}{2}},\\ f(y_\epsilon )< & {} f(x) + \epsilon ^{\frac{1}{2}}d(y_\epsilon , x),~~ \forall x \ne y_\epsilon . \end{aligned}$$

Proof

See [20]. $\square$

Theorem A.4

Let assumptions A–C and initial condition (42) be satisfied and $u_1,u_2\in C([0, 1]\times [0, T])$. Then, there exist positive constants $\delta _1$, $\delta _2$ and $\delta _3$ independent of $\tau$ such that

$$\begin{aligned} \int _0^1\rho ^2c^2(\rho ,\tau )\hbox {d}\rho\le & {} \left( \int _0^1\rho ^2c^2(\rho ,0){\text{ d }}\rho +\int _0^\tau \left( \delta _1 \left| \left( c(\rho ,s)\dfrac{\partial c(\rho ,s)}{\partial \rho }\right) \Big |_{\rho =1} \right| \right. \right. \\&\left. \left. +\delta _3\int _0^1\rho ^2({\eta } ^{2}f\left( c,p,q\right) -u_1)^2{\text{ d }}\rho \right) {\text{ ds }}\right) e^{\delta _2\int _0^\tau (|u\left( 1, s \right) |^2+1)\text{ ds }}, \end{aligned}$$

where c is the solution of (31)–(33). If in addition, $u_1(|x|,\tau ),u_2(|x|,\tau )\in C^{\alpha ,\alpha /2}(\overline{Q^1_T})$, then the problem (31)–(36) has a unique solution $(c(\rho ,\tau ), w(\rho ,\tau ))$ such that $\dfrac{\partial ^\kappa }{\partial \rho ^\kappa }\dfrac{\partial ^z}{\partial \tau ^z}c$ and $\dfrac{\partial ^\kappa }{\partial \rho ^\kappa }\dfrac{\partial ^z}{\partial \tau ^z}w$ are continuous functions on $]0,1[\times ]0,T[$, for each $\kappa ,z\in \mathbb {N}_0$ with $\kappa +2 z\le 2$.

Proof

From [17], we easily conclude that the problem (31)–(33) has a unique solution c such that $c(|x|,\tau )\in W^{2,1}_p(Q^1_T)$. Therefore,

$$\begin{aligned} \int _0^1\rho ^2c\dfrac{\partial c}{\partial \tau }{\text{ d }}\rho= & {} D_{1}\int _0^1c\dfrac{\partial }{\partial \rho }\left( {\rho }^{2}\dfrac{\partial c}{\partial \rho }\right) {\text{ d }}\rho +u\left( 1, \tau \right) \int _0^1 \rho ^3c \dfrac{\partial c}{\partial \rho }{\text{ d }}\rho \\&-\int _0^1\rho ^2c({\eta } ^{2}f\left( c,p,q\right) -u_1){\text{ d }}\rho \\= & {} -D_{1}\int _0^1\rho ^2(\dfrac{\partial c}{\partial \rho })^2{\text{ d }}\rho +D_{1}\left( c(\rho ,\tau )\dfrac{\partial c(\rho ,\tau )}{\partial \rho }\right) \Big |_{\rho =1}\\&+u\left( 1, \tau \right) \int _0^1 \rho ^3c \dfrac{\partial c}{\partial \rho }{\text{ d }}\rho -\int _0^1\rho ^2c({\eta } ^{2}f\left( c,p,q\right) -u_1){\text{ d }}\rho . \end{aligned}$$

Thus,

$$\begin{aligned} \int _0^\tau \int _0^1\rho ^2c\dfrac{\partial c}{\partial s }{\text{ d }}\rho {\text{ d }}s= & {} \int _0^\tau ( -D_{1}\int _0^1\rho ^2(\dfrac{\partial c}{\partial \rho })^2{\text{ d }}\rho +D_{1}\left( c(\rho ,s)\dfrac{\partial c(\rho ,s)}{\partial \rho }\right) \Big |_{\rho =1}\\&+u\left( 1, s \right) \int _0^1 \rho ^3c \dfrac{\partial c}{\partial \rho }{\text{ d }}\rho -\int _0^1\rho ^2c({\eta } ^{2}f\left( c,p,q\right) -u_1){\text{ d }}\rho ){\text{ d }}s. \end{aligned}$$

Therefore, we arrive at

$$\begin{aligned} \int _0^1\rho ^2c^2(\rho ,\tau ){\text{ d }}\rho= & {} \int _0^1\rho ^2c^2(\rho ,0){\text{ d }}\rho + 2\int _0^\tau ( -D_{1}\int _0^1\rho ^2\left( \dfrac{\partial c}{\partial \rho }\right) ^2{\text{ d }}\rho \\&+D_{1}\left( c(\rho ,s)\dfrac{\partial c(\rho ,s)}{\partial \rho }\right) \Big |_{\rho =1} +u\left( 1, s \right) \int _0^1 \rho ^3c \dfrac{\partial c}{\partial \rho }{\text{ d }}\rho \\&-\int _0^1\rho ^2c\left( {\eta } ^{2}f\left( c,p,q\right) -u_1){\text{ d }}\rho \right) {\text{ d }}s. \end{aligned}$$

Hence, there exist positive constants $\delta _1$, $\delta _2$ and $\delta _3$ independent of $\tau$ such that

$$\begin{aligned} \int _0^1\rho ^2c^2(\rho ,\tau ){\text{ d }}\rho\le & {} \int _0^1\rho ^2c^2(\rho ,0){\text{ d }}\rho +\delta _1\int _0^\tau \left| \left( c(\rho ,s)\dfrac{\partial c(\rho ,s)}{\partial \rho }\right) \Big |_{\rho =1}\right| {\text{ d }}s\\&+\int _0^\tau \left( \delta _2(|u\left( 1,s \right) |^2+1)\int _0^1 \rho ^2c^2{\text{ d }}\rho \right. \\&\left. +\delta _3\int _0^1\rho ^2({\eta } ^{2}f\left( c,p,q\right) -u_1)^2{\text{ d }}\rho \right) {\text{ d }}s. \end{aligned}$$

Now from the Gronwall’s inequality we conclude that

$$\begin{aligned} \int _0^1\rho ^2c^2(\rho ,\tau ){\text{ d }}\rho\le & {} \left( \int _0^1\rho ^2c^2(\rho ,0){\text{ d }}\rho +\int _0^\tau \left( \delta _1\left| \left( c(\rho ,s)\dfrac{\partial c(\rho ,s)}{\partial \rho }\right) \Big |_{\rho =1}\right| \right. \right. \\&\left. \left. +\delta _3\int _0^1\rho ^2({\eta } ^{2}f \left( c,p,q\right) -u_1)^2{\text{ d }}\rho \right) {\text{ d }}s\right) {e}^{\delta _2 \int _0^\tau (|u\left( 1, s \right) |^2+1){\text{ d }}s}. \end{aligned}$$

Also from (31)–(36), Theorem A.1 and t-Anisotropic Embedding Theorem [17], one can easily conclude that there exist functions $f^1(|x|,\tau )\in C^{\alpha ,\frac{\alpha }{2}}(\overline{Q^1_T})$ and $g^1(|x|,\tau )\in C^{\alpha ,\frac{\alpha }{2}}(\overline{Q^1_T})$, where $Q^1_T$ is defined in Lemma A.1, such that $\widehat{c}(x,\tau )=c(|x|,\tau )$ and $\widehat{w}(x,\tau )=w(|x|,\tau )$ for $|x|<1,0<\tau <T$ are the solutions of

$$\begin{aligned}&\dfrac{\partial \widehat{c}}{\partial \tau }-D_1\Delta \widehat{c}-u\left( 1, \tau \right) x.\nabla \widehat{c} =f^1(|x|,\tau ),~~|x|<1,~0<\tau \le T,\\&\widehat{c}(x,\tau )=c_1(\tau )+\overline{c},~~|x|=1,~0<\tau \le T,~\widehat{c}(x,0)=c_0(|x|)+\overline{c},~~|x|\le 1, \end{aligned}$$

and

$$\begin{aligned}&\dfrac{\partial \widehat{w}}{\partial \tau }-D_2\Delta \widehat{w}-u\left( 1, \tau \right) x.\nabla \widehat{w} =g^1(|x|,\tau ),~~|x|<1,~0<\tau \le T,\\&\widehat{w}(x,\tau )=w_1(\tau )+\overline{w},~~|x|=1,~0<\tau \le T,~\widehat{w}(x,0)=w_0(|x|)+\overline{w},~~|x|\le 1, \end{aligned}$$

respectively. Using Theorem A.2, assumptions A–C and Theorem A.1 one can easily arrive at

$$\begin{aligned} \widehat{c},~\dfrac{\partial \widehat{c}}{\partial \tau },~\dfrac{\partial \widehat{c}}{\partial x_j},~\dfrac{\partial ^2 \widehat{c}}{\partial x_i\partial x_j},~\widehat{w},~\dfrac{\partial \widehat{w}}{\partial \tau },~\dfrac{\partial \widehat{w}}{\partial x_j},~\dfrac{\partial ^2 \widehat{w}}{\partial x_i\partial x_j}\in C^{\alpha ,\frac{\alpha }{2}}\left( \overline{Q^1_T}\right) ,~~\forall i,j. \end{aligned}$$

Therefore, we get the desired result. $\square$

Theorem A.5

Let assumptions A–C and condition (42) be satisfied, also $(c_1,w_1,p_1,q_1,d_1,\eta _1)$ and $(c_2,w_2,p_2,q_2,d_2,\eta _2)$ be the exact solutions of the problem (31)–(41) corresponding to $(u^1_1,u^1_2,\overline{c}^1,\overline{w}^1)\in \mathcal {C}_\mathrm{{ad}}\times \mathcal {W}_\mathrm{{ad}}\times \mathcal {C}^1_\mathrm{{ad}}\times \mathcal {W}^1_\mathrm{{ad}}$ and$(u^2_1,u^2_2,\overline{c}^2,\overline{w}^2)\in \mathcal {C}_\mathrm{{ad}}\times \mathcal {W}_\mathrm{{ad}}\times \mathcal {C}^1_\mathrm{{ad}}\times \mathcal {W}^1_\mathrm{{ad}}$, respectively, where $\mathcal {C}_\mathrm{{ad}}\times \mathcal {W}_\mathrm{{ad}}\times \mathcal {C}^1_\mathrm{{ad}}\times \mathcal {W}^1_\mathrm{{ad}}$ is admissible control set for FOCP (or SOCP). Then, there exist positive constants $\mu _1$, $\mu _1^*$, $\beta _1$, $\lambda _0$ and $\lambda$, which are independent of $\eta _0$ such that for , where

(69)

we have

$$\begin{aligned}&{\Vert (c_1-c_2,w_1-w_2,p_1-p_2,q_1-q_2,d_1-d_2,\eta _1-\eta _2) \Vert }_{L^{\infty }}\\&\quad \le \lambda _0{\Vert \left( u^1_1-u^2_1,u^1_2-u^2_2, \overline{c}^1-\overline{c}^2,\overline{w}^1-\overline{w}^2\right) \Vert }_{L^{\infty }},\\&{\Vert \rho (c_1-c_2,w_1-w_2,p_1-p_2,q_1-q_2,d_1-d_2,\eta _1-\eta _2) \Vert }_{L^2}\\&\quad \le \lambda {\Vert \rho \left( u^1_1-u^2_1,u^1_2-u^2_2,\overline{c}^1 -\overline{c}^2,\overline{w}^1-\overline{w}^2\right) \Vert }_{L^2}. \end{aligned}$$

Proof

Let $X_1:=\{(c,w,p,q,d,\eta ):\alpha \in C([0, 1]\times [0, T]), ~for~\alpha =c,w,p,q,d,\eta \}$. Clearly $(X_1,\Vert .\Vert _{L^\infty })$ is a Banach space. Now, using (31)–(41) we can define the operator $F^{(u_1,u_2,\overline{c},\overline{w})}_1:X_1\rightarrow X_1$, such that $(c_1,w_1,p_1,q_1,d_1,\eta _1)$, the exact solution of the problem (31)–(41) corresponding to $(u^1_1,u^1_2,\overline{c}^1,\overline{w}^1)$, is the fixed point of $F^{(u^1_1,u^1_2,\overline{c}^1,\overline{w}^1)}_1$ and also $(c_2,w_2,p_2,q_2,d_2,\eta _2)$, the exact solution of the problem (31)–(41) corresponding to $(u^2_1,u^2_2,\overline{c}^2,\overline{w}^2)$, is the fixed point of $F^{(u^2_1,u^2_2,\overline{c}^2,\overline{w}^2)}_1$. Finally, by choosing appropriate $(c^{0}_1,w^{0}_1,p^{0}_1,q^{0}_1,d^{0}_1,\eta ^{0}_1)\in X_1$, $(c^{0}_2,w^{0}_2,p^{0}_2,q^{0}_2,d^{0}_2,\eta ^{0}_2)\in X_1$ and $F^{(u_1,u_2,\overline{c},\overline{w})}_1$, using Lemmas A.1 and A.2 and Theorem A.4, we can prove that the sequences $\{(c^n_1,w^n_1,p^n_1,q^n_1,d^n_1,\eta ^n_1)\}_{n=0}^{\infty }$ and $\{(c^n_2,w^n_2,p^n_2,q^n_2,d^n_2,\eta ^n_2)\}_{n=0}^{\infty }$ with

$$\begin{aligned} \left( c^{n+1}_1,w^{n+1}_1,p^{n+1}_1,q^{n+1}_1,d^{n+1}_1,\eta ^{n+1}_1\right)= & {} F^{\left( u^1_1,u^1_2,\overline{c}^1,\overline{w}^1\right) }_1\left( c^{n}_1, w^{n}_1,p^{n}_1,q^{n}_1,d^{n}_1,\eta ^{n}_1\right) ,\\ \left( c^{n+1}_2,w^{n+1}_2,p^{n+1}_2,q^{n+1}_2,d^{n+1}_2,\eta ^{n+1}_2\right)= & {} F^{\left( u^2_1,u^2_2,\overline{c}^2,\overline{w}^2\right) }_1\left( c^{n}_2, w^{n}_2,p^{n}_2,q^{n}_2,d^{n}_2,\eta ^{n}_2\right) , \end{aligned}$$

converge to $(c_1,w_1,p_1,q_1,d_1,\eta _1)$ and $(c_2,w_2,p_2,q_2,d_2,\eta _2)$, respectively, and there exist positive constants $\lambda ^n_0<\lambda _0$ and $\lambda ^n<\lambda$ such that

$$\begin{aligned}&\Vert \left( c^{n+1}_1-c^{n+1}_2,w^{n+1}_1-w^{n+1}_2,p^{n+1}_1-p^{n+1}_2,q^{n+1}_1 -q^{n+1}_2,\right. \\&\quad \left. d^{n+1}_1{-}d^{n+1}_2,\eta ^{n+1}_1-\eta ^{n+1}_2\right) \Vert _{L^{\infty }}\le \lambda ^n_0{\Vert \left( u^1_1-u^2_1,u^1_2{-}u^2_2,\overline{c}^1 -\overline{c}^2,\overline{w}^1-\overline{w}^2\right) \Vert }_{L^{\infty }},\\&\Vert \rho \left( c^{n+1}_1-c^{n+1}_2,w^{n+1}_1-w^{n+1}_2,p^{n+1}_1-p^{n+1}_2,q^{n+1}_1 -q^{n+1}_2,\right. \\&\left. \quad d^{n+1}_1-d^{n+1}_2,\eta _1^{n{+}1}{-}\eta _2^{n+1}\right) \Vert _{L^2} \le \lambda ^n{\Vert \rho \left( u^1_1{-}u^2_1,u^1_2-u^2_2,\overline{c}^1 {-}\overline{c}^2,\overline{w}^1-\overline{w}^2\right) \Vert }_{L^2}. \end{aligned}$$

Therefore, we can obtain the result. $\square$

The next theorem shows that there exists a unique solution for the problem (49)–(59).

Theorem A.6

Let assumptions A–C be satisfied. Then, for $t\ge 0$ the adjoint system (49)–(59) has a unique solution $\left( z_c,z_w,z_p,z_q,z_d,z_\eta \right)$ and for every $T>0$ we have

$$\begin{aligned}&z_c(|x|,\tau ),z_w(|x|,\tau )\in W^{2,1}_p\left( Q^1_T\right) ,~p>5,~z_\eta \left( \tau \right) \in C^1\left[ 0,T\right] ,\\&\quad _p,z_q,z_d\in C^1\left( [0,1]\times [0,T]\right) , \end{aligned}$$

where $Q^1_T$ is defined in Lemma A.1.

Proof

Using the change of variable $t=T-\tau$, employing Lemmas A.1 and A.2 and Theorem A.1, in a similar manner to the proof of the main theorem of [3], we obtain the result. $\square$

Theorem A.7

Let assumptions A–C and initial condition (42) be satisfied, also assume that $(z_c^1,z_w^1,z_p^1,z_q^1,z_d^1,z_\eta ^1)$ and $(z_c^2,z_w^2,z_p^2,z_q^2,z_d^2,z_\eta ^2)$ are the exact solutions of adjoint system (49)–(59) corresponding to the controls$(u^1_1,u^1_2,\overline{c}^1,\overline{w}^1)\in \mathcal {C}_\mathrm{{ad}}\times \mathcal {W}_\mathrm{{ad}}\times \mathcal {C}^1_\mathrm{{ad}}\times \mathcal {W}^1_\mathrm{{ad}}$ and $(u^2_1,u^2_2,\overline{c}^2,\overline{w}^2)\in \mathcal {C}_\mathrm{{ad}}\times \mathcal {W}_\mathrm{{ad}}\times \mathcal {C}^1_\mathrm{{ad}}\times \mathcal {W}^1_\mathrm{{ad}}$, respectively, where$\mathcal {C}_\mathrm{{ad}}\times \mathcal {W}_\mathrm{{ad}}\times \mathcal {C}^1_\mathrm{{ad}}\times \mathcal {W}^1_\mathrm{{ad}}$ is admissible control set for FOCP (or SOCP). Then, there exist positive constants $\mu _2$, $\mu _2^*$, $\beta _2$, $\lambda ^*_1$ and $\lambda ^*$, which are independent of $\eta _0$ such that for , where

(70)

and $K_1$ is defined in Theorem A.5, we have

$$\begin{aligned}&{\Vert \left( z_c^1-z_c^2,z_w^1-z_w^2,z_p^1-z_p^2,z_q^1-z_q^2,z_d^1-z_d^2,z_\eta ^1 -z_\eta ^2\right) \Vert }_{L^{\infty }}\\&\quad \le \lambda ^*{\Vert \left( u^1_1-u^2_1,u^1_2-u^2_2,\overline{c}^1 -\overline{c}^2,\overline{w}^1-\overline{w}^2\right) \Vert }_{L^{\infty }},\\&{\Vert \rho \left( z_c^1-z_c^2,z_w^1-z_w^2,z_p^1-z_p^2,z_q^1-z_q^2,z_d^1 -z_d^2,z_\eta ^1-z_\eta ^2\right) \Vert }_{L^2}\\&\quad \le \lambda ^*_1{\Vert \rho \left( u^1_1-u^2_1,u^1_2-u^2_2,\overline{c}^1 -\overline{c}^2,\overline{w}^1-\overline{w}^2\right) \Vert }_{L^2}, \end{aligned}$$

where $\lambda ^*_1,\lambda ^*\rightarrow 0$ if $T\rightarrow 0$.

Proof

Similar to the proof of Theorem A.5, the proof is concluded. $\square$

In the following, we provide the proofs of the theorems, which are addressed in Sect. 2.2.

Proof of Theorem 2.1

Let $\left( c^*,w^*,p^*,q^*,d^*,\eta ^*\right)$ and $\left( c^{\epsilon },w^{\epsilon },p^{\epsilon },q^{\epsilon },d^{\epsilon },\eta ^{\epsilon }\right)$ be the solutions of (31)–(41) corresponding to the controls $(u_1^*,u_2^*,{\overline{c}}^*,{\overline{w}}^*)$ and $(u_1^{\epsilon },u_2^{\epsilon },{\overline{c}}^\epsilon ,{\overline{w}}^\epsilon ) =(u_1^*+\epsilon u_1^0,u_2^*+\epsilon u_2^0,{\overline{c}}^*+\epsilon {\overline{c}}_0,{\overline{w}}^*+\epsilon {\overline{w}}_0) \in \mathcal {C}_\mathrm{{ad}}\times \mathcal {W}_\mathrm{{ad}}\times \mathcal {C}^1_\mathrm{{ad}}\times \mathcal {W}^1_\mathrm{{ad}}$, respectively, where $\mathcal {C}_\mathrm{{ad}}$, $\mathcal {W}_\mathrm{{ad}}$, $\mathcal {C}^1_\mathrm{{ad}}$ and $\mathcal {W}^1_\mathrm{{ad}}$ are defined in (28)–(30) and $\epsilon$ is a positive constant. Also assume that$J(u_1^*,u_2^*,{\overline{c}}^*,{\overline{w}}^*)=\min \{J(u_1,u_2,\overline{c},\overline{w}):(u_1,u_2,\overline{c},\overline{w})\in \mathcal {C}_\mathrm{{ad}}\times \mathcal {W}_\mathrm{{ad}}\times \mathcal {C}^1_\mathrm{{ad}}\times \mathcal {W}^1_\mathrm{{ad}}\}$, $c^{\epsilon }_1=\dfrac{c^{\epsilon }-c^*}{\epsilon },~w^{\epsilon }_1=\dfrac{w^{\epsilon }-w^*}{\epsilon },$ $p^{\epsilon }_1=\dfrac{p^{\epsilon }-p^*}{\epsilon },~q^{\epsilon }_1=\dfrac{q^{\epsilon }-q^*}{\epsilon },~d^{\epsilon }_1=\dfrac{d^{\epsilon }-d^*}{\epsilon }$ and $\eta ^{\epsilon }_1=\dfrac{\eta ^{\epsilon }-\eta ^*}{\epsilon }.$ Therefore, one can arrive at

$$\begin{aligned} \dfrac{J(u_1^{\epsilon },u_2^{\epsilon },{\overline{c}}^\epsilon ,{\overline{w}}^\epsilon ) -J(u_1^*,u_2^*,{\overline{c}}^*,{\overline{w}}^*)}{\epsilon }= & {} \Big (\int _0^1\int _0^T\rho ^2\Big (\lambda _1\Big (p^*+p^{\epsilon }\Big )p^{\epsilon }_1 \nonumber \\&+\lambda _2\Big (q^*+q^{\epsilon }\Big )q^{\epsilon _1}+\left( u_1^*+u_1^{\epsilon } -2r_1^*\right) u_1^0\nonumber \\&+\Big (u_2^*+u_2^{\epsilon }-2r_2^*\Big )u_2^0\Big ){\text{ d }}\rho {\text{ d }}\tau \nonumber \\&+\Big ({\overline{c}}^*+{\overline{c}}^\epsilon -2r_3^*\Big ){\overline{c}}_0 +\Big ({\overline{w}}^*+{\overline{w}}^\epsilon \nonumber \\&-2r_4^*\Big ){\overline{w}}_0\Big )\ge 0, \end{aligned}$$

(71)

and

$$\begin{aligned} \dfrac{\partial c^{\epsilon }_1}{\partial \tau }= & {} D_{1}\dfrac{1}{\rho ^2}\dfrac{\partial }{\partial \rho }\left( {\rho }^{2}\dfrac{\partial c^{\epsilon }_1}{\partial \rho }\right) +u^{\epsilon }\left( 1, \tau \right) \rho \dfrac{\partial c^{\epsilon }_1}{\partial \rho }+u^{\epsilon }_1\left( 1, \tau \right) \rho \dfrac{\partial c^*}{\partial \rho }\nonumber \\&-f^{\epsilon }_1\left( \eta ^{\epsilon },c^{\epsilon },p^{\epsilon }, q^{\epsilon },\eta ^*,c^*,p^*,q^*\right) +u^0_{1}, \mathrm{\ 0}<\rho <1,\ \tau \mathrm{>}0, \end{aligned}$$

(72)

$$\begin{aligned} \dfrac{\partial c^{\epsilon }_1}{\partial \rho }\left( 0,\tau \right)= & {} 0, ~c^{\epsilon }_1\left( \mathrm{1,}\tau \right) \mathrm{=}{\overline{c}}_0, ~~\ \tau \mathrm{>}0, \end{aligned}$$

(73)

$$\begin{aligned} c^{\epsilon }_1\left( \rho ,0\right)= & {} \overline{c}_0,~\mathrm{\ 0}\le \rho \le \mathrm{1,} \end{aligned}$$

(74)

$$\begin{aligned} \dfrac{\partial w^{\epsilon }_1}{\partial \tau }= & {} D_{2}\dfrac{1}{{\rho }^{2}}\dfrac{\partial }{\partial \rho }\left( {\rho }^{2}\dfrac{\partial w^{\epsilon }_1}{\partial \rho }\right) +u^{\epsilon }\left( 1, \tau \right) \rho \dfrac{\partial w^{\epsilon }_1}{\partial \rho }+u^{\epsilon }_1\left( 1, \tau \right) \rho \dfrac{\partial w^*}{\partial \rho }\nonumber \\&-g^{\epsilon }_1\left( \eta ^{\epsilon },w^{\epsilon },p^{\epsilon }, q^{\epsilon },\eta ^*,w^*,p^*,q^*\right) +u^0_{2}, \mathrm{\ 0}<\rho <1,\ \tau \mathrm{>}0, \end{aligned}$$

(75)

$$\begin{aligned} \dfrac{\partial w^{\epsilon }_1}{\partial \rho }\left( 0,\tau \right)= & {} 0,~w^{\epsilon }_1\left( \mathrm{1,}\tau \right) ={\overline{w}}_0, ~~\ \tau \mathrm{>}0, \end{aligned}$$

(76)

$$\begin{aligned} w^{\epsilon }_1\left( \rho ,0\right)= & {} {\overline{w}}_0,~\mathrm{\ \ 0}\le \rho \le \mathrm{1,} \end{aligned}$$

(77)

$$\begin{aligned} \dfrac{\partial p^{\epsilon }_1}{\partial \tau }\mathrm{+}v^{\epsilon }\dfrac{\partial p^{\epsilon }_1}{\partial \rho }+\mathrm{}v^{\epsilon }_1\dfrac{\partial p^*}{\partial \rho }= & {} G^{\epsilon }_1\left( \eta ^{\epsilon },c^{\epsilon },w^{\epsilon },p^{\epsilon },q^{\epsilon },d^{\epsilon },\eta ^*,c^*,w^*,p^*,q^*,d^*\right) \mathrm{, }\nonumber \\ \mathrm{0}\le & {} \rho \le \mathrm{1,\ }\tau \mathrm{>}0, \end{aligned}$$

(78)

$$\begin{aligned} \dfrac{\partial q^{\epsilon }_1}{\partial \tau }\mathrm{+}v^{\epsilon }\dfrac{\partial q^{\epsilon }_1}{\partial \rho }+\mathrm{}v^{{\epsilon }}_1\dfrac{\partial q^*}{\partial \rho }= & {} G^{\epsilon }_2\left( \eta ^{\epsilon },c^{\epsilon },w^{\epsilon }, p^{\epsilon },q^{\epsilon },d^{\epsilon },\eta ^*,c^*,w^*,p^*,q^*,d^*\right) \mathrm{, }\nonumber \\ \mathrm{0}\le & {} \rho \le \mathrm{1,\ }\tau \mathrm{>}0, \end{aligned}$$

(79)

$$\begin{aligned} \dfrac{\partial d^{\epsilon }_1}{\partial \tau }\mathrm{+}v^{\epsilon }\dfrac{\partial d^{\epsilon }_1}{\partial \rho }+\mathrm{}v^{\epsilon }_1\dfrac{\partial d^*}{\partial \rho }= & {} G^{\epsilon }_3\left( \eta ^{\epsilon },c^{\epsilon },w^{\epsilon }, p^{\epsilon },q^{\epsilon },d^{\epsilon },\eta ^*,c^*,w^*,p^*, q^*,d^*\right) \mathrm{, }\nonumber \\ \mathrm{0}\le & {} \rho \le \mathrm{1,\ }\tau \mathrm{>}0, \end{aligned}$$

(80)

$$\begin{aligned} p^\epsilon _1\left( \rho ,0\right)= & {} 0,~~~q^\epsilon _1\left( \rho ,0\right) =0,~~~ d^\epsilon _1\left( \rho ,0\right) =0,~~~~\mathrm{0}\le \rho \le \mathrm{1,\ } \end{aligned}$$

(81)

$$\begin{aligned} \dfrac{\mathrm{1}}{{\rho }^{\mathrm{2}}}\dfrac{\partial }{\partial \rho }\left( {\rho }^{\mathrm{2}}u^{\epsilon }_1\right)= & {} h^{\epsilon }_1\left( \eta ^{\epsilon },c^{\epsilon },w^{\epsilon }, p^{\epsilon },q^{\epsilon },d^{\epsilon },\eta ^*,c^*,w^*,p^*,q^*,d^*\right) ,\nonumber \\ \mathrm{0}< & {} \rho \le \mathrm{1,\ }\tau \mathrm{>}0,\nonumber \\ \dfrac{\mathrm{1}}{{\rho }^{\mathrm{2}}}\dfrac{\partial }{\partial \rho }\left( {\rho }^{\mathrm{2}}u^{\epsilon }\right)= & {} {\eta ^{\epsilon }}^2h\left( c^{\epsilon },w^{\epsilon },p^{\epsilon }, q^{\epsilon },d^{\epsilon }\right) ,~~~\mathrm{0}< \rho \le \mathrm{1,\ }\tau \mathrm{>}0,\nonumber \\ \dfrac{d\eta ^{\epsilon }_1 (\tau )}{d\tau }= & {} \eta ^{\epsilon }_1(\tau )u^\epsilon ( 1,\tau )+\eta ^* (\tau )u^{\epsilon }_1( 1,\tau ),~~~\tau>0,\\ \eta ^{\epsilon }_1(0)= & {} 0,~~u^{\epsilon }_1\left( 0,\tau \right) \mathrm{=0,\ \ }~~u^{\epsilon }\left( 0,\tau \right) \mathrm{=0,\ \ }~\tau>0,\nonumber \\ v^{\epsilon }_1\left( \rho ,\tau \right)= & {} u^{\epsilon }_1\left( \rho ,\tau \right) \mathrm{-}\rho u^{\epsilon }_1\left( \mathrm{1,}\tau \right) ,~~v^{\epsilon }\left( \rho ,\tau \right) \mathrm{=}u^{\epsilon }\left( \rho ,\tau \right) \mathrm{-}\rho u^{\epsilon }\left( \mathrm{1,}\tau \right) ,\nonumber \\ \mathrm{0}\le & {} \rho \le \mathrm{1,\ }\tau \mathrm{>}0,\nonumber \end{aligned}$$

(82)

where

$$\begin{aligned}&f^{\epsilon }_1\left( \eta ^{\epsilon },c^{\epsilon },p^{\epsilon }, q^{\epsilon },\eta ^*,c^*,p^*,q^*\right) =\dfrac{{\eta ^{\epsilon }}^2(\tau )f \left( c^{\epsilon },p^{\epsilon },q^{\epsilon }\right) -{\eta ^*}^2(\tau )f\left( c^*,p^*,q^*\right) }{\epsilon },\\&g^{\epsilon }_1\left( \eta ^{\epsilon },w^{\epsilon },p^{\epsilon }, q^{\epsilon },\eta ^*,w^*,p^*,q^*\right) =\dfrac{{\eta ^{\epsilon }}^2(\tau )g \left( w^{\epsilon },p^{\epsilon },q^{\epsilon }\right) -{\eta ^*}^2(\tau )g\left( w^*,p^*,q^*\right) }{\epsilon },\\&G_i^{\epsilon }\left( \eta ^{\epsilon },c^{\epsilon },w^{\epsilon }, p^{\epsilon },q^{\epsilon },d^{\epsilon },\eta ^*,c^*,w^*,p^*,q^*,d^*\right) \\&\quad = \dfrac{ G_i(\eta ^{\epsilon },c^{\epsilon },w^{\epsilon },p^{\epsilon }, q^{\epsilon },d^{\epsilon })-G_i(\eta ^*,c^*,w^*,p^*,q^*,d^*)}{\epsilon },~~ i=1,2,3,\\&h^{\epsilon }_1\left( \eta ^{\epsilon },c^{\epsilon },w^{\epsilon },p^{\epsilon }, q^{\epsilon },d^{\epsilon },\eta ^*,c^*,w^*,p^*,q^*,d^*\right) \\&\quad = \dfrac{{\eta ^{\epsilon }}^2 h\left( c^{\epsilon },w^{\epsilon }, p^{\epsilon },q^{\epsilon },d^{\epsilon }\right) -{\eta ^*}^2 h \left( c^*,w^*,p^*,q^*,d^*\right) }{\epsilon }. \end{aligned}$$

Therefore, using (49)–(59), assumption A, Theorem A.5 and Lemmas A.1 and A.2 we deduce that

$$\begin{aligned}&\int _0^T\int _0^1\rho ^2 \Big (c^\epsilon _1\dfrac{\partial z_c}{\partial \tau } +z_c\dfrac{\partial c^\epsilon _1}{\partial \tau } +w^\epsilon _1\dfrac{\partial z_w}{\partial \tau } +z_w\dfrac{\partial w^\epsilon _1}{\partial \tau } +p^\epsilon _1\dfrac{\partial z_p}{\partial \tau } +z_p\dfrac{\partial p^\epsilon _1}{\partial \tau } +q^\epsilon _1\dfrac{\partial z_q}{\partial \tau }\\&\quad +z_q\dfrac{\partial q^\epsilon _1}{\partial \tau } +d^\epsilon _1\dfrac{\partial z_d}{\partial \tau } +z_d\dfrac{\partial d^\epsilon _1}{\partial \tau }\Big ) {\text{ d }}\rho {\text{ d }}\tau +\int _0^T\left( \eta ^\epsilon _1\dfrac{d z_\eta }{d \tau }+z_\eta \dfrac{d \eta ^\epsilon _1}{d \tau }\right) {\text{ d }}\tau =\\&\quad -\int _0^T\int _0^1\rho ^2\Big (\lambda _1p^*p^\epsilon _1 +\lambda _2q^*q^\epsilon _1-z_cu_1^0- z_wu_2^0\Big ){\text{ d }}\rho {\text{ d }}\tau +\varpi (\epsilon ,\rho , T)\\&\quad -\int _0^TD_1\dfrac{\partial z_c}{\partial \rho }(1,\tau )\overline{c}_0+D_2\dfrac{\partial z_w}{\partial \rho }(1,\tau )\overline{w}_0{\text{ d }}\tau , \end{aligned}$$

where $\displaystyle {\lim _{\epsilon \rightarrow 0}}\varpi (\epsilon ,\rho , T)=0$. Thus, we have

$$\begin{aligned}&\int _0^T\int _0^1\rho ^2\Big (\lambda _1p^*p^\epsilon _1+\lambda _2q^*q^\epsilon _1-z_cu_1^0- z_wu_2^0\Big ){\text{ d }}\rho {\text{ d }}\tau -\varpi (\epsilon ,\rho , T)\nonumber \\&\quad +\int _0^TD_1\dfrac{\partial z_c}{\partial \rho }(1,\tau )\overline{c}_0+D_2\dfrac{\partial z_w}{\partial \rho }(1,\tau )\overline{w}_0{\text{ d }}\tau \nonumber \\&\quad =-\int _0^T\int _0^1\rho ^2 \Big (\dfrac{\partial z_cc^\epsilon _1}{\partial \tau } +\dfrac{\partial z_ww^\epsilon _1}{\partial \tau } +\dfrac{\partial z_pp^\epsilon _1}{\partial \tau } +\dfrac{\partial z_qq^\epsilon _1}{\partial \tau }+\dfrac{\partial z_dd^\epsilon _1}{\partial \tau }\Big ){\text{ d }}\rho {\text{ d }}\tau \nonumber \\&\qquad -\int _0^T\dfrac{{\hbox {d}} z_\eta \eta ^\epsilon _1}{\hbox {d} \tau } {\text{ d }}\tau =\int _0^1\rho ^2 \Big ( z_c(\rho ,0)\overline{c}_0 + z_w(\rho ,0)\overline{w}_0\Big ){\text{ d }}\rho . \end{aligned}$$

(83)

So using (71) and (83), we arrive at

$$\begin{aligned}&\int _0^1\int _0^T\rho ^2\Big (( z_c+u_1^*-r_1^*)u_1^0+( z_w+u_2^*-r_2^*)u_2^0\Big ){\text{ d }}\rho {\text{ d }}\tau \nonumber \\&\quad +\left( \overline{c}^*-r_3^*-\int _0^T D_1 \dfrac{\partial z_c}{\partial \rho }(1,\tau )d\tau +\int _0^1 \rho ^2 z_c(\rho ,0)d\rho \right) {\overline{c}}_0\nonumber \\&\quad +\left( \overline{w}^*-r_4^*-\int _0^T D_2 \dfrac{\partial z_w}{\partial \rho }(1,\tau )d\tau +\int _0^1\rho ^2 z_w(\rho ,0)d \rho \right) {\overline{w}}_0 \ge 0. \end{aligned}$$

(84)

Therefore, using tangent-normal cone techniques [21] and (28)–(30), $(u_1^*,u_2^*,{\overline{c}}^*,{\overline{w}}^*)$ is as follows:

$$\begin{aligned} u_1^*= & {} \mathcal {F}_1(-z_c+r_1^*),~u_2^*=\mathcal {F}_2(-z_w+r_2^*),\\ {\overline{c}}^*= & {} \mathcal {F}_3\left( \int _0^T D_1\dfrac{\partial z_c}{\partial \rho }(1,\tau ){\text{ d }}\tau +r_3^*-\int _0^1\rho ^2 z_c(\rho ,0){\text{ d }}\rho \right) ,\\ {\overline{w}}^*= & {} \mathcal {F}_4\left( \int _0^T D_2\dfrac{\partial z_w}{\partial \rho }(1,\tau ){\text{ d }}\tau +r_4^*-\int _0^1\rho ^2 z_w(\rho ,0){\text{ d }}\rho \right) . \end{aligned}$$

$\square$

Proof of Theorem 2.2

Using Theorem A.5, we conclude that for each

where

$$ \begin{aligned} \mathcal {C}^*_\mathrm{{ad}}:= & {} \Big \{ v\in L^\infty ([0, 1]\times [0, T]): l^{*}_1(\rho ,\tau )\le v(\rho ,\tau )\\&\le l^{**}_1(\rho ,\tau ),~a.e.~ (\rho ,\tau )\in [0, 1]\times [0, T]\Big \},\\ \mathcal {W}^*_\mathrm{{ad}}:= & {} \Big \{ v\in L^\infty ([0, 1]\times [0, T]): l^{*}_2(\rho ,\tau ) \le v(\rho ,\tau )\\&\le l^{**}_2(\rho ,\tau ),~a.e.~ (\rho ,\tau )\in [0, 1]\times [0, T]\Big \},\\ \mathcal {C}^{1*}_\mathrm{{ad}}:= & {} \Big \{ v\in L^\infty ( [0, T]):\exists c~\mathrm{{constant}}~s.t ~v(\tau )=c,~a.e.~\tau \in [0,T]~ \& ~ l^{*}_3\le c\\&\le l^{**}_3\Big \},\\ \mathcal {W}^{1*}_\mathrm{{ad}}:= & {} \Big \{ v\in L^\infty ( [0, T]):\exists c~\mathrm{{constant}}~s.t ~v(\tau )=c,~a.e.~\tau \in [0,T]~ \& ~ l^{*}_4\le c\\&\le l^{**}_4\Big \}, \end{aligned}$$

there exists a unique value ${J^*}(u_1,u_2,\overline{c},\overline{w})$ such that for every sequence $\Big \{(u^n_1,u^n_2,\overline{c}^n,\overline{w}^n)\Big \}_{n=0}^{\infty }$ in $\mathcal {C}_{ad}\times \mathcal {W}_{ad}\times \mathcal {C}^1_{ad}\times \mathcal {W}^1_{ad}$, which converges to $(u_1,u_2,\overline{c},\overline{w})$ in ${L^4_2([0, 1]\times [0, T])}:=L^2([0, 1]\times [0, T])\times L^2([0, 1]\times [0, T])\times L^2[0, T]\times L^2 [0, T]$, we have ${J}(u^n_1,u^n_2,\overline{c}^n,\overline{w}^n)\rightarrow {J^*}(u_1,u_2,\overline{c},\overline{w})$. Using the Lebesgue Dominated Convergence Theorem [22], we deduce that the function

$$\begin{aligned}&\mathcal {J}(u_1,u_2,\overline{c},\overline{w})\nonumber \\&={\left\{ \begin{array}{ll} {J}(u_1,u_2,\overline{c},\overline{w}),~~~~(u_1,u_2, \overline{c},\overline{w})\in \mathcal {C}_\mathrm{{ad}}\times \mathcal {W}_\mathrm{{ad}} \times \mathcal {C}^1_\mathrm{{ad}}\times \mathcal {W}^1_\mathrm{{ad}},\\ {J^*}(u_1,u_2,\overline{c},\overline{w}),~~~~(u_1,u_2, \overline{c},\overline{w})\in \mathcal {C}^*_\mathrm{{ad}}\times \mathcal {W}^{*}_\mathrm{{ad}} \times \mathcal {C}^{1*}_\mathrm{{ad}}\times \mathcal {W}^{1*}_\mathrm{{ad}}\setminus C_4([0, 1]\times [0, T]),\\ +\infty ,~~~~~~~~~~~~~~~~~(u_1,u_2,\overline{c},\overline{w}) \in (\mathcal {C}^*_\mathrm{{ad}}\times \mathcal {W}^*_\mathrm{{ad}}\times \mathcal {C}^{1*}_\mathrm{{ad}} \times \mathcal {W}^{1*}_\mathrm{{ad}})^c\cap L^{4,2}_1([0, 1]\times [0, T]),\\ \end{array}\right. }\nonumber \\ \end{aligned}$$

(85)

is lower semicontinuous with respect to $(u_1,u_2,{\overline{c}},{\overline{w}})$ in ${L_1^{4,2}([0, 1]\times [0, T])}$, where

$$\begin{aligned} {L^{4,2}_1([0, 1]\times [0, T])}:= & {} \Big \{(f_1,f_2,f_3,f_4):(\rho ^2 f_1, \rho ^2 f_2,f_3,f_4)\in L^1([0, 1]\\&\times [0, T])\times L^1([0, 1] \times [0, T])\times L^1[0, T]\times L^1 [0, T]\Big \}. \end{aligned}$$

Using Theorem A.3, we conclude that for each positive $\epsilon$ there exists $(u^\epsilon _1,u^\epsilon _2,\overline{c}^\epsilon ,\overline{w}^\epsilon )$ such that

$$\begin{aligned}&\mathcal {J}(u_1^\epsilon ,u^\epsilon _2,{\overline{c}}^\epsilon , {\overline{w}^\epsilon })<\mathcal {J}(u_1,u_2,{\overline{c}}, {\overline{w}})+\epsilon ^{\frac{1}{2}}\Big (\Vert \rho ^2(u_1-u_1^\epsilon )\Vert _{L^1} +\Vert \rho ^2(u_2-u_2^\epsilon )\Vert _{L^1}\nonumber \\&\quad +\Vert \overline{c} -\overline{c}^\epsilon \Vert _{L^1[0,T]}+\Vert \overline{w} -\overline{w}^\epsilon \Vert _{L^1[0,T]}\Big ), \forall (u_1,u_2,\overline{c},\overline{w})\not =(u_1^\epsilon ,u^\epsilon _2,\overline{c}^\epsilon ,\overline{w}^\epsilon ). \end{aligned}$$

(86)

In addition, each sequence $\Big \{(u_1^n,u^n_2,\overline{c}^n,\overline{w}^n)\Big \}_{n=0}^{\infty }$ in $\mathcal {C}_\mathrm{{ad}}\times \mathcal {W}_\mathrm{{ad}}\times \mathcal {C}^1_\mathrm{{ad}}\times \mathcal {W}^1_\mathrm{{ad}}$, which converges to $(u^\epsilon _1,u^\epsilon _2,\overline{c}^\epsilon ,\overline{w}^\epsilon )$ in $L_2^4([0, 1]\times [0, T])$, is a Cauchy sequence in $L_2^4([0, 1]\times [0, T])$. Also assume that for each $n\in \mathbb {N}_0$, $(c_{n},w_{n},p_{n},q_{n},d_{n},\eta _{n})$ is the solution of (31)–(41) corresponding to $(u_1^{n},u^{n}_2,\overline{c}^{n},\overline{w}^{n})$. Therefore, according to Theorem A.5, the sequence

$$\begin{aligned} \Big \{\rho (c_{n},w_{n},p_{n},q_{n},d_{n},\eta _{n})\Big \}_{n=0}^{\infty }, \end{aligned}$$

is a Cauchy sequence in

$$\begin{aligned} L^6_2([0, 1]\times [0, T]):= & {} L^2([0, 1]\times [0, T])\times L^2([0, 1]\times [0, T]) \times L^2([0, 1]\\&\times [0, T])\times L^2([0, 1] \times [0, T]) \times L^2([0, 1]\times [0, T])\\&\times L^2[0, T]. \end{aligned}$$

So using the Lebesgue Dominated Convergence Theorem [22], we derive that for each $p>5$ there exist subsequences $\Big \{(u_1^{n_k},u^{n_k}_2,\overline{c}^{n_k},\overline{w}^{n_k})\Big \}_{k=0}^{\infty }$ and $\Big \{(c_{n_k},w_{n_k},p_{n_k},q_{n_k},d_{n_k},\eta _{n_k})\Big \}_{k=0}^{\infty }$, which are Cauchy in

$$\begin{aligned} L_p^4([0, 1]\times [0, T]):= & {} L^p([0, 1]\times [0, T])\times L^p([0, 1] \times [0, T])\\&\times L^p[0, T]\times L^p [0, T], \end{aligned}$$

and

$$\begin{aligned} L^6_p([0, 1]\times [0, T]):= & {} L^p([0, 1]\times [0, T])\times L^p([0, 1] \times [0, T])\times L^p([0, 1]\\&\times [0, T])\times L^p([0, 1] \times [0, T]) \times L^p([0, 1]\times [0, T])\\&\times L^p[0, T], \end{aligned}$$

respectively. Thus, using Lemma A.1, it is easy to see that the sequences $\Big \{(c_{n_k},w_{n_k},p_{n_k},q_{n_k},d_{n_k},\eta _{n_k})\Big \}_{k=0}^{\infty }$ and $\Big \{(z_{c_{n_k}},z_{w_{n_k}},z_{p_{n_k}},z_{q_{n_k}},z_{d_{n_k}},z_{\eta _{n_k}})\Big \}_{k=0}^{\infty }$ are Cauchy in

$$\begin{aligned} W^{2,1,6}_p(Q^1_T):= & {} W^{2,1}_p(Q^1_T)\times W^{2,1}_p(Q^1_T) \times W^{2,1}_p(Q^1_T)\\&\times W^{2,1}_p(Q^1_T) \times W^{2,1}_p(Q^1_T)\times W^{2,1}_p(Q^1_T), \end{aligned}$$

where $(z_{c_{n_k}},z_{w_{n_k}},z_{p_{n_k}},z_{q_{n_k}},z_{d_{n_k}},z_{\eta _{n_k}})$ is the solution of (49)–(59) corresponding to $(u_1^{n_k},u^{n_k}_2,\overline{c}^{n_k},\overline{w}^{n_k})$. Therefore, there exist $A=(c^\epsilon ,w^\epsilon ,p^\epsilon ,q^\epsilon ,d^\epsilon ,\eta ^\epsilon ,z^\epsilon _{c},z^\epsilon _{w},z^\epsilon _{p},z^\epsilon _{q},z^\epsilon _{d},z^\epsilon _{\eta })$ and sequences and such that

$$\begin{aligned} \lim _{m\rightarrow \infty }A_m= & {} A,~in~W^{2,1,12}_p(Q^1_T),\\ \lim _{m\rightarrow \infty }B_m= & {} (u^\epsilon _1,u^\epsilon _2,\overline{c}^\epsilon , \overline{w}^\epsilon ),~in~L^4_p([0, 1]\times [0, T]). \end{aligned}$$

From t-Anisotropic Embedding Theorem [17], we conclude that for $0<\alpha <1$, each component of A belongs to $C^{\alpha ,\frac{\alpha }{2}}(\overline{Q^1_T})$. Using the proof of Theorem 2.1 and tangent-normal cone techniques [21], for $\epsilon$ small enough we have

$$\begin{aligned}&|u_1^\epsilon -\mathcal {F}_1(-z^\epsilon _c+r_1^*)|\le \epsilon ^{\frac{1}{2}},~|u_2^\epsilon -\mathcal {F}_2( -z^\epsilon _w+r_2^*)|\le \epsilon ^{\frac{1}{2}}, \end{aligned}$$

(87)

$$\begin{aligned}&\left| {\overline{c}}^\epsilon -\mathcal {F}_3\left( \int _0^T(D_1\dfrac{\partial z^\epsilon _c}{\partial \rho }(1,\tau )){\text{ d }}\tau +r_3^*-\int _0^1\rho ^2 z^\epsilon _c(\rho ,0){\text{ d }}\rho \right) \right| \le \epsilon ^{\frac{1}{2}}, \end{aligned}$$

(88)

$$\begin{aligned}&\left| {\overline{w}}^\epsilon -\mathcal {F}_4\left( \int _0^T(D_2\dfrac{\partial z^\epsilon _w}{\partial \rho }(1,\tau )){\text{ d }}\tau +r_4^*-\int _0^1\rho ^2 z^\epsilon _w(\rho ,0){\text{ d }}\rho \right) \right| \le \epsilon ^{\frac{1}{2}}. \end{aligned}$$

(89)

From Theorem A.7 and (49)–(54), it is clear that there exists a positive constant $\beta ^*_0$, which is independent of T and $\eta _0$ such that if $T<\beta _0^*\min \{T_0,T_1\}$, then the function

$$\begin{aligned}&\mathcal {H}:\mathcal {C}_\mathrm{{ad}}\times \mathcal {W}_\mathrm{{ad}}\times \mathcal {C}^1_\mathrm{{ad}} \times \mathcal {W}^1_\mathrm{{ad}}\rightarrow \mathcal {C}_\mathrm{{ad}}\times \mathcal {W}_\mathrm{{ad}} \times \mathcal {C}^1_\mathrm{{ad}}\times \mathcal {W}^1_\mathrm{{ad}},\\&\mathcal {H}(u_1,u_2,\overline{c},\overline{w})=\left( \mathcal {F}_1(-z_c+r_1^*), \mathcal {F}_2(-z_w+r_2^*),\mathcal {F}_3\left( \int _0^T D_1\dfrac{\partial z_c}{\partial \rho }(1,\tau ){\text{ d }}\tau +r_3^*\right. \right. \\&\quad \left. -\int _0^1\rho ^2 z_c(\rho ,0){\text{ d }}\rho \right) ,\\&\left. \mathcal {F}_4\left( \int _0^T D_2\dfrac{\partial z_w}{\partial \rho }(1,\tau ){\text{ d }}\tau +r_4^* -\int _0^1\rho ^2 z_w(\rho ,0){\text{ d }}\rho \right) \right) , \end{aligned}$$

where $z_c$ and $z_w$ are adjoint states corresponding to $(u_1,u_2,\overline{c},\overline{w})$, has a unique fixed point $(u_1^*,u_2^*,{\overline{c}}^*,{\overline{w}}^*)$. Also using (49)–(54), (86), Theorems A.5 and A.7, and the proof of Theorem A.4, it is easy to show that there exists a positive constant $\beta ^*$, which is independent of T and $\eta _0$ such that if $T<\beta ^*\min \{T_0,T_1\}$, then the sequence $\{(\rho u_1^\epsilon ,\rho u_2^\epsilon ,\overline{c}^\epsilon ,\overline{w}^\epsilon )\}$ has a subsequence, which converges to $(\rho u_1^*,\rho u_2^*,\overline{c}^*,\overline{w}^*)$ in $L_2^4([0, 1]\times [0, T])$, as $\epsilon$ converges to zero. Therefore, one can arrive at

$$\begin{aligned} \mathcal {J}(u_1^*,u_2^*,{\overline{c}}^*,{\overline{w}}^*)\le \mathcal {J}(u_1,u_2, {\overline{c}},{\overline{w}}),~~\forall (u_1,u_2,{\overline{c}}, {\overline{w}})\in \mathcal {C}^*_\mathrm{{ad}}\times \mathcal {W}^*_\mathrm{{ad}} \times \mathcal {C}^1_\mathrm{{ad}}\times \mathcal {W}^1_\mathrm{{ad}}, \end{aligned}$$

which results in

$$\begin{aligned} {J}(u_1^*,u_2^*,{\overline{c}}^*,{\overline{w}}^*)\le {J}(u_1,u_2, {\overline{c}},{\overline{w}}),~~\forall (u_1,u_2,{\overline{c}},{\overline{w}})\in \mathcal {C}_\mathrm{{ad}}\times \mathcal {W}_\mathrm{{ad}}\times \mathcal {C}^1_\mathrm{{ad}} \times \mathcal {W}^1_\mathrm{{ad}}. \end{aligned}$$

$\square$

Proof of Theorem 2.3

Let $\left( c^*,w^*,p^*,q^*,d^*,\eta ^*\right)$ and $\left( c^{\epsilon },w^{\epsilon },p^{\epsilon },q^{\epsilon },d^{\epsilon },\eta ^{\epsilon }\right)$ be the solutions of (31)–(41) such that in (33) and (36) $\overline{c}=0$ and $\overline{w}=0$, respectively, corresponding to the controls $(u_1^*,u_2^*,{\overline{c}}^*,{\overline{w}}^*)\in \mathcal {C}_\mathrm{{ad}}\times \mathcal {W}_\mathrm{{ad}}\times \mathcal {C}^1_\mathrm{{ad}}\times \mathcal {W}^1_\mathrm{{ad}}$ and$(u_1^{\epsilon },u_2^{\epsilon },{\overline{c}}^\epsilon ,{\overline{w}}^\epsilon ) =(u_1^*+\epsilon u_1^0,u_2^*+\epsilon u_2^0,{\overline{c}}^*+\epsilon {\overline{c}}_0,{\overline{w}}^*+\epsilon {\overline{w}}_0) \in \mathcal {C}_\mathrm{{ad}}\times \mathcal {W}_\mathrm{{ad}}\times \mathcal {C}^1_\mathrm{{ad}}\times \mathcal {W}^1_\mathrm{{ad}}$, respectively, where $\mathcal {C}_\mathrm{{ad}}$, $\mathcal {W}_\mathrm{{ad}}$, $\mathcal {C}^1_\mathrm{{ad}}$ and $\mathcal {W}^1_\mathrm{{ad}}$ are defined in (45)–(48) and $\epsilon$ is a positive constant. Also assume that

$$\begin{aligned} c^{\epsilon }_1= & {} \dfrac{c^{\epsilon }-c^*}{\epsilon },~w^{\epsilon }_1 =\dfrac{w^{\epsilon }-w^*}{\epsilon }, ~p^{\epsilon }_1=\dfrac{p^{\epsilon }-p^*}{\epsilon },~q^{\epsilon }_1 =\dfrac{q^{\epsilon }-q^*}{\epsilon },\\ d^{\epsilon }_1= & {} \dfrac{d^{\epsilon }-d^*}{\epsilon },~ \eta ^{\epsilon }_1=\dfrac{\eta ^{\epsilon }-\eta ^*}{\epsilon }, \end{aligned}$$

and $J_1(u_1^*,u_2^*,{\overline{c}}^*,{\overline{w}}^*)=\min \Big \{J_1(u_1,u_2,\overline{c},\overline{w}):(u_1,u_2,\overline{c},\overline{w})\in \mathcal {C}_\mathrm{{ad}}\times \mathcal {W}_\mathrm{{ad}}\times \mathcal {C}^1_\mathrm{{ad}}\times \mathcal {W}^1_\mathrm{{ad}}\Big \}.$ Therefore, we have

$$\begin{aligned}&\dfrac{J_1(u_1^{\epsilon },u_2^{\epsilon },{\overline{c}}^\epsilon , {\overline{w}}^\epsilon )-J_1(u_1^*,u_2^*,{\overline{c}}^*,{\overline{w}}^*)}{\epsilon } =\Big (\int _0^1\int _0^T\rho ^2\Big (\lambda _1(p^*+p^{\epsilon })p^{\epsilon }_1\nonumber \\&\quad +\lambda _2(q^*+q^{\epsilon })q^{\epsilon }_1 +(u_1^*+u_1^{\epsilon }-2r_1^*)u_1^0 +(u_2^*+u_2^{\epsilon }-2r_2^*)u_2^0\Big ){\text{ d }}\rho {\text{ d }}\tau \nonumber \\&\quad +\int _0^T({\overline{c}}^*+{\overline{c}}^\epsilon -2r_3^*){\overline{c}}_0 +({\overline{w}}^*+{\overline{w}}^\epsilon -2r_4^*){\overline{w}}_0{\text{ d }}\tau \Big )\ge 0, \end{aligned}$$

(90)

where $\left( c^{\epsilon }_1,w^{\epsilon }_1,p^{\epsilon }_1,q^{\epsilon }_1,d^{\epsilon }_1,\eta ^{\epsilon }_1\right)$ is the solution of (72)–(82) such that in (74) and (77), $\overline{c}_0=0$ and $\overline{w}_0=0$, respectively, also in (73) and (76), $\overline{c}_0=\overline{c}_0(\tau )$ and $\overline{w}_0=\overline{w}_0(\tau )$, respectively. Therefore, using (49)–(59), assumption A and Theorems A.4 and A.5, one can deduce that

$$\begin{aligned}&\int _0^T\int _0^1\rho ^2 \left( c^\epsilon _1\dfrac{\partial z_c}{\partial \tau }+z_c\dfrac{\partial c^\epsilon _1}{\partial \tau }+w^\epsilon _1 \dfrac{\partial z_w}{\partial \tau }+z_w\dfrac{\partial w^\epsilon _1}{\partial \tau } +p^\epsilon _1\dfrac{\partial z_p}{\partial \tau }+z_p\dfrac{\partial p^\epsilon _1}{\partial \tau }+q^\epsilon _1\dfrac{\partial z_q}{\partial \tau }\right. \\&\left. \quad +z_q\dfrac{\partial q^\epsilon _1}{\partial \tau }+d^\epsilon _1\dfrac{\partial z_d}{\partial \tau }+z_d\dfrac{\partial d^\epsilon _1}{\partial \tau }\right) {\text{ d }}\rho {\text{ d }}\tau +\int _0^T\left( \eta ^\epsilon _1\dfrac{d z_\eta }{d \tau }+z_\eta \dfrac{d \eta ^\epsilon _1}{d \tau }\right) {\text{ d }}\tau \\&\quad = -\int _0^T\int _0^1\rho ^2\Big (\lambda _1p^*p^\epsilon _1 +\lambda _2q^*q^\epsilon _1-z_cu_1^0 - z_wu_2^0\Big ){\text{ d }}\rho {\text{ d }}\tau +\varpi (\epsilon ,\rho , T)\\&\qquad -\int _0^TD_1\dfrac{\partial z_c}{\partial \rho }(1,\tau )\overline{c}_0 +D_2\dfrac{\partial z_w}{\partial \rho }(1,\tau )\overline{w}_0{\text{ d }}\tau , \end{aligned}$$

where $\displaystyle {\lim _{\epsilon \rightarrow 0}}\varpi (\epsilon ,\rho , T)=0$. Thus, we arrive at

$$\begin{aligned}&\int _0^T\int _0^1\rho ^2\Big (\lambda _1p^*p^\epsilon _1 +\lambda _2q^*q^\epsilon _1-z_cu_1^0- z_wu_2^0\Big ){\text{ d }}\rho {\text{ d }}\tau -\varpi (\epsilon ,\rho , T)\\&\quad +\int _0^TD_1\dfrac{\partial z_c}{\partial \rho }(1,\tau )\overline{c}_0 +D_2\dfrac{\partial z_w}{\partial \rho }(1,\tau ) \overline{w}_0{\text{ d }}\tau = -\int _0^T\int _0^1\rho ^2\left( \dfrac{\partial z_cc^\epsilon _1}{\partial \tau }+\dfrac{\partial z_ww^\epsilon _1}{\partial \tau }\right. \\&\quad \left. +\dfrac{\partial z_pp^\epsilon _1}{\partial \tau }+\dfrac{\partial z_qq^\epsilon _1}{\partial \tau }+\dfrac{\partial z_dd^\epsilon _1}{\partial \tau }\right) {\text{ d }}\rho {\text{ d }}\tau -\int _0^T\dfrac{{\text{ d }} z_\eta \eta ^\epsilon _1}{{\text{ d }} \tau } {\text{ d }}\tau =0. \end{aligned}$$

So using (90) we get

$$\begin{aligned}&\Big (\int _0^1\int _0^T\rho ^2\left( ( z_c+u_1^*-r_1^*)u_1^0 +( z_w+u_2^*-r_2^*)u_2^0\right) {\text{ d }}\rho {\text{ d }}\tau \\&\quad +\int _0^T\left( \overline{c}^*-r_3^*-\left( D_1\dfrac{\partial z_c}{\partial \rho } (1,\tau )\right) \right) {\overline{c}}_0(\tau ){\text{ d }}\tau + \int _0^T\left( \overline{w}^*-r_4^*\right. \\&\quad \left. -\left( D_2\dfrac{\partial z_w}{\partial \rho }(1,\tau )\right) \right) {\overline{w}}_0(\tau ){\text{ d }}\tau \Big ) \ge 0. \end{aligned}$$

Therefore, from tangent-normal cone techniques [21] and (45)–(48), $(u_1^*,u_2^*,{\overline{c}}^*,{\overline{w}}^*)$ is as follows:

$$\begin{aligned}&u_1^*=\mathcal {F}_1(-z_c+r_1^*),~u_2^*=\mathcal {F}_2(-z_w+r_2^*),~ {\overline{c}}^*=\mathcal {F}_3\left( D_1\dfrac{\partial z_c}{\partial \rho }(1,\tau )+r_3^*\right) ,\\&{\overline{w}}^* =\mathcal {F}_4\left( D_2\dfrac{\partial z_w}{\partial \rho }(1,\tau )+r_4^*\right) . \end{aligned}$$

Finally, in a similar manner to the proof of Theorem 2.2, the desired result is obtained. $\square$