The solutions
\(q_{123},q_{12},q_1,q_{23},q_{2},q_{21},q_3,q_{32},q_{321}\) have the explicit forms
$$\begin{aligned} q_{123}(t)= & {} q^1(t;1/\lambda ,T_{12},T)I_{[\,T_{12},T\,]} +q^2(t;\overline{c}^{p-1},T_{123},T_{12})I_{[\,T_{123},T_{12}\,]}\\&+\,q^3(t;\underline{c}^{p-1},0,T_{123})I_{[\,0,T_{123}\,]};\\ q_{12}(t)= & {} q^1(t;1/\lambda ,T_{12},T)I_{[\,T_{12},T\,]} +q^2(t;\overline{c}^{p-1},0,T_{12})I_{[\,0,T_{12}\,]};\\ q_{1}(t)= & {} q^1(t;1/\lambda ,0,T);\qquad q_2(t)=q^2(t;1/\lambda ,0,T);\qquad q_3(t)=q^3(t;1/\lambda ,0,T);\\ q_{23}(t)= & {} q^2(t;1/\lambda ,T_{23},T)I_{[\,T_{23},T\,]}+q^3(t;\underline{c}^{p-1},0,T_{23})I_{[\,0,T_{23}\,]};\\ q_{21}(t)= & {} q^2(t;1/\lambda ,T_{21},T)I_{[\,T_{21},T\,]}+q^1(t;\overline{c}^{p-1},0,T_{21})I_{[\,0,T_{21}\,]};\\ q_{32}(t)= & {} q^3(t;1/\lambda ,T_{32},T)I_{[\,T_{32},T\,]}+q^2(t;\underline{c}^{p-1},0,T_{32})I_{[\,0,T_{32}\,]};\\ q_{321}(t)= & {} q^3(t;1/\lambda ,T_{32},T)I_{[\,T_{32},T\,]}+q^2(t;\underline{c}^{p-1},T_{321},T_{32})I_{[\,T_{321},T_{32}\,]}\\&+\,q^1(t;\overline{c}^{p-1},0,T_{321})I_{[\,0,T_{321}\,]}, \end{aligned}$$
where
\(I_{[\underline{T},\overline{T}]}\) is an indicator function of the set
\([\underline{T},\overline{T}]\), and the functions
\(q^1(t;A,\underline{T},\overline{T}),q^2(t;A,\underline{T},\overline{T})\),
\(q^3(t;A,\underline{T},\overline{T})\) in the interval
\([\,\underline{T},\overline{T}\,]\) are given as
$$\begin{aligned}&q^1(t;A,\underline{T},\overline{T})\nonumber \\&\quad =\ln \lambda + \left\{ \begin{array}{l@{\quad }l} \ln \left[ \,\left( \,A-{\overline{c}^p\over \rho +p\overline{c}-pK}\,\right) e^{(\rho +p\overline{c}-pK)(t-\overline{T})}+{\overline{c}^p\over \rho +p\overline{c}-pK}\,\right] , &{}\rho -pK\ne -p\overline{c};\\ \ln \Big [\,A+\overline{c}^p(\overline{T}-t)\,\Big ], &{}\rho -pK=-p\overline{c}; \end{array} \right. \nonumber \\ \end{aligned}$$
(37)
$$\begin{aligned}&q^2(t;A,\underline{T},\overline{T})\nonumber \\&\quad =\ln \lambda + \left\{ \begin{array}{l@{\quad }l} (1-p)\ln \left[ \,\left( \,A^{1/(1-p)}-{1-p\over \rho -pK}\,\right) e^{{\rho -pK\over 1-p}\left( \,t-\overline{T}\,\right) }+{1-p\over \rho -pK}\,\right] , &{}\rho -pK\ne 0;\\ (1-p)\ln \left[ \,A^{1/(1-p)}+\overline{T}-t\,\right] , &{}\rho -pK=0; \end{array} \right. \nonumber \\ \end{aligned}$$
(38)
$$\begin{aligned}&q^3(t;A,\underline{T},\overline{T})\nonumber \\&\quad =\ln \lambda + \left\{ \begin{array}{l@{\quad }l} \ln \left[ \,\left( \,A-{\underline{c}^p\over \rho +p\underline{c}-pK}\,\right) e^{(\rho +p\underline{c}-pK)(t-\overline{T})}+{\underline{c}^p\over \rho +p\underline{c}-pK}\,\right] , &{}\rho -pK\ne -p\underline{c};\\ \ln \left[ \,A+\underline{c}^p(\overline{T}-t)\,\right] , &{}\rho -pK=-p\underline{c}, \end{array} \right. \nonumber \\ \end{aligned}$$
(39)
and
\(T_{12},T_{123},T_{23},T_{21},T_{32},T_{321}\) are given as
$$\begin{aligned} T_{12}= & {} \left\{ \begin{array}{l@{\quad }l} T+{1\over \rho +p\overline{c}-pK}\left[ \, \ln \left| \,\overline{c}^{p-1}-{\overline{c}^p\over \rho +p\overline{c}-pK}\,\right| -\ln \left| \,{1\over \lambda }-{\overline{c}^p\over \rho +p\overline{c}-pK}\,\right| \,\right] ,\; \;&{}\rho -pK\ne -p\overline{c};\\ T-1/\overline{c}+1/(\lambda \overline{c}^p),\;&{}\rho -pK=-p\overline{c}; \end{array} \right. \nonumber \\ \end{aligned}$$
(40)
$$\begin{aligned} T_{123}= & {} \left\{ \begin{array}{l@{\quad }l} T_{12}+{1-p\over \rho -pK}\left[ \, \ln \left| \,{1\over \underline{c}}-{1-p\over \rho -pK}\,\right| -\ln \left| \,{1\over \overline{c}}-{1-p\over \rho -pK}\,\right| \,\right] ,\quad &{}\rho -pK\ne 0;\\ T_{12}+1/\overline{c}-1/\underline{c}\,, \;&{}\rho -pK=0; \end{array} \right. \nonumber \\ \end{aligned}$$
(41)
$$\begin{aligned} T_{23}= & {} \left\{ \begin{array}{l@{\quad }l} T+{1-p\over \rho -pK}\left[ \, \ln \left| \,{1\over \underline{c}}-{1-p\over \rho -pK}\,\right| -\ln \left| \,\lambda ^{1/(p-1)}-{1-p\over \rho -pK}\,\right| \,\right] ,\quad &{}\rho -pK\ne 0;\\ T+\lambda ^{1/(p-1)}-1/\underline{c}\,, \;&{}\rho -pK=0; \end{array} \right. \nonumber \\ \end{aligned}$$
(42)
$$\begin{aligned} T_{21}= & {} T+{1-p\over \rho -pK}\left[ \, \ln \left( \,{1\over \overline{c}}-{1-p\over \rho -pK}\,\right) -\ln \left( \,\lambda ^{1/(p-1)}-{1-p\over \rho -pK}\,\right) \,\right] ; \end{aligned}$$
(43)
$$\begin{aligned} T_{32}= & {} T+{1\over \rho +p\underline{c}-pK}\left[ \, \ln \left( \,\underline{c}^{p-1}-{\underline{c}^p\over \rho +p\underline{c}-pK}\,\right) -\ln \left( \,{1\over \lambda }-{\underline{c}^p\over \rho +p\underline{c}-pK}\,\right) \,\right] ; \nonumber \\ \end{aligned}$$
(44)
$$\begin{aligned} T_{321}= & {} T_{32}+{1-p\over \rho -pK}\left[ \, \ln \left( \,{1\over \overline{c}}-{1-p\over \rho -pK}\,\right) -\ln \left( \,{1\over \underline{c}}-{1-p\over \rho -pK}\,\right) \,\right] . \end{aligned}$$
(45)
It is routine to check that for any
\(A>0\) and
\(0\le \underline{T}\le \overline{T}\), the functions
\(q^1(t;A,\underline{T},\overline{T})\),
\(q^2(t;A,\underline{T},\overline{T})\) and
\(q^3(t;A,\underline{T},\overline{T})\) solve the following ODEs, respectively,
$$\begin{aligned} {q_P}(t)= & {} \ln \lambda +\ln A+\int _t^{\overline{T}} \Big [\,-\rho +\lambda \overline{c}^pe^{-{q_P}(s)}-p\overline{c}+pK\,\Big ]ds,\;\forall \;t\in [\,\underline{T},\overline{T}\,]; \end{aligned}$$
(46)
$$\begin{aligned} {q_P}(t)= & {} \ln \lambda +\ln A+\int _t^{\overline{T}} \Bigg [\,-\rho +(1-p)\lambda ^{1\over (1-p)}\exp \left\{ {{q_P}(s)\over p-1}\right\} +pK\,\Bigg ]ds,\;\forall \;t\in [\,\underline{T},\overline{T}\,]; \nonumber \\ \end{aligned}$$
(47)
$$\begin{aligned} {q_P}(t)= & {} \ln \lambda +\ln A+\int _t^{\overline{T}} \Big [\,-\rho +\lambda \underline{c}^pe^{-{q_P}(s)}-p\underline{c}+pK\,\Big ]ds,\;\forall \;t\in [\,\underline{T},\overline{T}\,]. \end{aligned}$$
(48)
When
\(\underline{c}>0\),
\(q^1(0;A,0,\overline{T}),\, q^2(0;A,0,\overline{T})\) and
\(q^3(0;A,0,\overline{T})\) have the following asymptotic properties,
$$\begin{aligned}&\lim \limits _{\overline{T}\rightarrow \infty }q^1(0;A,0,\overline{T})= \left\{ \begin{array}{ll} \ln \lambda +\ln \left( \,{\overline{c}^p\over \rho +p\overline{c}-pK}\,\right) , &{}\rho -pK>-p\overline{c};\\ +\infty , &{}\rho -pK\le -p\overline{c}; \end{array} \right. \nonumber \\&\lim \limits _{\overline{T}\rightarrow \infty }q^1(0;A,0,\overline{T})\le (p-1)\ln \overline{c}+\ln \lambda \Leftrightarrow \rho -pK\ge (1-p)\overline{c}; \end{aligned}$$
(49)
$$\begin{aligned}&\lim \limits _{\overline{T}\rightarrow \infty }q^2(0;A,0,\overline{T})=\left\{ \begin{array}{ll} \ln \lambda +(1-p)\ln \left( \,{1-p\over \rho -pK}\,\right) , &{}\rho -pK>0; \\ +\infty , &{}\rho -pK\le 0; \end{array} \right. \nonumber \\&\lim \limits _{\overline{T}\rightarrow \infty }q^2(0;A,0,\overline{T})\ge (p-1)\ln \overline{c}+\ln \lambda \Leftrightarrow \rho -pK\le (1-p)\overline{c};\end{aligned}$$
(50)
$$\begin{aligned}&\lim \limits _{\overline{T}\rightarrow \infty }q^2(0;A,0,\overline{T}) \le (p-1)\ln \underline{c}+\ln \lambda \Leftrightarrow \rho -pK\ge (1-p)\underline{c}; \end{aligned}$$
(51)
$$\begin{aligned}&\lim \limits _{\overline{T}\rightarrow \infty }q^3(0;A,0,\overline{T})= \left\{ \begin{array}{ll} \ln \lambda +\ln \left( \,{\underline{c}^p\over \rho +p\underline{c}-pK}\,\right) , &{}\rho -pK>-p\underline{c};\\ +\infty , &{}\rho -pK\le -p\underline{c}; \end{array} \right. \nonumber \\&\lim \limits _{\overline{T}\rightarrow \infty }q^3(0;A,0,\overline{T})\ge (p-1)\ln \underline{c}+\ln \lambda \Leftrightarrow \rho -pK\le (1-p)\underline{c}. \end{aligned}$$
(52)
Proof of Theorem 4.5
Case (1) \(0\le \underline{c}<\overline{c}<\lambda ^{1/(1-p)}\).
In this case,
\((p-1)\ln \overline{c}+\ln \lambda >0\). Since
\(q_P(T)=0\), then, when
t is close to
T,
\(q_P(t)<(p-1)\ln \overline{c}+\ln \lambda \) and
\(c^*(t)=\widetilde{x}_{c,P}^*(q_P(t))=\overline{c}.\) Thus
\(q_P(t)\) satisfies ODE (
46) with
\(A=1/\lambda \) and
\(\overline{T}=T\) in the interval
\([\,\underline{T},T\,]\), until
\(\underline{T}=0\) or
\(q_P(\underline{T})=(p-1)\ln \overline{c}+\ln \lambda \).
(1.1) If
\(\rho -pK<(1-p)\underline{c}\ne 0\), solving ODE (
46), we obtain
\(q_P(t)=q^1(t;1/\lambda ,\underline{T},T)\) taking the form of (
37) in the interval
\([\,\underline{T},T\,]\). According to (
49),
\(q^1(0;1/\lambda ,0,T)>(p-1)\ln \overline{c}+\ln \lambda \) provided
T is large enough. Thus, there exists a positive constant
\(T_{12}\) such that
\(q^1(T_{12};1/\lambda ,T_{12},T)=(p-1)\ln \overline{c}+\ln \lambda \), and
\(T_{12}\) is given in (
40). Hence, we derive
\(q_P(t)=q^1(t;1/\lambda ,T_{12},T)\), and
\(c^*(t)=\overline{c}\) in the interval
\([\,T_{12},T\,]\).
Since
$$\begin{aligned} q^\prime _P(T_{12})=\rho -pf_P(q_P(T_{12}),c^*(T_{12}))-pg(x^*_\pi ;x^*_\mu ,{x_\sigma ^*})=\rho -(1-p)\overline{c}-pK<0, \end{aligned}$$
then, when
\(t<T_{12}\) and
t is close to
\(T_{12}\),
\((p-1)\ln \overline{c}+\ln \lambda<q_P(t)<(p-1)\ln \underline{c}+\ln \lambda \), and
\(c^*(t)=\widehat{c}(t)\) taking the form of (
27). Thus
\(q_P(t)\) satisfies ODE (
47) with
\(A=\overline{c}^{p-1}\) and
\(\overline{T}=T_{12}\) in the interval
\([\,\underline{T},T_{12}\,]\), until
\(\underline{T}=0\) or
\(q_P(\underline{T})=(p-1)\ln \underline{c}+\ln \lambda \) (where we have used the fact that the sign of
\(q_P^\prime (t)\) does not change (cf. Proposition
4.6), and
\(q_P(t)>(p-1)\ln \overline{c}+\ln \lambda \) for any
\(t<T_{12}\)).
Solving ODE (
47), we obtain
\(q_P(t)=q^2(t;\overline{c}^{p-1},\underline{T},T_{12})\) taking the form of (
38) in the interval
\([\,\underline{T},T_{12}\,]\). According to (
51),
\(q^2(0;\overline{c}^{p-1},0,T_{12})>(p-1)\ln \underline{c}+\ln \lambda \) provided
T is large enough. Thus, there exists a positive constant
\(T_{123}\) such that
\(q^2(T_{123};\overline{c}^{p-1},T_{123},T_{12})=(p-1)\ln \underline{c}+\ln \lambda \), and
\(T_{123}\) is given in (
41). Hence, we derive
\(q_P(t)=q^2(t;\overline{c}^{p-1},T_{123},T_{12})\), and
\(c^*(t)=\widehat{c}(t)\) in the interval
\([\,T_{123},T_{12}\,]\).
Recalling the fact that the sign of
\(q_P^\prime (t)\) does not change (cf. Proposition
4.6), we deduce that
\(q_P(t)\ge (p-1)\ln \underline{c}+\ln \lambda ,\,c^*(t)=\underline{c}\), and
\(q_P(t)\) satisfies ODE (
48) with
\(A=\underline{c}^{p-1}\) and
\(\overline{T}=T_{123}\) in the interval
\([\,0,T_{123}\,]\). Solving ODE (
48), we have
\(q_P(t)=q^3(t;\underline{c}^{p-1},0,T_{123})\) as in (
39).
(1.2) If
\(\rho -pK<(1-p)\underline{c}=0\) or
\((1-p)\underline{c}\le \rho -pK<(1-p)\overline{c}\), repeating the same argument as in Case (1.1), we have
\(q_P(t)=q^1(t;1/\lambda ,T_{12},T)\) as in (
37), and
\(c^*=\overline{c}\) in the interval
\([\,T_{12},T\,]\), and
\(q_P(t)=q^2(t;\overline{c}^{p-1},\underline{T},T_{12})\) as in (
38) until
\(\underline{T}=0\) or
\(q_P(\underline{T})=(p-1)\ln \underline{c}+\ln \lambda \).
In the case of
\(\rho -pK\le (1-p)\underline{c}=0\),
\((p-1)\ln \underline{c}+\ln \lambda =+\infty >q^2(t;\overline{c}^{p-1},0,T_{12})\), and
\(\underline{T}=0\). In the other case, since
\(\rho -pK>0\) and
\(\rho -pK-(1-p)\overline{c}<0\), then (
38) implies that
$$\begin{aligned} q^2(t;\overline{c}^{p-1},0,T_{12})< & {} \ln \lambda +(1-p)\ln {1-p\over \rho -pK} \\\le & {} \ln \lambda +(1-p)\ln {1\over \underline{c}}=(p-1)\ln \underline{c}+\ln \lambda ,\;\;\forall \;t\in [\,0,T_{12}\,]. \end{aligned}$$
Thus, we deduce that
\(\underline{T}=0\). Therefore,
\(q_P(t)=q^2(t;\overline{c}^{p-1},0,T_{12})\) and
\(c^*(t)=\widehat{c}(t)\) in the interval
\([\,0,T_{12}\,]\).
(1.3) If
\((1-p)\overline{c}\le \rho -pK<\lambda \overline{c}^p-p\overline{c}\), solving ODE (
46), we have
\(q_P(t)=q^1(t;1/\lambda ,\underline{T},T)\) as in (
37) until
\(\underline{T}=0\) or
\(q_P(\underline{T})=(p-1)\ln \overline{c}+\ln \lambda \). Since
$$\begin{aligned} q^1(t;1/\lambda ,0,T)< & {} \ln \lambda +\ln {\overline{c}^p\over \rho +p\overline{c}-pK} \\\le & {} \ln \lambda +\ln \overline{c}^{p-1}=(p-1)\ln \overline{c}+\ln \lambda ,\;\;\forall \;t\in [\,0,T\,], \end{aligned}$$
then
\(q_P(t)=q^1(t;1/\lambda ,0,T)\) and
\(c^*(t)=\overline{c}\) in the interval
\([\,0,T\,]\).
(1.4) If
\(\rho -pK\ge \lambda \overline{c}^p-p\overline{c}\), solving ODE (
46), we derive that
\(q_P(t)=q^1(t;1/\lambda ,\underline{T},T)\) until
\(\underline{T}=0\) or
\(q_P(\underline{T})=(p-1)\ln \overline{c}+\ln \lambda \).
Since \(\rho +p\overline{c}-pK\ge 0\) and \(\rho +p\overline{c}-pK-\lambda \overline{c}^p\ge 0\), then \(q^1(t;1/\lambda ,0,T)\) is nondecreasing with respect to t, thus for \(t\in [\,0,T\,]\), \(q^1(t;1/\lambda ,0,T)\le q^1(T;1/\lambda ,0,T)=0<(p-1)\ln \overline{c}+\ln \lambda \). Hence, \(q_P(t)=q^1(t;1/\lambda ,0,T)\) and \(c^*(t)=\overline{c}\) in the interval \([\,0,T\,]\).
Case (2) \(0\le \underline{c}<\overline{c}=\lambda ^{1/(1-p)}\). In this case, note that \((p-1)\ln \overline{c}+\ln \lambda =0\).
(2.1) If
\(\rho -pK<(1-p)\underline{c}\ne 0\), since
$$\begin{aligned} q_P^\prime (T-0)= & {} \rho -pf_P(q_P(T),c^*(T))-pg(x^*_\pi ;x^*_\mu ,{x_\sigma ^*})\\= & {} \rho -(1-p)\overline{c}-pK\le \rho -pK-(1-p)\underline{c}<0, \end{aligned}$$
then, when
t is close to
T,
\(q_P(t)>0=(p-1)\ln \overline{c}+\ln \lambda ,\,q_P(t)<(p-1)\ln \underline{c}+\ln \lambda \), and
\(c^*(t)=\widehat{c}(t)\). Thus
\(q_P(t)\) satisfies ODE (
47) with
\(A=1/\lambda \) and
\(\overline{T}=T\) in the interval
\([\,\underline{T},T\,]\), until
\(\underline{T}=0\) or
\(q_P(\underline{T})=(p-1)\ln \underline{c}+\ln \lambda \).
Solving ODE (
47), we obtain
\(q_P(t)=q^2(t;1/\lambda ,\underline{T},T)\) as in (
38) in the interval
\([\,\underline{T},T\,]\). According to (
51),
\(q^2(0;1/\lambda ,0,T)>(p-1)\ln \underline{c}+\ln \lambda \) provided that
T is large enough. Thus, there exists a positive constant
\(T_{23}\) such that
\(q^2(T_{23};1/\lambda ,T_{23},T)=(p-1)\ln \underline{c}+\ln \lambda \), and
\(T_{23}\) is given in (
42). Hence, we derive that
\(q_P(t)=q^2(t;1/\lambda ,T_{23},T)\), and
\(c^*(t)=\widehat{c}(t)\) in the interval
\([\,T_{23},T\,]\).
Recalling the fact that the sign of
\(q_P^\prime (t)\) does not change, we deduce that in the interval
\([\,0,T_{23}\,]\),
\(q_P(t)\ge q_P(T_{23})= (p-1)\ln \underline{c}+\ln \lambda ,\,c^*(t)=\underline{c}\), and
\(q_P(t)\) satisfies ODE (
48) with
\(A=\underline{c}^{p-1}\) and
\(\overline{T}=T_{23}\). Solving ODE (
48), we have
\(q_P(t)=q^3(t;\underline{c}^{p-1},0,T_{23})\) as in (
39) in the interval
\([\,0,T_{23}\,]\).
(2.2) If
\(\rho -pK<(1-p)\underline{c}=0\) or
\((1-p)\underline{c}\le \rho -pK<(1-p)\overline{c}\), since
$$\begin{aligned} q_P^\prime (T-0)=\rho -pf_P(q_P(T),c^*(T))-pg(x^*_\pi ;x^*_\mu ,{x_\sigma ^*})=\rho -(1-p)\overline{c}-pK<0 \end{aligned}$$
still holds, then repeating the similar argument as in Case (2.1), we deduce that
\(c^*(t)=\widehat{c}(t)\) and
\(q_P(t)=q^2(t;1/\lambda ,\underline{T},T)\) in the interval
\([\,\underline{T},T\,]\), until
\(\underline{T}=0\) or
\(q_P(\underline{T})=(p-1)\ln \underline{c}+\ln \lambda \).
In the case of
\(\rho -pK\le (1-p)\underline{c}=0\),
\((p-1)\ln \underline{c}+\ln \lambda =+\infty >q^2(t;1/\lambda ,0,T)\), and
\(\underline{T}=0\). In the other case, since
\(\rho -pK>0\) and
\((\rho -pK)-(1-p)\lambda ^{1/(1-p)}=(\rho -pK)-(1-p)\overline{c}<0\), then for any
\(t\in [\,0,T\,]\), we still have
$$\begin{aligned} q^2(t;1/\lambda ,0,T)<\ln \lambda +(1-p)\ln {1-p\over \rho -pK} \le \ln \lambda +(1-p)\ln {1\over \underline{c}}=(p-1)\ln \underline{c}+\ln \lambda .\nonumber \\ \end{aligned}$$
(53)
Therefore,
\(q_P(t)=q^2(t;1/\lambda ,0,T)\) and
\(c^*(t)=\widehat{c}(t)\) in the interval
\([\,0,T\,]\).
(2.3) If
\(\rho -pK\ge (1-p)\overline{c}\). We first discuss the case when
\(\rho -pK>(1-p)\overline{c}\). Combining the following calculation
$$\begin{aligned} q_P^\prime (T-0)=\rho -pf_P(q_P(T),c^*(T))-pg(x^*_\pi ;x^*_\mu ,{x_\sigma ^*})=\rho -(1-p)\overline{c}-pK>0, \end{aligned}$$
and the fact that the sign of
\(q_P^\prime (t)\) does not change, we drive that
\(q_P(t)<q_P(T)=0=(p-1)\ln \overline{c}+\ln \lambda , c^*(t)=\overline{c}\), and
\(q_P(t)\) satisfies ODE (
46) with
\(A=1/\lambda \) and
\(\overline{T}=T\) in the interval
\([\,0,T\,]\). Solving ODE (
46), we have
\(q_P(t)=q^1(t;1/\lambda ,0,T)\) and
\(c^*(t)=\overline{c}\) in the interval
\([\,0,T\,]\).
On the other hand, if \(\rho -pK=(1-p)\overline{c}\), then \(\rho -pK=\lambda \overline{c}^p-p\overline{c}\), and for \(t\in [\,0,T\,]\), we have \(q_P(t)=0\), thus still have \(q_P(t)=q^1(t;1/\lambda ,0,T)\).
Case (3) \(0\le \underline{c}<\lambda ^{1/(1-p)}<\overline{c}\).
In this case, note that
\((p-1)\ln \overline{c}+\ln \lambda<0<(p-1)\ln \underline{c}+\ln \lambda \). Since
\(q_P(T)=0\), then, when
t is close to
T,
\((p-1)\ln \overline{c}+\ln \lambda<q_P(t)<(p-1)\ln \underline{c}+\ln \lambda \),
\(c^*(t)=\widehat{c}(t)\) and
\(q_P(t)\) satisfies ODE (
47) with
\(A=1/\lambda \) and
\(\overline{T}=T\) in the interval
\([\,\underline{T},T\,]\), until
\(\underline{T}=0\) or
\(q_P(\underline{T})=(p-1)\ln \overline{c}+\ln \lambda \) or
\(q_P(\underline{T})=(p-1)\ln \underline{c}+\ln \lambda \).
(3.1) If
\(\rho -pK<(1-p)\underline{c}\ne 0\), solving ODE (
47), we have
\(q_P(t)=q^2(t;1/\lambda ,\underline{T},T)\) and
\(c^*(t)=\widehat{c}(t)\) in the interval
\([\,\underline{T},T\,]\).
Since
$$\begin{aligned} q_P^\prime (T-0)= & {} \rho -pf_P(q_P(T),c^*(T))-pg(x^*_\pi ;x^*_\mu ,{x_\sigma ^*})\\= & {} \rho -(1-p)\lambda ^{1/(1-p)}-pK<\rho -(1-p)\underline{c}-pK<0, \end{aligned}$$
then we deduce
\(q_P(t)\) is nonincreasing with respect to
t from the fact that the sign of
\(q_P^\prime (t)\) does not change. Hence, we have
\(q_P(t)>(p-1)\ln \overline{c}+\ln \lambda \) for any
\(t\in [\,0,T\,]\). Moreover, (
51) implies that
\(q^2(0;1/\lambda ,0,T)>(p-1)\ln \underline{c}+\ln \lambda \) provided that
T is large enough. Thus, there exists a positive constant
\(T_{23}\) such that
\(q^2(T_{23};1/\lambda ,T_{23},T)=(p-1)\ln \underline{c}+\ln \lambda \), and
\(T_{23}\) is given in (
42). Hence, we derive that
\(q_P(t)=q^2(t;1/\lambda ,T_{23},T)\), and
\(c^*(t)=\widehat{c}(t)\) in the interval
\([\,T_{23},T\,]\).
Since
$$\begin{aligned} q_P^\prime (T_{23})=\rho -pf_P(q_P(T_{23}),c^*(T_{23}))-pg(x^*_\pi ;x^*_\mu ,{x_\sigma ^*}) =\rho -(1-p)\underline{c}-pK<0, \end{aligned}$$
then for any
\(t\in [\,0,T_{23})\), we have
\(q_P(t)>(p-1)\ln \underline{c}+\ln \lambda \) , and
\(q_P(t)\) satisfies ODE (
48) with
\(A=\underline{c}^{p-1}\) and
\(\overline{T}=T_{23}\) in the interval
\([\,0,T_{23}\,]\). Solving ODE (
48), we obtain
\(q_P(t)=q^3(t;\underline{c}^{p-1},0,T_{23})\) and
\(c^*(t)=\underline{c}\) in the interval
\([\,0,T_{23}\,]\).
(3.2) If \(\rho -pK<(1-p)\underline{c}=0\) or \((1-p)\underline{c}\le \rho -pK<(1-p)\lambda ^{1/(1-p)}\), repeating the similar argument as in case (3.1), we deduce that \(q_P(t)=q^2(t;1/\lambda ,\underline{T},T),\,c^*(t)=\widehat{c}(t)\) in the interval \([\,\underline{T},T\,]\), until \(\underline{T}=0\) or \(q_P(\underline{T})=(p-1)\ln \underline{c}+\ln \lambda \).
For the case of
\(\rho -pK\le (1-p)\underline{c}=0\),
\((p-1)\ln \underline{c}+\ln \lambda =+\infty >q^2(t;1/\lambda ,0,T)\), and
\(\underline{T}=0\). For the other case, since
\(\rho -pK>0\) and
\(\rho -pK-(1-p)\lambda ^{1/(1-p)}<0\), then (
53) still holds. Therefore,
\(q_P(t)=q^2(t;1/\lambda ,0,T)\) and
\(c^*(t)=\widehat{c}(t)\) in the interval
\([\,0,T\,]\).
(3.3) If
\((1-p)\lambda ^{1-p}\le \rho -pK\le (1-p)\overline{c}\), solving ODE (
47), we have
\(q_P(t)=q^2(t;1/\lambda ,\underline{T},T)\) and
\(c^*(t)=\widehat{c}(t)\) in the interval
\([\,\underline{T},T\,]\). Since
\(\rho -pK>0\) and
\(\rho -pK-(1-p)\lambda ^{1-p}\ge 0\), then
\(q^2(t;1/\lambda ,0,T)\) is nondecreasing and for
\(t\in [\,0,T\,]\), we have
$$\begin{aligned} (p-1)\ln \underline{c}+\ln \lambda> & {} 0=q^2(T;1/\lambda ,0,T)\ge q^2(t;1/\lambda ,0,T)\\\ge & {} (1-p)\ln \left( \,{1-p\over \rho -pK}\,\right) +\ln \lambda \\\ge & {} (1-p)\ln {1\over \overline{c}}+\ln \lambda =(p-1)\ln \overline{c}+\ln \lambda . \end{aligned}$$
Therefore,
\(q_P(t)=q^2(t;1/\lambda ,0,T)\) and
\(c^*(t)=\widehat{c}(t)\) in the interval
\([\,0,T\,]\).
(3.4) If
\(\rho -pK>(1-p)\overline{c}\), solving ODE (
47), we have
\(q_P(t)=q^2(t;1/\lambda ,\underline{T},T)\) and
\(c^*(t)=\widehat{c}(t)\) in the interval
\([\,\underline{T},T\,]\).
Since
$$\begin{aligned} q_P^\prime (T-0)= & {} \rho -pf_P(q_P(T),c^*(T))-pg(x^*_\pi ;x^*_\mu ,{x_\sigma ^*})\\= & {} \rho -(1-p)\lambda ^{1/(1-p)}-pK>\rho -(1-p)\overline{c}-pK>0, \end{aligned}$$
then we deduce
\(q_P(t)\) is nondecreasing with respect to
t from the fact that the sign of
\(q_P^\prime (t)\) does not change. Hence, we have
\(q_P(t)<(p-1)\ln \underline{c}+\ln \lambda \) for any
\(t\in [\,0,T\,]\). Moreover, (
50) implies that
\(q^2(0;1/\lambda ,0,T)<(p-1)\ln \overline{c}+\ln \lambda \) provided that
T is large enough. Thus, there exists a positive constant
\(T_{21}\) such that
\(q^2(T_{21};1/\lambda ,T_{21},T)=(p-1)\ln \overline{c}+\ln \lambda \), and
\(T_{21}\) is given in (
43). Hence, we derive that
\(q_P(t)=q^2(t;1/\lambda ,T_{21},T)\), and
\(c^*(t)=\widehat{c}(t)\) in the interval
\([T_{21},T]\).
Since
$$\begin{aligned} q_P^\prime (T_{21})=\rho -pf_P(q_P(T_{21}),c^*(T_{21}))-pg(x^*_\pi ;x^*_\mu ,{x_\sigma ^*}) =\rho -(1-p)\overline{c}-pK>0, \end{aligned}$$
then for any
\(t\in [\,0,T_{12})\), we have
\(q_P(t)<(p-1)\ln \overline{c}+\ln \lambda \) , and
\(q_P(t)\) satisfies ODE (
46) with
\(A=\overline{c}^{p-1}\) and
\(\overline{T}=T_{12}\) in the interval
\([\,0,T_{12}]\). Solving ODE (
46), we obtain
\(q_P(t)=q^1(t;\overline{c}^{p-1},0,T_{12})\) and
\(c^*(t)=\overline{c}\) in the interval
\([\,0,T_{12}]\).
Case (4) \(\lambda ^{1/(1-p)}=\underline{c}<\overline{c}\). In this case, note that \((p-1)\ln \overline{c}+\ln \lambda <0=(p-1)\ln \underline{c}+\ln \lambda \).
(4.1) If
\(\rho -pK\le (1-p)\underline{c}\), we first consider the case where
\(\rho -pK<(1-p)\underline{c}\). Combining the following calculation
$$\begin{aligned} q_P^\prime (T-0)=\rho -pf_P(q_P(T),c^*(T))-pg(x^*_\pi ;x^*_\mu ,{x_\sigma ^*}) =\rho -(1-p)\underline{c}-pK<0, \end{aligned}$$
and the fact that the sign of
\(q_P^\prime (t)\) does not change, we deduce that
\(q_P(t)>0=(p-1)\ln \underline{c}+\ln \lambda , c^*(t)=\underline{c}\), and
\(q_P(t)\) satisfies ODE (
48) with
\(A=1/\lambda \) and
\(\overline{T}=T\) in the interval
\([\,0,T]\). Solving ODE (
48), we have
\(q_P(t)=q^3(t;1/\lambda ,0,T)\) and
\(c^*(t)=\underline{c}\) in the interval
\([\,0,T]\).
When \(\rho -pK=(1-p)\underline{c}\), it is easy to see that for \(t\in [\,0,T]\), we have \(q_P(t)=0\), and we still have \(q_P(t)\) equal to \(q^3(t;1,0,T)\) and \(c^*(t)=\underline{c}\) in the interval \([\,0,T]\).
(4.2) If
\((1-p)\underline{c}<\rho -pK\le (1-p)\overline{c}\), since
\(q(T)=0=(p-1)\ln \underline{c}+\ln \lambda \), and
$$\begin{aligned} q_P^\prime (T-0)=\rho -pf_P(q_P(T),c^*(T))-pg(x^*_\pi ;x^*_\mu ,{x_\sigma ^*}) =\rho -(1-p)\underline{c}-pK>0, \end{aligned}$$
then, when
t is close to
T, we have
\((p-1)\ln \overline{c}+\ln \lambda<q_P(t)<(p-1)\ln \underline{c}+\ln \lambda \). Thus,
\(q_P(t)\) satisfies ODE (
47) with
\(A=1/\lambda \) and
\(\overline{T}=T\) in the interval
\([\,\underline{T},T]\), until
\(\underline{T}=0\) or
\(q_P(\underline{T})=(p-1)\ln \overline{c}+\ln \lambda \) or
\(q_P(\underline{T})=(p-1)\ln \underline{c}+\ln \lambda \). Recalling the fact that the sign of
\(q_P^\prime (t)\) does not change, we deduce that
\(q_P(t)\) is nondecreasing with respect to
t. Thus, it is impossible that
\(q_P(\underline{T})=(p-1)\ln \underline{c}+\ln \lambda \) for some
\(\underline{T}\in [\,0,T)\).
Solving ODE (
47), we have
\(q_P(t)=q^2(t;1/\lambda ,\underline{T},T)\) and
\(c^*(t)=\widehat{c}(t)\) in the interval
\([\,\underline{T},T]\). Since
\((\rho -pK)-(1-p)\lambda ^{1/(1-p)}=(\rho -pK)-(1-p)\underline{c}>0\), then
\(q^2(t;1/\lambda ,0,T)\) is increasing with respect to
t, thus for
\(t\in [\,0,T)\), we have
$$\begin{aligned} (p-1)\ln \underline{c}+\ln \lambda= & {} 0=q^2(T;1/\lambda ,0,T)>q^2(t;1/\lambda ,0,T) >(1-p)\ln {1-p\over \rho -pK}+\ln \lambda \\\ge & {} (1-p)\ln {1\over \overline{c}}+\ln \lambda =(p-1)\ln \overline{c}+\ln \lambda . \end{aligned}$$
Therefore,
\(q_P(t)=q^2(t;1/\lambda ,0,T)\) and
\(c^*(t)=\widehat{c}(t)\) in the interval
\([\,0,T]\).
(4.3) If \(\rho -pK>(1-p)\overline{c}\), repeating the similar argument as in case (4.2), we deduce that \(q_P(t)=q^2(t;1/\lambda ,\underline{T},T)\) and \(c^*(t)=\widehat{c}(t)\) in the interval \([\,\underline{T},T]\), until \(\underline{T}=0\) or \(q_P(\underline{T})=(p-1)\ln \overline{c}+\ln \lambda \).
According to (
50),
\(q^2(0;1/\lambda ,0,T)<(p-1)\ln \overline{c}+\ln \lambda \) provided that
T is large enough. Thus, there exists a positive constant
\(T_{21}\) such that
\(q^2(T_{21};1/\lambda ,T_{21},T)=(p-1)\ln \overline{c}+\ln \lambda \), and
\(T_{21}\) is given in (
43). Hence, we derive that
\(q_P(t)=q^2(t;1/\lambda ,T_{21},T)\), and
\(c^*(t)=\widehat{c}(t)\) in the interval
\([\,T_{21},T]\).
Combining
$$\begin{aligned} q_P^\prime (T_{21})=\rho -pf_P(q_P(T_{21}),c^*(T_{21}))-pg(x^*_\pi ;x^*_\mu ,{x_\sigma ^*}) =\rho -pK-(1-p)\overline{c}>0, \end{aligned}$$
and the fact that the sign of
\(q_P^\prime (t)\) does not change, we deduce that in the interval
\([\,0,T_{21})\),
\(q_P(t)<(p-1)\ln \overline{c}+\ln \lambda ,\,c^*(t)=\overline{c}\), and
\(q_P(t)\) satisfies ODE (
46) with
\(A=\overline{c}^{p-1}\) and
\(\overline{T}=T_{21}\). Solving ODE (
46), we obtain
\(q_P(t)=q^1(t;\overline{c}^{p-1},0,T_{21})\) in the interval
\([\,0,T_{21}]\).
Case (5) \(\lambda ^{1/(1-p)}<\underline{c}<\overline{c}\).
Since
\(q_P(T)=0>(p-1)\ln \underline{c}+\ln \lambda \), then
\(q_P(t)\) satisfies ODE (
48) with
\(A=1/\lambda \) and
\(\overline{T}=T\) in the interval
\([\,\underline{T},T]\), until
\(\underline{T}=0\) or
\(q_P(\underline{T})=(p-1)\ln \underline{c}+\ln \lambda \). Solving ODE (
48), we obtain
\(q_P(t)=q^3(t;1/\lambda ,\underline{T},T)\) and
\(c^*(t)=\underline{c}\) in the interval
\([\,\underline{T},T]\).
(5.1) If \(\rho -pK\le \lambda \underline{c}^p-p\underline{c}\), then \(\rho +p\underline{c}-pK-\lambda \underline{c}^p\le 0\), and \(q^3(t;1/\lambda ,0,T)\) is nonincreasing with respect to t, thus for \(t\in [\,0,T]\), we have \(q^3(t;1/\lambda ,0,T)\ge q^3(T;1/\lambda ,0,T)=0>(p-1)\ln \underline{c}+\ln \lambda \). Therefore, \(q_P(t)=q^3(t;1/\lambda ,0,T)\) and \(c^*(t)=\underline{c}\) in the interval \([\,0,T]\).
(5.2) If
\(\lambda \underline{c}^p-p\underline{c}<\rho -pK\le (1-p)\underline{c}\), then
\(\rho +p\underline{c}-pK-\lambda \underline{c}^p>0\), and
$$\begin{aligned} q^3(t;1/\lambda ,0,T)\ge & {} \ln \lambda +\ln {\underline{c}^p\over \rho +p\underline{c}-pK} \\\ge & {} \ln {\underline{c}^p\over \underline{c}}+\ln \lambda =(p-1)\ln \underline{c}+\ln \lambda ,\;\;\forall \; t\in [\,0,T]. \end{aligned}$$
Therefore we still have
\(q_P(t)=q^3(t;1/\lambda ,0,T)\) and
\(c^*(t)=\underline{c}\) in the interval [0,
T].
(5.3) If
\((1-p)\underline{c}<\rho -pK\le (1-p)\overline{c}\), then (
52) implies that
\(q^3(0;1/\lambda ,0,T)<(p-1)\ln \underline{c}+\ln \lambda \) provided that
T is large enough. Thus, there exists a positive constant
\(T_{32}\) such that
\(q^3(T_{32};1/\lambda ,T_{32},T)=(p-1)\ln \underline{c}+\ln \lambda \), and
\(T_{32}\) is given in (
44). Hence, we derive
\(q_P(t)=q^3(t;1/\lambda ,T_{32},T)\), and
\(c^*(t)=\underline{c}\) in the interval
\([\,T_{32},T]\).
Since
$$\begin{aligned} q_P^\prime (T_{32})=\rho -pf_P(q_P(T_{32}),c^*(T_{32}))-pg(x^*_\pi ;x^*_\mu ,{x_\sigma ^*})=\rho -(1-p)\underline{c}-pK>0, \end{aligned}$$
then, when
\(t<T_{32}\) and
t is close to
\(T_{32}\), we have
\(q_P(t)<(p-1)\ln \underline{c}+\ln \lambda \) and
\(q_P(t)>(p-1)\ln \overline{c}+\ln \lambda \), and
\(q_P(t)\) is nondecreasing with respect to
t, and
\(q_P(t)\) satisfies ODE (
47) with
\(A=\underline{c}^{p-1}\) and
\(\overline{T}=T_{32}\) in the interval
\([\,\underline{T},\,T_{32}]\), until
\(\underline{T}=0\) or
\(q_P(\underline{T})=(p-1)\ln \overline{c}+\ln \lambda \). Solving ODE (
47), we obtain
\(q_P(t)=q^2(t;\underline{c}^{p-1},\underline{T},T_{32})\) and
\(c^*(t)=\widehat{c}(t)\) in the interval
\([\,\underline{T},\,T_{32}]\).
Since
\(\rho -pK-(1-p)\lambda ^{1/(1-p)}>\rho -pK-(1-p)\underline{c}>0\), then in this case
\(q^2(t;\underline{c}^{p-1},0,T_{32})\) is increasing with respect to
t, thus for
\(t\in [\,0,T_{32})\), we have
$$\begin{aligned} (p-1)\ln \underline{c}+\ln \lambda= & {} q^2(T_{32};\underline{c}^{p-1},0,T_{32})> q^2(t;\underline{c}^{p-1},0,T_{32})\\> & {} \ln \lambda +(1-p)\ln {1-p\over \rho -pK}\\\ge & {} (1-p)\ln {1\over \overline{c}}+\ln \lambda =(p-1)\ln \overline{c}+\ln \lambda . \end{aligned}$$
Therefore,
\(q_P(t)=q^2(t;\underline{c}^{p-1},0,T_{32})\) and
\(c^*(t)=\widehat{c}(t)\) in the interval
\([\,0,\,T_{32}]\).
(5.4) If \(\rho -pK>(1-p)\overline{c}\), repeating the similar argument as in case (5.3), we deduce that \(q_P(t)=q^3(T_{32};1/\lambda ,T_{32},T)\), and \(c^*(t)=\underline{c}\) in the interval \([\,T_{32},T]\), and \(q_P(t)=q^2(t;\underline{c}^{p-1},\underline{T},T_{32})\) and \(c^*(t)=\widehat{c}(t)\) in the interval \([\,\underline{T},\,T_{32}]\), until \(\underline{T}=0\) or \(q_P(\underline{T})=(p-1)\ln \overline{c}+\ln \lambda \).
According to (
50),
\(q^2(0;\underline{c}^{p-1},0,T_{32})<(p-1)\ln \overline{c}+\ln \lambda \) provided that
T is large enough. Thus, there exists a positive constant
\(T_{321}\) such that
\(q^2(T_{321};\underline{c}^{p-1},T_{321},T_{32})=(p-1)\ln \overline{c}+\ln \lambda \), and
\(T_{321}\) is given in (
45). Hence, we derive that
\(q_P(t)=q^2(t;\underline{c}^{p-1},T_{321},T_{32})\), and
\(c^*(t)=\widehat{c}(t)\) in the interval
\([\,T_{321},T_{32}]\).
Combining
$$\begin{aligned} q_P^\prime (T_{321})=\rho -pf_P(q_P(T_{321}),c^*(T_{321}))-pg(x^*_\pi ;x^*_\mu ,{x_\sigma ^*})=\rho -(1-p)\overline{c}-pK>0, \end{aligned}$$
and the fact that the sign of
\(q_P^\prime (t)\) does not change sign, we deduce that
\(q_P(t)\) is nondecreasing with respect to
t, and
\(q_P(t)<(p-1)\ln \overline{c}+\ln \lambda \) for any
\(t\in [\,0,T_{321})\). Thus,
\(c^*(t)=\overline{c}\) and
\(q_P(t)\) satisfies ODE (
46) in the interval
\(t\in [\,0,T_{321})\). Solving ODE (
46), we obtain
\(q_P(t)=q^1(t;\overline{c}^{p-1},0,T_{321})\) in the interval
\(t\in [\,0,T_{321}]\).
\(\square \)