As a first step, we notice that the existence of an optimal control
\(l^*\) can be proved by employing the results outlined for a general minimum Bolza problem in Theorem 4.1 and Corollary 4.1, page 68, in Fleming and Rishel (
1975). With this aim, we write the problem in equivalent form with similar notation as in Fleming and Rishel (
1975): the state variables are arranged in vector form such that
\(x(t)=(S(t), W(t))^{\scriptscriptstyle T}\) and the corresponding differential system is revisited as
$$\begin{aligned} \dot{x}(t)=f(l(t)), \end{aligned}$$
where
\(x(0)=x_0:=(S_0,W_0)^{\scriptscriptstyle T}\) and the right-hand side is defined as
\(f(u)= \alpha + \beta l\) with vector coefficients given by
\(\alpha = (0, (\gamma _2-1)m_2L)^{\scriptscriptstyle T}\) and
\(\beta =(-m_1,\gamma _1m_1+(1-\gamma _2)m_2)^{\scriptscriptstyle T}\). In addition, by interpreting the definition of feasible class
\({\mathcal {F}}'\) given in Fleming and Rishel (
1975), here
\({\mathcal {F}}'\) merely corresponds to the class of all pairs
\((x_0,l)\) such that
\(l \in {\mathcal {A}}\) and the state variable can be integrated as
$$\begin{aligned} x(t)= x(0)+ \int _0^t f(l(s)) \, ds, \qquad 0 \le t \le T, \end{aligned}$$
(21)
by prescribing the initial condition
\(x(0)=x_0\) which is given and fixed in our problem. Furthermore, using definitions in (
20) of benefit and damage functions, we consider the functional
$$\begin{aligned} {\mathcal {L}}(t,x(t),l(t))=e^{- \delta t} \left( c_W \frac{[x^{(2)}(t)]^2}{2}-2c_S \sqrt{x^{(1)}(t)}-\frac{[C(l(t))]^{1- \sigma }}{1- \sigma } \right) , \end{aligned}$$
with
\(x^{(1)}(t)\) and
\(x^{(2)}(t)\) corresponding to the entries of vector
x(
t) for any time
t. In this framework, the so-called performance index can be defined as
$$\begin{aligned} \widetilde{J}(x_0,l) = \int _0^T {\mathcal {L}}(t,x(t),l(t)) \, dt - \Psi (x^{(1)}(T),x^{(2)}(T),T). \end{aligned}$$
We remark that our functional
J(
l) in (
11) takes on the same value as
\(\widetilde{J}(x_0,l)\) but opposite in sign (i.e.,
\(\widetilde{J}(x_0,l)=-J(l)\)). As a consequence, our model (
11) is equivalent to the problem which consists of finding a control
\(l^* \in {\mathcal {A}}\) such that the corresponding performance index
\(\widetilde{J}(x_0,l^*)\) is minimized in the class
\({\mathcal {F}}'\). The result provided by Theorem 4.1 and Corollary 4.1 in Fleming and Rishel (
1975) represents a sufficient condition for the existence of a solution, and it is employed for the specific problem we face. Actually, in our case, it can be applied since
f(
u) is continuous and there exists a positive constant
\(C_1= \max \{ |\alpha |, |\beta |\}\) such that
\(|f(l) |\le C_1 (1+ |l |)\) for all
\(l \in {\mathcal {A}}\). In addition, Theorem 4.1 in Fleming and Rishel (
1975) relies on further assumptions (a), (b), (c), (d), (e) which hold in our case according to the following items:
-
The assumption (a) stated in Theorem 4.1 of Fleming and Rishel (
1975) is merely satisfied; indeed, the feasible class
\({\mathcal {F}}'\) is not empty due to the fact that it is possible to pick any
\(l \in {\mathcal {A}}\) which is Lebesgue integrable, then the state system admits a unique solution which is obtained by the integration in (
21).
-
The subset of
\(\mathbb R\) involved in the assumption (b) stated in Theorem 4.1 of Fleming and Rishel (
1975) corresponds to the interval [0,
L] where each labor function
l(
t) has its value. The same assumption is verified as [0,
L] is a closed set.
-
Due to (
19),
x(
t) is bounded for any
t; therefore, each labor
\(l \in {\mathcal {A}}\) steers
x(
T) to a compact set where the scrap function
\(\Psi \) is continuous with respect to its inputs. Then, the assumption (c) stated in Theorem 4.1 of Fleming and Rishel (
1975) is satisfied.
-
The assumption (d) in Theorem 4.1 is replaced by the corresponding (d’) stated in Corollary 4.1 of Fleming and Rishel (
1975); it is satisfied since
\({\mathcal {L}}(t,x, \cdot )\) is convex in [0,
L]. Indeed, it is not so difficult to verify that
$$\begin{aligned} \frac{\partial {\mathcal {L}}}{\partial l}=e^{- \delta t} [C(l)]^{- \sigma } \left( \sigma [C(l)]^{-1} \left( \frac{d C}{d l} \right) ^2- \frac{d^2C}{d l^2} \right) >0, \end{aligned}$$
since
$$\begin{aligned} \frac{d^2C}{d l^2} = (\rho -1) [C(l)]^{1-2 \rho } \, \theta (1 - \theta ) L^2 [m_1 \, m_2]^{ \rho } \cdot [l(L-l)]^{ \rho -2 } <0, \end{aligned}$$
under the assumption
\(0< \rho <1\).
-
Also, the assumption (e) in Theorem 4.1 is replaced by the corresponding (e’) stated in Corollary 4.1 of Fleming and Rishel (
1975). In this respect, due to the bounds in (
19), for any
\(t \in [0,T]\) we have
\(0 < \sqrt{x^{(1)}(t)} \le \sqrt{S_0}\) and
$$\begin{aligned}{}[x^{(2)}(t)]^2 \ge \underline{W}^2 \ge (1- \gamma _2)^2 \, (m_2 T)^2 \, [l(t)]^2-2W_0 (1- \gamma _2) m_2 LT. \end{aligned}$$
Furthermore, due to the fact that the consumption function
\(C( \cdot )\) is continuous, we may consider its maximum value
\(\overline{C}= \max _{0 \le z \le L} C(z)\). It follows that there exist two constants
\(c_1= e^{- \delta T}(c_W/2) (1 - \gamma _2)^{2} \, (m_2T)^{2}>0\) and
\(c_2=e^{- \delta T} (c_W W_0 (1- \gamma _2) m_2 LT +2 c_S \sqrt{S_0}+\overline{C}^{1-\sigma }/(1- \sigma ) )>0\) such that
$$\begin{aligned} \mathcal {L} (t,x(t),l(t)) \ge c_1 |l(t) |^2 -c_2. \end{aligned}$$
Thus, assumption (e’) stated in Corollary 4.1 of Fleming and Rishel (
1975) is satisfied.
Under the previous argument, all the assumptions stated in Theorem 4.1 of Fleming and Rishel (
1975) hold; as a consequence, there exists
\(l^* \in {\mathcal {A}}\) minimizing the performance index
\(\widetilde{J} (x_0,l)\) on the class
\(\mathcal {F}'\). Due to the equivalence between the two problems,
\(l^*\) represents an optimal control which solves also our model (
11). Therefore, the existence of a solution is proved for the problem at hand.
As the next step, we focus on the uniqueness proof. We notice that, in correspondence with any optimal solution
\(l^*\) of (
11), the Hamiltonian function
\(H(l, S, W, \varphi _1, \varphi _2)\) defined in the previous Sect.
4 is maximized with respect to
l, according to the Maximum Principle statement (
i). In this respect, we remark that it is not so difficult to get the second derivative:
$$\begin{aligned} \frac{\partial ^2 H}{\partial l^2}= [C(l)]^{1- \sigma -2 \rho } \, \left( - \sigma \, [ \Upsilon (l)]^2+ (\rho -1) \theta (1 - \theta ) L^2 [m_1 m_2]^{ \rho } [l(L-l)]^{ \rho -2 } \right) . \end{aligned}$$
Since
\(\partial ^2 H/ \partial l^2<0\), then the Hamiltonian function is strictly concave in
l and it can admit no more than one maximum value. As a conclusion, problem (
11) can admit no more than one solution.
\(\square \)