(p.194) Appendix: Three Useful Theorems from Dynamic Optimization
(p.194) Appendix: Three Useful Theorems from Dynamic Optimization
In this book, we have presumed that economic agents behave intertemporally and perform dynamic optimization. In this appendix, we present some basics of the method of dynamic optimization using Pontryagin's maximum principle and the Hamiltonian.
Let an intertemporal optimization problem be given by
F(x(t), u(t)), f_{i}(x(t), u(t)), and ∂f_{i}(x(t), u(t))/∂x_{j}(t), ∂F(x(t), u(t))/∂x_{j}(t) are continuous with respect to all n + m variables for i, j = 1, …, n Further, u(t) is said to be admissible if it is a piecewise continuous function on [0,∞) with u(t) ∈ Ω.
Define the currentvalue Hamiltonian ℋ(x(t)), u(t), λ(t), λ_{0}) as follows:
Theorem A.1 Let u ^{○}(t) be an admissible control and x ^{0}(t) is the trajectory belonging to u ^{0}(t). For u ^{0}(t) to be optimal, it is necessary that there (p.195) exists a continuous vector function λ(t) = (λ_{1}(t),…,λ_{n}(t) with piecewise continuous derivatives and a constant scalar λ_{0} such that

a. λ(t) and x ^{0}(t) are solutions of the canonical system
$$\begin{array}{l}{\dot{x}}^{0}(t)=\frac{\partial}{\partial \lambda}\mathcal{H}({x}^{0}(t),\text{}{u}^{0}(t),\lambda (t),{\lambda}_{0}),\\ \dot{\lambda}(t)=\rho \lambda (t)\frac{\partial}{\partial x}\mathcal{H}({x}^{0}(t),\text{}{u}^{0}(t),\lambda (t),{\lambda}_{0}).\end{array}$$ 
b. For all t ∈ [0,∞) where u ^{0}(t) is continuous, the following inequality must hold: H(x ^{0}(t), u ^{0}(t), λ(t), λ_{0}) ≥ H(x ^{0}(t), u(t), λ(t), λ_{0}),

c. (λ_{0}, λ(t) ≠ (0,0) and λ_{0} = 1 or λ_{0} = 0.
Remarks:

1. If the maximum with respect to u(t) is in the interior of Ω, ∂ℋ(·)/∂u(t) can be used as a necessary condition for a local maximum of ℋ(·).

2. It is implicitly assumed that the objective functional (A.1) takes on a finite value, that is, ${\int}_{0}^{\infty}{e}^{\rho t}}F({x}^{0}\left(t\right),\text{}{u}^{0}\left(t\right))\text{}<\text{}\infty $. if x ^{0} and u ^{0} grow without an upper bound F(·) must not grow faster than ρ.
Theorem A.1 provides only necessary conditions. The next theorem gives sufficient conditions.
Theorem A.2 If the Hamiltonian with λ_{0} = 1 is concave in (x(t), u(t)) jointly and if the transversality condition lim_{t→∞} e ^{ρ t}λ(t)(x(t  x ^{0}(t)) ≥ 0 holds, conditions a and b from theorem A.1 are also sufficient for an optimum. If the Hamiltonian is strictly concave in (x(t), u(t)) the solution is unique.
Remarks:

1. If the state and costate variables are positive the transversality condition can be written as stated in the foregoing chapters, that is, as lim_{t→∞} e ^{ρt}λ(t) x ^{0}(t) = 0.^{1}

2. Given some technical conditions, it can be shown that the transversality condition is also a necessary condition.
Theorem A.2 requires joint concavity of the currentvalue Hamiltonian in the control and state variables. A less restrictive theorem is the following.
Because the joint concavity of ℋ(x(t), u(t), λ(t), λ_{0}) with respect to (x(t), u(t)) implies concavity of ℋ^{0}(x(t), λ(t, λ_{0} with respect to x(t), but the reverse does not necessarily hold, theorem A.3 may be applicable where theorem A.2 cannot be applied.Theorem A.3 If the maximized Hamiltonian
(p.196) with λ_{0} = 1 is concave in x(t) and if the transversality condition lim_{t→∞} e ^{ρt}λ(t)(x(t)  x ^{0}(t)) ≥ 0 holds, conditions a and b from theorem A.1 are also sufficient for an optimum. If the maximized Hamiltonian ℋ^{0}(x(t), λ(t), λ_{0}) is strictly concave in x(t) for all t, x ^{0}(t) is unique (but not necessarily u ^{0}(t)).$${\mathcal{H}}^{0}(x(t),\text{}\lambda (t),{\lambda}_{0})=\underset{u(t)\in \text{\Omega}}{\mathrm{max}}\text{}\mathcal{H}(x(t),\text{}\lambda (t),{\lambda}_{0})$$
The three theorems demonstrate how optimal control theory can be applied to solve dynamic optimization problems. The main role is played by the Hamiltonian (A.3). It should be noted that in most economic applications, as in this book, interior solutions are optimal so that ∂ℋ(·)/∂u(t) = 0 can be presumed. For further reading and more details concerning optimal control theory, we refer to the books by Feichtinger and Hartl (1986) or Seierstad and Sydsaeter (1987).
Notes:
(1) Note that in the book we did not indicate optimal values by^{0}.