Can someone here knowing Finnish help me with this question in mathematical physics? I am trying to understand the solution to the problem 2:
[here] (https://dl.dropboxusercontent.com/u/96742826/School/DY_ratkaisut190214.pdf) that uses the Hamilton conditions for the HJB. I don't understand how the Hamilton conditions can be used with the Hamilton-Jacobian-Bellman equations.
Questions
1. How are the HJB conditions and Hamilton conditions related to each other?
2. When are the HJB conditions and Hamilton conditions the same?
3. When is the Euler equation sufficient condition? With fixed end points' problem? Euler equation is below (Kirk p.148). $$\frac{\partial g}{\partial x}=\frac{d}{dt}\frac{\partial g}{\partial \dot x}$$
Necessary Conditions for the Hamilton equation
$\frac{\partial H}{\partial u}=0$
$\frac{\partial H}{\partial x}=-\dot p^*(t)$
$f=\dot x^*(t)=\frac{\partial H}{\partial p}$
Equations
Hamilton equation
$H(x,u,p)=g(x,u)+p'f(x,u)$
where $f(x,u)=\dot x$ is the system function, $p$ is the "liittomuuttuja"
HJB equation (continuous case, compare to Discreate DP case p.3)
$0=\min_{u\in U}\{g(x,u)+\nabla_t V(t,x)+\nabla_x V(T,x)'f(x,u)\}$