Dynamic programming and Bellman Equation

We consider the following minimization problem

subject to

where we do not assume any function. Here, we have two kinds of costs, x and v.

We convert it into the following Bellman equation

subject to at time t

We guess .

Then we get

Taking the derivative with respect to v_t, we get

So

To find A, we plug v_t back into Bellman equation and get

Factoring out xt²,

Matching the coefficients,

(*)

We cannot solve out A explicitly but we can get A by some nonlinear optimization.

The value function we get is

using the solution of A.