\(\def\Real{\mathbb{R}}

\def\Comp{\mathbb{C}}

\def\Rat{\mathbb{Q}}

\def\Field{\mathbb{F}}

\def\Fun{\mathbf{Fun}}

\def\e{\mathbf{e}}

\def\f{\mathbf{f}}

\)

Consider the optimal control problem

\[

\dot{x}=u; x(0)=x_o; \int_0^T u^2/2dt+M(x(T))\to\min.

\]

The task is to find the cost function \(S(x_o)\) (the optimal achievable value).

It can be solved as follows. Assume for …