Closed-form solutions

According to (14.1), the action trajectory must be integrated to produce a state trajectory. In some cases, this integration leads to a closed-form expression. For example, if the system is a chain of integrators, then a polynomial expression can easily be obtained for $ x(t)$. For example, suppose $ q$ is a scalar and $ {\ddot q}
= u$. If $ q(0) = {\dot q}(0) = 0$ and a constant action $ u=1$ is applied, then $ x(t) = t^2/2$. If $ {\dot x}=
f(x,u)$ is a linear system (which includes chains of integrators; recall the definition from Section 13.2.2), then a closed-form expression for the state trajectory can always be obtained. This is based on matrix exponentials and is given in many control theory texts (e.g, [192]).

Steven M LaValle 2012-04-20