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]).