If I have a quadratic cost functional such as \begin{equation} J(x_0, \ u) = \frac{1}{2}\int_0^\infty \bigg(x^TQx + u^TRu \bigg)\ dt \end{equation} $Q, R$ are positive-definite, and the dynamics $\dot{x} = f(x)+g(x)u$, $x(0) = x_0$ are such that $g(0) = 0$, and also that $f$ is not dissipative with respect to any positive definite storage function, for example $f = Ax$ with $A$ having only imaginary eigenvalues, so that trajectories may decay very slowly around the origin, how can I prove or disprove that the minimum cost functional is actually finite?
My first try with this was knowing that state components that decay at a rate of $\frac{1}{x^p}$ where $p>.5$ would lead to a finite $J$, but I'm not sure how to bound solutions to ODEs I don't know the solution to, much less ones that aren't fully determined (the $u$ part).
Any advice toward how to begin solving a problem like this would be appreciated. I'm also aware that the optimum may not exist in the sense that one can always find a smaller number by perturbing the control by a vanishingly small amount - something like the concept of an open set in the space of controllers - this information would be welcome although isn't my primary aim as I intend to address the problem of finding the "practically optimal" control via numerical methods.