On Controllability and Approximate Sturm-Liouville Problems in Time-Varying Second-Order Differential Equations

Abstract

This paper considers a time-varying second-order Sturm-Liouville boundary value problem on a finite time interval as a nominal version of another perturbed differential equation that is not related to a Sturm-Liouville problem. It is proven that, under controllability conditions of the nominal system, there is a bounded continuous control function, for each real constant eigenvalue of the nominal Sturm-Liouville system, which plays its role in the current perturbed one such that the same two-point boundary values as those of the nominal Sturm-Liouville system are matched by the trajectory solution. In a more general context, it is possible to fix arbitrary finite two point boundary values for the perturbed current one which are not necessarily coincident with those of the nominal Sturm-Liouville system by an appropriate synthesis of such a function. This property is a direct consequence of the controllability property. Since the mentioned control function is not constant, the perturbed system is not a Sturm-Liouville one even if the two-boundary values are identical to those of the nominal Sturm-Liouville system. It is also characterized, in terms of norms, the worst-case errors of the two-point boundary values of the current perturbed differential system with respect to those of the nominal Sturm-Liouville system provided that the nominal constant eigenvalue is not replaced by the appropriate time-varying function based on nominal controllability conditions. Under small deviations of the parameterization of the perturbed system with respect to the nominal one, such a worst-error characterization makes the current perturbed system to be an approximate Sturm-Liouville system of the nominal one.