Relative controllability of neutral systems with ψ-Prabhakar derivatives and constant delay: minimum-energy infusion design in pharmacokinetics

We study the relative controllability of a neutral semilinear system driven by a ψ -Prabhakar Caputo-type derivative with a constant retardation, together with the design of the infusion protocol that achieves a prescribed terminal state at least cost. The neutral structure is removed by an algebraic reduction, and the two delayed exponential kernels that represent the solution are derived, rather than postulated, by inverting the ψ -Laplace transform of the reduced system. Three findings follow. First, the controllability Gramian of such a system exists as a finite matrix precisely when the order of the derivative exceeds one half; below that threshold the Gramian diverges and the method fails, a restriction that is invisible in the integer-order theory. Second, above the threshold the Gramian is the composition of an explicitly identified reachability operator with its adjoint, from which a necessary and sufficient criterion follows for the linear system, and from which the steering control is characterised as the protocol of minimum energy among all protocols achieving the transfer. Third, for the semilinear system an explicit and checkable smallness condition on the nonlinearity yields relative controllability through Krasnoselskii’s theorem. The theory is applied to a two-compartment pharmacokinetic model with saturable hepatic elimination and a distribution lag. For theophylline data the criterion is satisfied with a wide margin, and the computed protocol steers plasma and tissue concentrations to their therapeutic targets. We also show that the optimal infusion rate, although of finite energy and everywhere nonnegative, diverges at the terminal time at a rate fixed by the order of the derivative, so that an implementable protocol must saturate near the end of the horizon.