PhronesisIn practical systems, the controller rarely knows the system's absolute state S or the desired state S_* with certainty. Instead, the controller observes only the displacement = S - S_*: the gap between current and desired. We develop a control-theoretic framework where the control input u(t) depends solely on measured displacement (t), not on estimates of S or S_* separately. This is the problem of output feedback control with a single measured output.
Using Pontryagin's maximum principle, we characterize the optimal control law and show a surprising result: under relative measurements, optimal control is smooth (proportional to displacement), never bang-bang, and incurs zero asymptotic penalty. The control law is u^*() = g where g is a proportional gain, identical in form to the unconstrained linear control of absolute displacement, but derived under the constraint of relative-only feedback.
We show that the periodic-returns strategy emerges naturally as a special case when control effort becomes expensive, and we discuss the implications for systems where only relative state information is available: interpersonal relationships, organizational feedback, and ecosystem monitoring. The theory explains why proportional (P) control without integral correction often suffices in practice despite relying solely on current error.