boundedness of solutions

In this paper we study feedback interconnections of positive nonlinear SISO systems and introduce two complementary properties: persistent-input/persistent-output (PIPO) for the plant and persistent-input/transient-output (PITO) for the controller. Assuming forward completeness of the closed loop and a one-sided affine growth bound on the controller output, we show that PIPO and PITO jointly imply boundedness of the control signal. The result is input-output in nature and does not require linearity or monotonicity of the interconnected subsystems, although it does not in general guarantee boundedness of the full controller state. We provide a structural PIPO condition for positive monotone plants with a class-K_infinity steady-state characteristic, establish PITO for the antithetic integral controller with explicit gains, and illustrate the framework on a nonlinear integral-feedback motif with multiplicative controller growth.

Antithetic feedback controllers have become a key experimental and theoretical tool in synthetic biology. Introduced by Khammash and collaborators about 10 years ago, they are employed in order to achieve the practical regulation of protein expression, including tracking and robust disturbance rejection. In closed-loop, there are unique equilibria which, depending on parameter values, can be unstable. It had been shown, however, that this instability is not arbitrary: any bounded trajectory that stays away from the equilibrium must converge to a periodic orbit. This motivated a long-standing open question: is every trajectory bounded? In other words, even if the equilibrium is unstable, can nonlinear effects prevent unbounded excursions in the state space? This paper provides an affirmative answer, establishing the boundedness of all solutions. Previous attempts to prove this fact using Lyapunov functions had no success. Instead, this paper takes a completely different approach, specific to antithetic configurations, in which the key idea is to think of the controller as providing a ``persistently negative feedback'' which acts far away from the equilibrium in such a way so as to keep trajectories from diverging. This new approach, although tailored to the antithetic controller, might be useful in other applications as well.