| Publications of Eduardo D. Sontag jointly with M.K. Wafi |
| Articles in journal or book chapters |
| This paper studies whether solutions of a class of nonlinear feedback systems remain bounded over time. The systems we consider arise naturally in synthetic biology, where the antithetic feedback controller regulates a biological process through a delayed feedback loop. Our main result is that every trajectory of such a system is bounded. The key insight is simple: if the regulated state grows too large for too long, the feedback loop will eventually respond and push it back down. More precisely, we show that whenever the state exceeds a threshold and remains there long enough, the feedback signal becomes strong enough to force the state to decrease. We then show that once this happens, the feedback remains strong enough to keep the state from growing unbounded. The proof works directly with differential inequalities and does not require constructing a Lyapunov function, making the mechanism transparent and easy to interpret. The boundedness result can be understood as a time-domain small-gain effect, where the delayed feedback ultimately counteracts any persistent growth in the system. |
| Conference articles |
| In this work we study the convergence of gradient methods for nonconvex optimization problems -- specifically the effect of the problem formulation to the convergence behavior of the solution of a gradient flow. We show through a simple example that, surprisingly, the gradient flow solution can be exponentially or asymptotically convergent, depending on how the problem is formulated. We then deepen the analysis and show that a policy optimization strategy for the continuous-time linear quadratic regulator (LQR) (which is known to present only asymptotic convergence globally) presents almost global exponential convergence if the problem is overparameterized through a linear feed-forward neural network (LFFNN). We prove this qualitative improvement always happens for a simplified version of the LQR problem and derive explicit convergence rates for the gradient flow. Finally, we show that both the qualitative improvement and the quantitative rate gains persist in the general LQR through numerical simulations. |
| We study the monotonicity of the cumulative dose response (cDR) for a class of incoherent feedforward motif (IFFMs) systems with linear intermediate dynamics and nonlinear output dynamics. While the instantaneous dose response (DR) may be nonmonotone with respect to the input, the cDR can still be monotone. To analyze this phenomenon, we derive an integral representation of the sensitivity of cDR with respect to the input and establish general sufficient conditions for both monotonicity and non-monotonicity. These results reduce the problem to verifying qualitative sign properties along system trajectories. We apply this framework to four canonical IFFM systems and obtain a complete characterization of their behavior. In particular, IFFM1 and IFFM3 exhibit monotone cDR despite potentially non-monotone DR, while IFFM2 is monotone already at the level of DR, which implies monotonicity of cDR. In contrast, IFFM4 violates these conditions, leading to a loss of monotonicity. Numerical simulations indicate that these properties persist beyond the structured initial conditions used in the analysis. Overall, our results provide a unified framework for understanding how network structure governs monotonicity in cumulative input–output responses. |
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