Minimax Linear Regulator Problems for Positive Systems

Explicit solutions to optimal control problems are rarely obtainable. Of particular interest are the explicit solutions derived for minimax problems, providing a framework to address adversarial conditions and uncertainty. This work considers a multi-disturbance minimax Linear Regulator (LR) framework for positive linear time-invariant systems in continuous time, which, analogous to the Linear-Quadratic Regulator (LQR) problem, can be utilized for the stabilization of positive systems. The problem is studied for nonnegative and state-bounded disturbances. Dynamic programming theory is leveraged to derive explicit solutions to the minimax LR problem for both finite and infinite time horizons. In addition, a fixed-point method is proposed that computes the solution for the infinite horizon case, and the minimum L1-induced gain of the system is studied. We motivate the prospective scalability properties of our framework with a large-scale water management network.

💡 Research Summary

The paper addresses a gap in optimal control theory by providing explicit solutions for a class of minimax linear regulator (LR) problems applied to continuous‑time positive linear time‑invariant (LTI) systems. Positive systems are characterized by Metzler dynamics and the property that non‑negative inputs and initial conditions generate non‑negative states and outputs. This structural property enables the authors to replace the traditionally intractable Hamilton‑Jacobi‑Isaacs (HJI) partial differential equation with either an ordinary differential equation (ODE) for a finite horizon or an algebraic equation for an infinite horizon, both of which are linear in the costate vector.

The system under study is
\