Families of Control-Cost-Parametrized Inverse-Optimal Universal Stabilizers

2026-06-08

AI Summary

This work addresses the lack of adaptability to user-specified control cost preferences in conventional universal stabilizers. It proposes a “semi-direct optimal” framework that constructs a family of inverse-optimal universal stabilizers parametrized explicitly by a user-defined control effort cost. The approach employs a three-step nonlinear operator—incorporating cost differentiation and functional inversion—to generate an extended feedback law that solves an infinite-horizon optimal control problem with state-dependent costs. For the first time, it enables arbitrary specification of the control cost and yields an explicit parameterization of the stabilizer, while establishing Lipschitz continuity of the mapping from cost parameters to controllers. This property facilitates unified neural operator approximation and online adaptation. Theoretical guarantees include semi-global practical asymptotic stability and second-order suboptimality, and numerical experiments validate the efficacy of operator learning for stable control synthesis.

Abstract

A classical universal stabilization formula offers the practitioner no design freedom: it is a single, parameter-free object. We introduce a cost-parametrized family of stabilizing feedback laws, where (1) the user chooses a function that serves as the running cost on control in an inverse-optimal cost functional, and (2) obtains, through a formula, a nonlinear "expander" of a pre-existing universal controller, which solves an infinite-horizon optimal control problem with a meaningful cost on the state. The cost-to-expander formula is a three-step construction, involving, inter alia, cost differentiation and function inversion-overall, a nonlinear infinite-dimensional operator. The cost-to-expander operator is proven Lipschitz, which enables uniform neural operator approximation of the entire family and supports both offline performance exploration and online adaptation. Semiglobal practical asymptotic stability and second-order suboptimality bounds are established under the approximation. The operator learning and its use in semiglobal stabilization are illustrated numerically. We call the result 'half-direct-optimal' because the paper's design is less than a general 'direct optimal' (HJB-inducing) control, but more than the fully inverse optimal, since the user performs minimization for an arbitrary given cost on control. The dual to the half-direct problem we solve is the problem in which the cost on the state is arbitrary and given. This dual problem is easier and outside of the scope of the paper.

Problem

• universal stabilization
• inverse-optimal control
• control-cost parametrization
• feedback design freedom
• optimal control

Innovation

• inverse-optimal control
• universal stabilizers
• cost-parametrized feedback
• neural operator approximation
• half-direct-optimal