Minimal norm control problems for linear mean-field control systems

Manfen An; Rui Gao; Yanqing Wang

Home/Archive/arXiv

arXiv · 2610.17825

Minimal norm control problems for linear mean-field control systems

Manfen An·Rui Gao·Yanqing Wang

View original source ↗Read source document ↗Export metadata ↓

Abstract

This paper studies controllability and minimal norm control problems (NP problems) for mean-field stochastic differential equation (MF-SDE) systems. We first establish an equivalent rank criterion for terminal controllability, namely, that the diffusion matrices $D$ and $D+\bar D$ are both of full row rank. We then derive an explicit Riccati-based representation of the minimal norm control by applying a parameterization technique. A key step is to identify the value $P(0)$ of an associated Riccati solution with the controllability Gramian of a controllable auxiliary system, which yields the parameter required in the construction. The resulting representation provides a basis for the future development and analysis of numerical methods for NP problems.

Explore related subjects

math.OC

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Manfen An, Rui Gao, Yanqing Wang. 2026-10-13. Minimal norm control problems for linear mean-field control systems. https://arxiv.org/abs/2610.17825

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Optimality conditions with cone-generalized convex vector functions

The object of investigation in this paper are generalized convex vector nonlinear programming problems with cone constraints. We prove that a quasiconvex vector function is pseudoconvex with respect to the same cone if and only if every vector critical point of the function is a weak global minimizer, a result, which is a natural extension of a known characterization of pseudoconvex scalar functions. We introduce a new notion of a Fréchet differentiable pseudoconvex problem. We derive that a problem with quasiconvex vector data is pseudoconvex if and only if every Fritz John vector critical point is a weakly efficient global solution. Thus, we generalize a lot of previous optimality conditions, concerning the scalar case and the multiobjective Paretian one. We also introduce the notion of a Fritz John cone-pseudoinvex vector problem. We prove that a problem with cone constraints is Fritz John pseudoinvex if and only if every vector critical point of Fritz John type is a weak global minimizer with respect to the considered cone..

math.OC↗

arXiv2026

Projecting onto a Capped Rotated Second-Order Cone

We derive a closed-form expression for the projection onto a capped rotated second-order cone, a convex set that arises in perspective relaxations of nonlinear programs with binary indicator variables. The closed-form solution involves four distinct cases, one of which coincides with the classical projection onto a secondorder cone. The remaining three cases involve a nontrivial analysis in determining the necessary and sufficient conditions under which the solution lies on the intersection of the cone and a facet of a box. Beyond theoretical interest, this result enables the development of efficient projection-based first-order methods for optimization problems involving perspective reformulations. As an illustration, we consider the continuous perspective relaxation of sparse logistic regression, whose feasible region contains a Cartesian product of capped rotated second-order cones. Computational experiments on synthetic datasets demonstrate that accelerated projected-gradient, together with computing our closed-form projection at each iteration, improves on the running time of a general first-order conic solver while obtaining solutions of similar accuracy. These results highlight the practical value of the proposed projection formula for solving continuous relaxations of perspective formulations of mixed-integer second-order cone problems.

math.OC↗

arXiv2026

Time-Invariant Linear Quadratic Regulators: A New Look

This paper demonstrates that linear ordinary differential equations with constant coefficients serve as the main building block for time-invariant linear quadratic regulators (LQRs). By applying the Euler-Lagrange equations to a flat output, i.e., to a basis of the free finitely-generated module over the commutative principal ideal ring of time-invariant linear differential operators with respect to the time derivation, this result emerges directly. In module-theoretic terms. The freeness of this structure corresponds to the classical Kalman controllability condition (M. Fliess, Some basic structural properties of generalized linear systems, Syst. Contr. Lett., 15 (1990) 391-396). Within this framework, the two-point boundary value problem, including the control variables, becomes analytically tractable, enabling the derivation of optimal time horizons, optimal parameter design, and rest-to-rest trajectories. Furthermore, a novel turnpike-like phenomenon is established via nonstandard analysis. The feedback loop is closed thanks to the HEOL approach (C. Join, E. Delaleau, M. Fliess, Flatness-based control revisited: The HEOL setting, C.R. Math., 362 (2024) 1693-1706), an extension of model-free control (MFC) that leverages intelligent controllers. Our viewpoint, which significantly simplifies the presentation by avoiding the conventional Kalman state-space representation, relies on foundational algebraic concepts that, while elementary, are rarely used in today's control theory. To aid intuition, some rudimentary examples are presented, most of which include computer simulations.

math.OC↗