Constructive reachability for linear control problems under conic constraints

TL;DR

This paper develops a constructive duality-based method to analyze reachability in linear infinite-dimensional control systems with conic constraints, providing new necessary and sufficient conditions for approximate and exact reachability.

Contribution

It introduces a novel duality approach for reachability under conic constraints, extending existing methods and covering both convex and nonconvex cases with new theoretical results.

Findings

Derived new sufficient and necessary conditions for reachability.

Extended the approach to nonconvex conic constraints with bang-bang properties.

Unified and generalized previous results in the literature.

Abstract

Motivated by applications requiring sparse or nonnegative controls, we investigate reachability properties of linear infinite-dimensional control problems under conic constraints. Relaxing the problem to convex constraints if the initial cone is not already convex, we provide a constructive approach based on minimising a properly defined dual functional, which covers both the approximate and exact reachability problems. Our main results heavily rely on convex analysis, Fenchel duality and the Fenchel-Rockafellar theorem. As a byproduct, we uncover new sufficient conditions for approximate and exact reachability under convex conic constraints. We also prove that these conditions are in fact necessary. When the constraints are nonconvex, our method leads to sufficient conditions ensuring that the constructed controls fulfill the original constraints, which is in the flavour of bang-bang type properties. We show that our approach encompasses and generalises several works, and we obtain new results for different types of conic constraints and control systems.