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Positivity of infinite-dimensional linear systems

Yassine El Gantouh

TL;DR

Addresses positivity and well-posedness of infinite-dimensional linear systems with unbounded input/output operators in Banach lattices. Develops $L^{p}$-admissible positive control/observation operator theory, characterizes internal/external positivity of input–output systems, and proves equivalence of weak and strong regularity for positive $L^{p}$‑well‑posed systems, along with Desch–Schappacher and Staffans–Weiss perturbation results. The theory is illustrated by a Boltzmann/transport equation on finite networks, yielding existence and uniqueness of positive mild solutions. The work provides a rigorous framework for positivity-preserving control and observation in PDEs and network models, with implications for stability and perturbation analysis.

Abstract

In this paper, we investigate the well-posedness and positivity property of infinite-dimensional linear system with unbounded input and output operators. In particular, we characterize the internal and external positivity for this class of systems. This latter effort is motivated in part by a complete description of well-posed positive control/observation systems. An interesting feature of positive well-posed linear systems is that weak regularity and strong regularity, in the sense of Salamon and Weiss, are equivalent. Moreover, we provide sufficient conditions for zero-class admissibility for positive semigroups. In the context of positive perturbations of positive semigroups, we establish two perturbation results, namely the Desch-Schappacher perturbation and the Staffans-Weiss perturbation. As for illustration, these findings are applied to investigate the existence and uniqueness of a positive mild solution of the linear Boltzmann equation with non-local boundary conditions on finite network.

Positivity of infinite-dimensional linear systems

TL;DR

Addresses positivity and well-posedness of infinite-dimensional linear systems with unbounded input/output operators in Banach lattices. Develops -admissible positive control/observation operator theory, characterizes internal/external positivity of input–output systems, and proves equivalence of weak and strong regularity for positive ‑well‑posed systems, along with Desch–Schappacher and Staffans–Weiss perturbation results. The theory is illustrated by a Boltzmann/transport equation on finite networks, yielding existence and uniqueness of positive mild solutions. The work provides a rigorous framework for positivity-preserving control and observation in PDEs and network models, with implications for stability and perturbation analysis.

Abstract

In this paper, we investigate the well-posedness and positivity property of infinite-dimensional linear system with unbounded input and output operators. In particular, we characterize the internal and external positivity for this class of systems. This latter effort is motivated in part by a complete description of well-posed positive control/observation systems. An interesting feature of positive well-posed linear systems is that weak regularity and strong regularity, in the sense of Salamon and Weiss, are equivalent. Moreover, we provide sufficient conditions for zero-class admissibility for positive semigroups. In the context of positive perturbations of positive semigroups, we establish two perturbation results, namely the Desch-Schappacher perturbation and the Staffans-Weiss perturbation. As for illustration, these findings are applied to investigate the existence and uniqueness of a positive mild solution of the linear Boltzmann equation with non-local boundary conditions on finite network.
Paper Structure (6 sections, 18 theorems, 114 equations)

This paper contains 6 sections, 18 theorems, 114 equations.

Table of Contents

  1. Introduction
  2. Admissible positive control operators
  3. Admissible positive observation operators
  4. Positivity of input-output system
  5. Application
  6. Conclusion

Key Result

Proposition 2.1

Let $X , U$ be Banach lattices, let $A$ generates a strongly continuous semigroup $\mathbb{T}$ on $X$ and $B\in \mathcal{L}(U,X_{-1})$. Then, the following assertions are equivalent: In addition, if one of these conditions is satisfied then the differential equation S2.1 has a unique positive mild solution $z(\cdot)\in C(\mathbb R_+;X)$ given by $z(t)=T(t)x+\Phi_tu$ for all $t\ge 0$ and $(x,u)\in

Theorems & Definitions (53)

  • Definition 2.1
  • Proposition 2.1
  • proof
  • Remark 2.1
  • Lemma 2.1
  • proof
  • Remark 2.2
  • Remark 2.3
  • Theorem 2.1
  • proof
  • ...and 43 more