On the Lyapunov equation with the state matrix in companion form

Technical Summary: On the Lyapunov Equation with the State Matrix in Companion Form

Problem Statement
This paper investigates the entrywise nonnegativity of the unique solution $X$ to the continuous-time Lyapunov equation:
$$XA + A^\top X = -Q$$
where $A \in \mathbb{R}^{n \times n}$ is a Hurwitz companion matrix associated with a polynomial $a(s)$, and $Q = Q^\top \succeq 0$ is a positive semidefinite matrix.

While standard Lyapunov theory guarantees that $X$ is symmetric and positive semidefinite (in the Loewner order), this property does not imply that $X$ is entrywise nonnegative ($X_{ij} \geq 0$ for all $i,j$). The paper addresses Conjecture 1.1, proposed by Prof. Luigi Fortuna, which posits that if $A$ is a Hurwitz companion matrix and $Q \succeq 0$, then the solution $X$ is entrywise nonnegative. This property is distinct from positive semidefiniteness and is of particular interest in the theory of positive systems, where componentwise nonnegativity ensures the preservation of physical quantities such as masses, concentrations, or populations.

Methodology
The proof strategy relies on reducing the general problem to a specific algebraic structure involving elementary symmetric polynomials and Cauchy-like matrices. The methodology proceeds through the following steps:

1. Problem Simplification:

   • Rank-1 Reduction: By the linearity of the Lyapunov equation and the decomposition of any positive semidefinite $Q$ into a sum of rank-1 matrices ($Q = \sum q_i q_i^\top$), the authors assume without loss of generality that $Q = qq^\top$.
   • Diagonalizability: The authors argue that the solution $X$ is a continuous function of $A$. Thus, it suffices to prove the result for diagonalizable companion matrices (where eigenvalues are distinct). Non-diagonalizable cases can be approximated by diagonalizable ones with arbitrarily close coefficients.

2. Spectral Decomposition:

   • Assuming $A$ is diagonalizable with distinct real eigenvalues $-\lambda_i$ (ordered $0 < \lambda_1 < \dots < \lambda_n$), the authors utilize the eigenvectors of $A^\top$. These eigenvectors, denoted $v_j$, are constructed using elementary symmetric polynomials $\sigma_m^{(j)}$ of the eigenvalues.
   • The forcing vector $q$ is expanded in this eigenbasis: $q = \sum a_i v_i$.

3. Integral Representation and Quadratic Forms:

   • Using the integral formula for the Lyapunov solution, $X = \int_0^\infty e^{A^\top t} Q e^{At} dt$, the authors derive an explicit expression for the entries $X_{kl}$.
   • The entry X_{kl is shown to be a quadratic form $a^\top M a$, where the matrix $M$ has elements defined by:
     $$M_{ij} = \frac{\sigma_{n-k}^{(i)}\sigma_{n-l}^{(j)} + \sigma_{n-k}^{(j)}\sigma_{n-l}^{(i)}}{\lambda_i + \lambda_j}$$
   • This matrix $M$ is factorized as $M = \Delta C \Delta$, where $\Delta$ is a diagonal matrix and $C$ is a Cauchy-like matrix with entries $C_{ij} = \frac{r_i + r_j}{\lambda_i + \lambda_j}$.

4. Positivity Verification:

   • The nonnegativity of $X_{kl}$ follows if the matrix $M$ is positive semidefinite. Since $\Delta$ is nonsingular, this is equivalent to proving $C \succeq 0$.
   • The authors cite established results [8] confirming that such Cauchy-like matrices are positive semidefinite.

Key Contributions and Results

• Theorem 1 (Main Result): The paper proves Conjecture 1.1 under the specific condition that the companion matrix $A$ has only real eigenvalues. Under this assumption, the unique solution $X$ is entrywise nonnegative.
• Total Positivity Analysis: The authors investigate a stronger property: total nonnegativity (all minors are nonnegative).
  • They provide a counterexample showing that $X$ is not generally totally nonnegative, even when eigenvalues are real.
  • Theorem 2: They establish a sufficient condition for total positivity: if the forcing vector $q$ has a nonnegative expansion in the eigenbasis of $A^\top$ (specifically, $q = \sum \alpha_j v_j$ with $\alpha_j \geq 0$), then $X$ is totally positive. This is proven by factorizing $X$ into a product of totally positive matrices (involving Vandermonde matrices and Cauchy matrices).

Significance and Claims
The paper claims to resolve a long-standing conjecture regarding the structural properties of Lyapunov solutions for companion matrices, specifically within the realm of positive systems.

• Motivation: The entrywise nonnegativity of $X$ is motivated by applications in optimal control for positive systems. Specifically, for single-input systems with nonnegative input matrices, an entrywise nonnegative solution to the associated algebraic Riccati equation (rewritten as a Lyapunov equation) implies that the optimal regulator can be implemented using hardware with only positive voltage supplies.
• Scope and Limitations: The authors are modest regarding the generality of their proof. They explicitly state that the assumption of real eigenvalues is a restriction. The paper leaves open the question of whether the conjecture holds for Hurwitz companion matrices with complex eigenvalues.
• Theoretical Value: The work connects Lyapunov theory with the theory of Cauchy-like matrices and elementary symmetric polynomials, demonstrating that the entrywise nonnegativity of $X$ arises from the positive semidefiniteness of a specific class of matrices dual to Kwong (or anti-Loewner) matrices.

In summary, the paper provides a rigorous proof that Lyapunov solutions for Hurwitz companion matrices with real spectra are entrywise nonnegative, while clarifying that total positivity requires additional constraints on the forcing vector.