Punit Sharma

Nicolas Gillis, Volker Mehrmann, Punit Sharma

In this paper, we study the nearest stable matrix pair problem: given a square matrix pair $(E,A)$, minimize the Frobenius norm of $(Δ_E,Δ_A)$ such that $(E+Δ_E,A+Δ_A)$ is a stable matrix pair. We propose a reformulation of the problem with a simpler feasible set by introducing dissipative Hamiltonian (DH) matrix pairs: A matrix pair $(E,A)$ is DH if $A=(J-R)Q$ with skew-symmetric $J$, positive semidefinite $R$, and an invertible $Q$ such that $Q^TE$ is positive semidefinite. This reformulation has a convex feasible domain onto which it is easy to project. This allows us to employ a fast gradient method to obtain a nearby stable approximation of a given matrix pair.

Nicolas Gillis, Punit Sharma

The notion of positive realness for linear time-invariant (LTI) dynamical systems, equivalent to passivity, is one of the oldest in system and control theory. In this paper, we consider the problem of finding the nearest positive-real (PR) system to a non PR system: given an LTI control system defined by $E \dot{x}=Ax+Bu$ and $y=Cx+Du$, minimize the Frobenius norm of $(Δ_E,Δ_A,Δ_B,Δ_C,Δ_D)$ such that $(E+Δ_E,A+Δ_A,B+Δ_B,C+Δ_C,D+Δ_D)$ is a PR system. We first show that a system is extended strictly PR if and only if it can be written as a strict port-Hamiltonian system. This allows us to reformulate the nearest PR system problem into an optimization problem with a simple convex feasible set. We then use a fast gradient method to obtain a nearby PR system to a given non PR system, and illustrate the behavior of our algorithm on several examples. This is, to the best of our knowledge, the first algorithm that computes a nearby PR system to a given non PR system that (i) is not based on the spectral properties of related Hamiltonian matrices or pencils, (ii) allows to perturb all matrices $(E,A,B,C,D)$ describing the system, and (iii) does not make any assumption on the original given system.

Nicolas Gillis, Michael Karow, Punit Sharma

In this paper, we consider the problem of stabilizing discrete-time linear systems by computing a nearby stable matrix to an unstable one. To do so, we provide a new characterization for the set of stable matrices. We show that a matrix $A$ is stable if and only if it can be written as $A=S^{-1}UBS$, where $S$ is positive definite, $U$ is orthogonal, and $B$ is a positive semidefinite contraction (that is, the singular values of $B$ are less or equal to 1). This characterization results in an equivalent non-convex optimization problem with a feasible set on which it is easy to project. We propose a very efficient fast projected gradient method to tackle the problem in variables $(S,U,B)$ and generate locally optimal solutions. We show the effectiveness of the proposed method compared to other approaches.

Christian Mehl, Volker Mehrmann, Punit Sharma

Eigenvalue and eigenpair backward errors are computed for matrix pencils arising in optimal control. In particular, formulas for backward errors are developed that are obtained under block-structure-preserving and symmetry-structure-preserving perturbations. It is shown that these eigenvalue and eigenpair backward errors are sometimes significantly larger than the corresponding backward errors that are obtained under perturbations that ignore the special structure of the pencil.

Nicolas Gillis, Michael Karow, Punit Sharma

Consider a discrete-time linear time-invariant descriptor system $Ex(k+1)=Ax(k)$ for $k \in \mathbb Z_{+}$. In this paper, we tackle for the first time the problem of stabilizing such systems by computing a nearby regular index one stable system $\hat E x(k+1)= \hat A x(k)$ with $\text{rank}(\hat E)=r$. We reformulate this highly nonconvex problem into an equivalent optimization problem with a relatively simple feasible set onto which it is easy to project. This allows us to employ a block coordinate descent method to obtain a nearby regular index one stable system. We illustrate the effectiveness of the algorithm on several examples.

Peter Benner, Volker Mehrmann, Anshul Prajapati et al.

We study linear time-invariant dissipative Hamiltonian differential-algebraic systems. We characterize when the systems are robustly asymptotically stable and derive exact conditions and bounds when this property is lost under structure-preserving perturbations.

Rajat Vashistha, Arif Gulzar, Parveen Kundu et al.

India faces a significant cancer burden, with an incidence-to-mortality ratio indicating that nearly three out of five individuals diagnosed with cancer succumb to the disease. While the limitations of physical healthcare infrastructure are widely acknowledged as a primary challenge, concerted efforts by government and healthcare agencies are underway to mitigate these constraints. However, given the country's vast geography and high population density, it is imperative to explore alternative soft infrastructure solutions to complement existing frameworks. Artificial Intelligence agents are increasingly transforming problem-solving approaches across various domains, with their application in medicine proving particularly transformative. In this perspective, we examine the potential role of AI agents in advancing nuclear medicine for cancer research, diagnosis, and management in India. We begin with a brief overview of AI agents and their capabilities, followed by a proposed agent-based ecosystem that can address prevailing sustainability challenges in India nuclear medicine.

Nicolas Gillis, Punit Sharma

The positive semidefinite Procrustes (PSDP) problem is the following: given rectangular matrices $X$ and $B$, find the symmetric positive semidefinite matrix $A$ that minimizes the Frobenius norm of $AX-B$. No general procedure is known that gives an exact solution. In this paper, we present a semi-analytical approach to solve the PSDP problem. First, we characterize completely the set of optimal solutions and identify the cases when the infimum is not attained. This characterization requires the unique optimal solution of a smaller PSDP problem where $B$ is square and $X$ is diagonal with positive diagonal elements. Second, we propose a very efficient strategy to solve the PSDP problem, combining the semi-analytical approach, a new initialization strategy and the fast gradient method. We illustrate the effectiveness of the new approach, which is guaranteed to converge linearly, compared to state-of-the-art methods.