arXiv subjects

Omri Dalin

Publications and source records attributed to Omri Dalin.

5 recordsLinked to original sources

arXiv2026

On the Asymptotic Switching Density in Time-Optimal Control of Linear Systems

We study the time-optimal control of a controllable linear system on a time horizon [0,T], focusing on the asymptotic switching density for large T. When the system matrix has only real eigenvalues, it is well-known that the number of switches is upper bounded uniformly in T; when it has complex eigenvalues, no such uniform bound exists, and the switching count instead typically grows with T. We characterize this growth for a system matrix with an arbitrary spectrum, allowing simultaneously for real eigenvalues, complex eigenvalues, and a non-trivial Jordan structure. If the dominant mode is complex, the number of switches grows at least linearly in T, with an explicit lower bound expressed via the mean motion problem and the Bohl-Weyl-Wintner formula. We illustrate the theory on a linearized aircraft pitch/altitude model, showing close agreement between the predicted asymptotic switching rate and numerically computed time-optimal controls.

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arXiv2025

An application of the mean motion problem to time-optimal control

We consider time-optimal controls of a controllable linear system with a scalar control on a long time interval. It is well-known that if all the eigenvalues of the matrix describing the linear system dynamics are real then any time-optimal control has a bounded number of switching points, where the bound does not depend on the length of the time interval. We consider the case where the governing matrix has purely imaginary eigenvalues, and show that then, in the generic case, the number of switching points is bounded from below by a linear function of the length of the time interval. The proof is based on relating the switching function in the optimal control problem to the mean motion problem that dates back to Lagrange and was solved by Hermann Weyl.

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arXiv2023

On special quadratic Lyapunov functions for linear dynamical systems with an invariant cone

We consider a continuous-time linear time-invariant dynamical system that admits an invariant cone. For the case of a self-dual and homogeneous cone we show that if the system is asymptotically stable then it admits a quadratic Lyapunov function with a special structure. The complexity of this Lyapuonv function scales linearly with the dimension of the dynamical system. In the particular case when the cone is the nonnegative orthant this reduces to the well-known and important result that a positive system admits a diagonal Lyapunov function. We demonstrate our theoretical results by deriving a new special quadratic Lyapunov function for systems that admit the ice-cream cone as an invariant set.

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arXiv2022

Verifying $k$-Contraction without Computing $k$-Compounds

Compound matrices have found applications in many fields of science including systems and control theory. In particular, a sufficient condition for $k$-contraction is that a logarithmic norm (also called matrix measure) of the $k$-additive compound of the Jacobian is uniformly negative. However, this may be difficult to check in practice because the $k$-additive compound of an $n\times n$ matrix has dimensions $\binom{n}{k}\times \binom{n}{k}$. For an $n\times n$ matrix $A$, we prove a duality relation between the $k$ and $(n-k)$ compounds of $A$. We use this duality relation to derive a sufficient condition for $k$-contraction that does not require the computation of any $k$-compounds. We demonstrate our results by deriving a sufficient condition for $k$-contraction of an $n$-dimensional Hopfield network that does not require to compute any compounds. In particular, for $k=2$ this sufficient condition implies that the network is $2$-contracting and this implies a strong asymptotic property: every bounded solution of the network converges to an equilibrium point, that may not be unique. This is relevant, for example, when using the Hopfield network as an associative memory that stores patterns as equilibrium points of the dynamics.

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arXiv2022

Compound matrices in systems and control theory: a tutorial

The multiplicative and additive compounds of a matrix play an important role in several fields of mathematics including geometry, multi-linear algebra, combinatorics, and the analysis of nonlinear time-varying dynamical systems. There is a growing interest in applications of these compounds, and their generalizations, in systems and control theory. The goal of this tutorial paper is to provide a gentle and self-contained introduction to these topics with an emphasis on the geometric interpretation of the compounds, and to describe some of their recent applications including several non-trivial generalizations of positive systems, cooperative systems, contracting systems, and more.

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