We consider a continuous-time bilinear control system (BCS) with Metzler matrices. Each entry in the transition matrix of such a system is nonnegative, making the positive orthant an invariant set of the dynamics. Motivated by the stability analysis of positive linear switched systems (PLSSs), we define a control as optimal if, for a fixed final time, it maximizes the spectral radius of the transition matrix. Our main result is a first-order necessary condition for optimality in the form of a maximum principle (MP). The proof of this MP combines the standard needle variation with a basic result from the Perron--Frobenius theory of nonnegative matrices. We describe several applications of this MP to the stability analysis of PLSSs under arbitrary switching.

A Maximum Principle for the Stability Analysis of Positive Bilinear Control Systems with Applications to Positive Linear Switched Systems

This paper considers the Aizerman problem for second-order systems with delays. It is proved that for retarded systems with a single delay the Aizerman conjecture is true. For systems with multiple delays, a delay-dependent class of systems is found, for which the Aizerman conjecture is true. The proof is based on the Popov's frequency-domain criterion for absolute stability.

/pdf/a-partial-solution-of-the-aizerman-problem-for-second-order-4p2sswz3cu.pdf

A Partial Solution of the Aizerman Problem for Second-Order Systems With Delays

Linear time-delay systems with transcendental characteristic equations have infinitelymany eigenvalues which are generally hard to compute completely. However, the spectrum of first-order linear time-delay systems can be analyzed with the Lambert function. This paper studies the stability and state feedback stabilization of first-order linear time-delay system in detail via the Lambert function. The main issues concerned are the rightmost eigenvalue locations, stability robustness with respect to delay time, and the response performance of the closed-loop system. Examples and simulations are presented to illustrate the analysis results.

/pdf/stability-and-performance-of-first-order-linear-time-delay-1o3h4wyft9.pdf

Stability and performance of first-order linear time-delay feedback systems: an eigenvalue approach

http://www.eng.tau.ac.il/~michaelm/posi.pdf

Stability Analysis of Positive Linear Switched Systems: A Variational Approach

Graphical methods are a key tool to analyze Lur’e systems with time delay. In this letter we revisit clockwise properties of the Nyquist plot and extend results in the literature to critically stable systems and time-delayed systems. It is known that rational transfer functions with no resonant poles and no zeros satisfy the Kalman conjecture. We show that the same class of transfer functions in series with a time delay also satisfies the Kalman conjecture. Furthermore the same class of transfer functions in series with an integrator and delay (which may be zero) satisfies a suitably relaxed form of the Kalman conjecture. Useful results are also obtained where the delay is constant but unknown. Results in this letter can be used as benchmarks to test sufficient stability conditions for the Lur’e problem with time-delay systems.

/pdf/absolute-stability-of-systems-with-integrator-and-or-time-2j1hjdxfi9.pdf

Vladimir Răsvan

Papers

Problem 6.6 Delay independent and delay dependent Aizerman problem