Hongli Yang

In this brief paper, we investigate the practical stabilization problem of multiple time-delayed positive linear time-varying systems. Firstly, a sufficient condition for the practical stability of the system is derived using the maximal separable Lyapunov–Krasovskii functional. Then, a state feedback controller is designed for the considered positive linear time-varying system with multiple time delays. Building on this method, the study further addresses the practical stabilization problem for positive linear time-varying systems with multiple state time delays and input time delays. The positivity of the linear time-varying system with input time delays is verified by designing a state feedback controller, and the practical stability is proven using the maximal separable Lyapunov–Krasovskii functional method. Finally, numerical examples are presented to validate the feasibility of the proposed method.

This paper addresses the controller design problem for linear systems with time-varying delays. By constructing a novel Lyapunov–Krasovskii functional incorporating delay-partitioning techniques, we establish delay-dependent stability criteria for the solvability of the robust stabilization problem. The derived conditions are formulated as linear matrix inequalities (LMIs) that become affine upon fixing a single scalar parameter, thereby facilitating efficient numerical computation. Furthermore, these criteria guarantee that the reachable set of the closed-loop system remains bounded within a prescribed ellipsoid under zero initial conditions. The effectiveness and superiority of the proposed approach are demonstrated through two comparative numerical examples, including a benchmark problem with varying delay.

This paper presents both sufficient and necessary conditions for polyhedral sets and symmetric polyhedral sets to be robust positively invariant sets within perturbed linear discrete-time systems. These conditions are derived through the application of optimization and dual optimization theory. By leveraging the definition of a robust positively invariant set and employing the Pontryagin difference, we have obtained robust positively invariant conditions in optimized forms. Through the use of dual optimization theory, various equivalent forms are introduced, offering additional tools for verifying that polyhedral sets are indeed robust positively invariant sets for perturbed linear discrete-time dynamic systems. The efficacy of these conclusions is further evidenced by numerical examples.

This paper studies the estimation of reachable sets for discrete-time singular systems with time-varying delays and bounded peak inputs. A novel linear matrix inequality condition for the reachable set estimation of the time-varying time-delay discrete singular system is derived using an inverse convex combination and the discrete form of the Wirtinger inequality. Furthermore, the symmetric matrix involved in the obtained results does not need to be positively definite. Compared to decomposing the time-delay discrete singular system under consideration into fast and slow subsystems, the method presented in this paper is simpler and involves fewer variables. Two numerical examples are provided to illustrate the proposed method.

A direct data-driven approach for computing the robust positively invariant sets of a discrete-time system is presented in this study. By transforming the invariance conditions into a set of tractable linear matrix inequality problems and combining them with a semidefinite programming problem, we maximize the volume of invariant sets without violating state constraints. Based on two equivalence conditions of invariance, we investigate two algorithms using the one-step method to maximize the volume of the invariant sets. Ultimately, we opted for Algorithm 1, which is more succinct and effective. To further reduce conservatism, we propose an iterative algorithm based on Algorithm 1. The effectiveness of the proposed algorithm is verified through numerical examples.

In this paper, we investigate a Lyapunov trajectory tracking design method that incorporates a Schrödinger equation with a dipole subterm and polarizability. Our findings suggest that the proposed control law can overcome the limitations of certain existing control laws that do not converge. By integrating a quadratic performance index, we introduce an optimal control law, which we subsequently analyze for stability and optimality. We also simulate the spin-1/2 particle system to illustrate our results. These findings are further validated through numerical illustrations involving a 3D, 5D system, and a spin-1/2 particle system.