The Optimal Regulator Problem for a Stationary Linear System With State-Dependent Noise

1970 ◽  
Vol 92 (2) ◽  
pp. 363-368 ◽  
Author(s):  
P. J. McLane
Keyword(s):  
Linear System ◽  
Sufficient Conditions ◽  
Quadratic Functional ◽  
Optimal Controls ◽  
Final Time ◽  
State Dependent ◽  
Continuous Dynamic ◽  
The Stability ◽  
And Control ◽  
Stationary Linear System

The problem of minimizing a quadratic functional of the system outputs and control for a stationary linear system with state-dependent noise is solved in this paper. Both the finite final time and infinite final time versions of the problem are treated. For the latter case existence conditions are obtained using the second method of Lyapunov. The optimal controls for both problems are obtained using Bellman’s continuous dynamic programming. In light of this, the system dynamics are assumed to determine a diffusion process. For the infinite final time version of the problem noted above, sufficient conditions are obtained for the stability of the optimal system and uniqueness of the optimal control law. In addition, for this problem, an example is treated. The computational results for this example illustrate some of the qualitative features of regulators for linear, stationary systems with state-dependent disturbances.

On the Necessary and Sufficient Conditions for the Stability of Linear Generalized Ordinary Differential, Linear Impulsive and Linear Difference Systems

2009 ◽  
Vol 16 (4) ◽  
pp. 597-616
Author(s):  
Shota Akhalaia ◽  
Malkhaz Ashordia ◽  
Nestan Kekelia
Keyword(s):  
Linear System ◽  
Difference Equations ◽  
Closed Interval ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Difference Systems ◽  
The Stability ◽  
Necessary And Sufficient ◽  
Generalized Ordinary Differential Equations ◽  
Lyapunov Sense

Abstract Necessary and sufficient conditions are established for the stability in the Lyapunov sense of solutions of a linear system of generalized ordinary differential equations 𝑑𝑥(𝑡) = 𝑑𝐴(𝑡) · 𝑥(𝑡) + 𝑑𝑓(𝑡), where and are, respectively, matrix- and vector-functions with bounded total variation components on every closed interval from . The results are realized for the linear systems of impulsive, ordinary differential and difference equations.

On general transformations and variational principles for the magnetohydrodynamics of ideal fluids. Part 4. Generalized isovorticity principle for three-dimensional flows

1999 ◽  
Vol 390 ◽  
pp. 127-150 ◽  
Author(s):  
V. A. VLADIMIROV ◽  
H. K. MOFFATT ◽  
K. I. ILIN
Keyword(s):  
Steady State ◽  
Variational Principles ◽  
Three Dimensional ◽  
Sufficient Conditions ◽  
Steady States ◽  
Mhd Equations ◽  
Quadratic Functional ◽  
The First Variation ◽  
The Stability ◽  
And Variational Principles

The equations of magnetohydrodynamics (MHD) of an ideal fluid have two families of topological invariants: the magnetic helicity invariants and the cross-helicity invariants. It is first shown that these invariants define a natural foliation (described as isomagnetovortical, or imv for short) in the function space in which solutions {u(x, t), h(x, t)} of the MHD equations reside. A relaxation process is constructed whereby total energy (magnetic plus kinetic) decreases on an imv folium (all magnetic and cross-helicity invariants being thus conserved). The energy has a positive lower bound determined by the global cross-helicity, and it is thus shown that a steady state exists having the (arbitrarily) prescribed families of magnetic and cross-helicity invariants.The stability of such steady states is considered by an appropriate generalization of (Arnold) energy techniques. The first variation of energy on the imv folium is shown to vanish, and the second variation δ2E is constructed. It is shown that δ2E is a quadratic functional of the first-order variations δ1u, δ1h of u and h (from a steady state U(x), H(x)), and that δ2E is an invariant of the linearized MHD equations. Linear stability is then assured provided δ2E is either positive-definite or negative-definite for all imv perturbations. It is shown that the results may be equivalently obtained through consideration of the frozen-in ‘modified’ vorticity field introduced in Part 1 of this series.Finally, the general stability criterion is applied to a variety of classes of steady states {U(x), H(x)}, and new sufficient conditions for stability to three-dimensional imv perturbations are obtained.

On stability of state-dependent queues and acyclic queueing networks

1989 ◽  
Vol 21 (3) ◽  
pp. 681-701 ◽  
Author(s):  
Nicholas Bambos ◽  
Jean Walrand
Keyword(s):  
Queueing Networks ◽  
Sufficient Conditions ◽  
Service Rate ◽  
Single Server ◽  
General Function ◽  
State Dependent ◽  
The Stability ◽  
Necessary And Sufficient ◽  
Periodic Inputs ◽  
Networks Of Queues

We consider a single server first-come-first-served queue with a stationary and ergodic input. The service rate is a general function of the workload in the queue. We provide the necessary and sufficient conditions for the stability of the system and the asymptotic convergence of the workload process to a finite stationary process at large times. Then, we consider acyclic networks of queues in which the service rate of any queue is a function of the workloads of this and of all the preceding queues. The stability problem is again studied. The results are then extended to analogous systems with periodic inputs.

Stability and constrained control for positive two-dimensional systems with delays in the second FM model

2018 ◽  
Vol 36 (2) ◽  
pp. 515-536
Author(s):  
Jian Shen ◽  
Weiqun Wang
Keyword(s):  
Closed Loop ◽  
Sufficient Conditions ◽  
Multiple Delays ◽  
Two Dimensional ◽  
Bounded Control ◽  
Delayed Systems ◽  
Closed Loop Systems ◽  
The Stability ◽  
And Control ◽  
Time Systems

Abstract This paper addresses the stability and control problem of linear positive two-dimensional discrete-time systems with multiple delays in the second Fornasini–Marchesini model. The contribution lies in three aspects. First, a novel proof is provided to establish necessary and sufficient conditions of asymptotic stability for positive two-dimensional delayed systems. Then, a state-feedback controller is designed to ensure the non-negativity and stability of the closed-loop systems. Finally, a sufficient condition for the existence of constrained controllers is developed under the additional constraint of bounded control, which means that the control inputs and the states of the closed-loop systems are bounded. Two examples are given to validate the proposed methods.

On stability of state-dependent queues and acyclic queueing networks

1989 ◽  
Vol 21 (03) ◽  
pp. 681-701 ◽  
Author(s):  
Nicholas Bambos ◽  
Jean Walrand
Keyword(s):  
Queueing Networks ◽  
Sufficient Conditions ◽  
Service Rate ◽  
Single Server ◽  
General Function ◽  
State Dependent ◽  
The Stability ◽  
Necessary And Sufficient ◽  
Periodic Inputs ◽  
Networks Of Queues

We consider a single server first-come-first-served queue with a stationary and ergodic input. The service rate is a general function of the workload in the queue. We provide the necessary and sufficient conditions for the stability of the system and the asymptotic convergence of the workload process to a finite stationary process at large times. Then, we consider acyclic networks of queues in which the service rate of any queue is a function of the workloads of this and of all the preceding queues. The stability problem is again studied. The results are then extended to analogous systems with periodic inputs.

scholarly journals Bifurcation and control of a predator-prey system with unfixed functional responses

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Lizhi Fei ◽  
Xingwu Chen
Keyword(s):  
Fixed Points ◽  
Hybrid Control ◽  
Sufficient Conditions ◽  
Rate Function ◽  
Functional Responses ◽  
Predator Prey ◽  
Prey System ◽  
The Stability ◽  
Necessary And Sufficient ◽  
And Control

<p style='text-indent:20px;'>In this paper we investigate a discrete-time predator-prey system with not only some constant parameters but also unfixed functional responses including growth rate function of prey, conversion factor function and predation probability function. We prove that the maximal number of fixed points is <inline-formula><tex-math id="M1">\begin{document}$ 3 $\end{document}</tex-math></inline-formula> and give necessary and sufficient conditions of exactly <inline-formula><tex-math id="M2">\begin{document}$ j $\end{document}</tex-math></inline-formula>(<inline-formula><tex-math id="M3">\begin{document}$ j = 1,2,3 $\end{document}</tex-math></inline-formula>) fixed points, respectively. For transcritical bifurcation and Neimark-Sacker bifurcation, we provide bifurcation conditions depending on these unfixed functional responses. In order to regulate the stability of this biological system, a hybrid control strategy is used to control the Neimark-Sacker bifurcation. Finally, we apply our main results to some examples and carry out numerical simulations for each example to verify the correctness of our theoretical analysis.</p>

scholarly journals Existence and control of weighted pseudo almost periodic solutions for abstract nonlinear vibration systems

2018 ◽  
Vol 38 (2) ◽  
pp. 765-773
Author(s):  
Yingxin Guo ◽  
Fei Wang ◽  
Luyao Xin
Keyword(s):  
Nonlinear Vibration ◽  
Periodic Solutions ◽  
Sufficient Conditions ◽  
Almost Periodic Solutions ◽  
Almost Periodic ◽  
Pseudo Almost Periodic Solutions ◽  
Precise Bound ◽  
The Stability ◽  
Pseudo Almost Periodic ◽  
And Control

In many vibration problems, it is very important to know precisely the bounds of the stability/instability frequencies and the associated amplitude ranges. This paper investigates the solvability and control of some weighted pseudo almost periodic solutions of abstract nonlinear vibration differential systems. Some sufficient conditions for the solvability and exponential stability of these systems are obtained. Moreover, the precise bound of Lyapunov exponents is estimated.

A near-optimal decentralized control approach for polynomial nonlinear interconnected systems

2020 ◽  
pp. 014233122097743
Author(s):  
Mohamed Sadok Attia ◽  
Mohamed Karim Bouafoura ◽  
Naceur Benhadj Braiek
Keyword(s):  
Optimal Control ◽  
Closed Loop ◽  
Sufficient Conditions ◽  
Open Loop ◽  
System Stability ◽  
Gronwall Lemma ◽  
Interconnected System ◽  
Control Approach ◽  
State Dependent ◽  
And Control

This article tackles the decentralized near-optimal control problem for the class of nonlinear polynomial interconnected system based on a shifted Legendre polynomials direct approach. The proposed method converts the interconnected optimal control problems into a nonlinear programming one with multiple constraints. In light of the formulated NLP optimization, state and control coefficients are used to design a nonlinear decentralized state feedback controller. Overall closed-loop system stability sufficient conditions are investigated with the help of Grönwall lemma. The triple inverted pendulum case is considered for simulation. Satisfactory results are obtained in both open-loop and closed-loop schemes with comparison to collocation and state-dependent Riccati equation techniques.

scholarly journals Synchronization and Control of Linearly Coupled Singular Systems

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Fang Qingxiang ◽  
Peng Jigen ◽  
Cao Feilong
Keyword(s):  
Singular Systems ◽  
Dynamical Behavior ◽  
Sufficient Conditions ◽  
Lyapunov Stability Theory ◽  
Pinning Control ◽  
State Feedback Control ◽  
Linear Matrix ◽  
Coupling Matrix ◽  
The Stability ◽  
And Control

The synchronization and control problem of linearly coupled singular systems is investigated. The uncoupled dynamical behavior at each node is general and can be chaotic or, otherwise the coupling matrix is not assumed to be symmetrical. Some sufficient conditions for globally exponential synchronization are derived based on Lyapunov stability theory. These criteria, which are in terms of linear matrix inequality (LMI), indicate that the left and right eigenvectors corresponding to eigenvalue zero of the coupling matrix play key roles in the stability analysis of the synchronization manifold. The controllers are designed for state feedback control and pinning control, respectively. Finally, a numerical example is provided to illustrate the effectiveness of the proposed conditions.

HETEROCLINIC BIFURCATIONS OF A PREY–PREDATOR FISHERY MODEL WITH IMPULSIVE HARVESTING

2013 ◽  
Vol 06 (05) ◽  
pp. 1350031 ◽  
Author(s):  
CHUNJIN WEI ◽  
LANSUN CHEN
Keyword(s):  
Dynamic Systems ◽  
Vector Fields ◽  
Sufficient Conditions ◽  
Heteroclinic Cycle ◽  
Heteroclinic Bifurcation ◽  
Perturbed System ◽  
State Dependent ◽  
Continuous Dynamic ◽  
Fishery Model ◽  
Impulsive Harvesting

In this paper, we consider a prey–predator fishery model with Allee effect and state-dependent impulsive harvesting. First, we investigate the existence of order-1 heteroclinic cycle. Second, choosing p as a control parameter, we obtain the sufficient conditions for the existence and uniqueness of order-1 periodic solution of system (2.3) by using the geometry theory of semi-continuous dynamic systems. Finally, on the basis of the theory of rotated vector fields, heteroclinic bifurcation to perturbed system of system (2.3) is also studied. The methods used in this paper are novel to prove the existence of order-1 heteroclinic cycle and heteroclinic bifurcations.

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