Chaos synchronization and chaos control of quantum-CNN chaotic system by variable structure control and impulse control

In this paper, we derive some less stringent conditions for the exponential and asymptotic stability of impulsive control systems with impulses at fixed times. These conditions are then used to design an impulsive control law for the Quantum Cellular Neural Network chaotic system, which drives the chaotic state to zero equilibrium and synchronizes two chaotic systems. An active sliding mode control method is synchronizing two chaotic systems and controlling chaotic state to periodic motion state. And a sufficient condition is drawn for the robust stability of the error dynamics, and is applied to guiding the design of the controllers. Finally, numerical results are used to show the robustness and effectiveness of the proposed control strategy.     

Optimal control for a population dynamics with age-dependent and diffusion in a periodic environment

Here we consider the optimal harvesting problem for a periodic age-dependent population dynamics with diffusion. Namely, we investigate the model with periodic vital rates and a periodic forcing term that sustains oscillations. By using Mazur’s theorem, we demonstrate existence of solutions of the optimal control problem and by the conception of normal cone, we also obtain the first order necessary conditions of optimality for the problem. Our results extend some known criteria.     

Data-driven optimal tracking control of switched linear systems

This paper addresses the optimal tracking control for switched linear systems with unknown dynamics. We convert the problem into an optimal control problem of the augmented switched systems. In view of the augmented systems, we propose a data-driven switched linear quadratic regular algorithm for obtaining the optimal switching signal under unknown system dynamics. It is proved that the optimal switching signal will not cause Zeno behavior and can make the system stable. Besides, with the proposed algorithm, we just need to identify an autonomous system instead of the original systems, which has fewer parameters to be determined. A numerical example is given to illustrate the validity of the main results.     

A Gradient Flow Approach to Optimal Model Reduction of Discrete-Time Periodic Systems

This paper is concerned with the optimal model reduction for linear discrete periodic time-varying systems and digital filters. Specifically, for a given stable periodic time-varying model, we shall seek a lower order periodic time-varying model to approximate the original model in an optimal H ₂ norm sense. By orthogonal projections of the original model, we convert the optimal periodic model reduction problem into an unconstrained optimization problem. Two effective algorithms are then developed to solve the optimization problem. The algorithms ensure that the H ₂ cost decreases monotonically and converges to an optimal (local) solution. Numerical examples are given to demonstrate the computational efficiency of the proposed method. The present paper extends the optimal model reduction for linear time invariant systems to linear periodic discrete time-varying systems.     

A sequential computational approach to optimal control problems for differential-algebraic systems based on efficient implicit Runge–Kutta integration

Efficient and reliable integrators are indispensable for the design of sequential solvers for optimal control problems involving continuous dynamics, especially for real-time applications. In this paper, optimal control problems for systems represented by index-1 differential-algebraic equations are investigated. On the basis of a time-scaling transformation, the control is parameterized as a piecewise constant function with variable heights and switching time instants. Compared with control parameterization with fixed time grids, the flexibility of adjusting switching time instants increases the chance of finding the optimal solution. Furthermore, error constraints are introduced in the optimization procedure such that the optimal control obtained has a guarantee of integration accuracy. For the derived approximate nonlinear programming problem, a function evaluation and forward sensitivity propagation algorithm is proposed with an embedded implicit Runge–Kutta integrator, which executes one Newton iteration in the limit by employing a predictor-corrector strategy. This algorithm is combined with a nonlinear programming solver Ipopt to construct the optimal control solver. Numerical experiments for the solution of the optimal control problem for a Delta robot demonstrate that the computational speed of this solver is increased by a factor of 0.5–2 when compared with the same solver without the predictor-corrector strategy, and increased by a factor of 20–40 when compared with solver embedding IDAS, the Implicit Differential-Algebraic solver with Sensitivity capabilities developed by Lawrence Livermore National Laboratory. Meanwhile, the accuracy loss compared with the one using IDAS is small and admissible.     

具有年龄结构的线性周期种群动力系统的最优收获控制问题

研究一类具有年龄结构的线性周期种群动力系统的最优收获控制问题,即讨论了具有周期的生死率和周期变化的收获项的Lotka Mckendrick模型.利用Mazur's定理,作者证明了控制问题最优解的存在性,同时借助于法锥概念,还得到了控制问题最优解存在的必要条件。最后，在适当的假设下，得到了最优控制问题的唯一解。该文的结论推广了某些已有的结果.      

Quadratic control for linear periodic systems

We propose a new quadratic control problem for linear periodic systems which can be finite or infinite dimensional. We consider both deterministic and stochastic cases. It is a generalization of average cost criterion, which is usually considered for time-invariant systems. We give sufficient conditions for the existence of periodic solutions.Under stabilizability and detectability conditions we show that the optimal control is given by a periodic feedback which involves the periodic solution of a Riccati equation. The optimal closed-loop system has a unique periodic solution which is globally exponentially asymptotically stable. In the stochastic case we also consider the quadratic problem under partial observation. Under two sets of stabilizability and detectability conditions we obtain the separation principle. The filter equation is not periodic, but we show that it can be effectively replaced by a periodic filter. The theory is illustrated by simple examples.This work was done while this author was a visiting professor at the Scuola Normale Superiore, Pisa.     

Stochastic optimal time-delay control of quasi-integrable Hamiltonian systems

A nonlinear stochastic optimal time-delay control strategy for quasi-integrable Hamiltonian systems is proposed. First, a stochastic optimal control problem of quasi-integrable Hamiltonian system with time-delay in feedback control subjected to Gaussian white noise is formulated. Then, the time-delayed feedback control forces are approximated by the control forces without time-delay and the original problem is converted into a stochastic optimal control problem without time-delay. After that, the converted stochastic optimal control problem is solved by applying the stochastic averaging method and the stochastic dynamical programming principle. As an example, the stochastic time-delay optimal control of two coupled van der Pol oscillators under stochastic excitation is worked out in detail to illustrate the procedure and effectiveness of the proposed control strategy.     

基于actor-critic算法的分数阶多自主体系统最优主-从一致性控制

研究了分数阶多自主体系统的最优主-从一致性问题.在考虑控制器周期间歇的前提下,将分数阶微分的一阶近似逼近式、事件触发机制和强化学习中的actor-critic算法有机整合,设计了基于周期间歇事件触发策略的强化学习算法结构.最后,通过数值仿真实验证明了该算法的可行性和有效性.     

Dynamics and control of a system of two non‐interacting preys with common predator

In this paper, we consider a Holling type model, which describes the interaction between two preys with a common predator. First, we give some sufficient conditions for the globally asymptotic stability and prove that local stability implies global stability. Then, we present a set of sufficient conditions for the existence of a positive periodic solution with strictly positive components. Finally, the optimal control strategy is developed to minimize the number of predator and maximize the number of preys. We also show the existence of an optimal control for the optimal control problem and derive the optimality system. The technical tool used to determine the optimal strategy is the Pontryagin Maximum Principle. Finally, the numerical simulations of global stability and the optimal problem are given as the conclusion of this paper. Copyright © 2011 John Wiley & Sons, Ltd.     

Banach空间中一类周期非线性受控系统最优控制的存在性

研究一类强非线性发展方程的周期解及相应的最优控制问题的存在性，首先，证明了Banach空间中一类包含非线性单调算子和非线性非单调扰动的强非线性发展方程周期解的存在性；其次，给出了保证相应的Lagrange最优控制的充分条件；最后，举例说明理论结果在拟线笥抛物方程周期问题及相应的最优控制问题中的应用。     

Stabilization and tracking controller for a class of nonlinear discrete-time systems

In this paper, stabilization and tracking control problem for parametric strict feedback class of discrete time systems is addressed. Recursive design of control function based on contraction theory framework is proposed instead of traditional Lyapunov based method. Explicit structure of controller is derived for the addressed class of nonlinear discrete-time systems. Conditions for exponential stability of system states are derived in terms of controller parameters. At each stage of recursive procedure a specific structure of Jacobian matrix is ensured so as to satisfy conditions of stability. The closed loop dynamics in this case remains nonlinear in nature. The proposed algorithm establishes global stability results in quite a simple manner as it does not require formulation of error dynamics. Problem of stabilization and output tracking control in case of single link manipulator system with actuator dynamics is analyzed using the proposed strategy. The proposed results are further extended to stabilization of discrete time chaotic systems. Numerical simulations presented in the end show the effectiveness of the proposed approach.     

Optimal synchronization of two different in‐commensurate fractional‐order chaotic systems with fractional cost function

This article investigates the optimal synchronization of two different fractional‐order chaotic systems with two kinds of cost function. We use calculus of variations for minimizing cost function subject to synchronization error dynamics. We introduce optimal control problem to solve fractional Euler–Lagrange equations. Optimal control signal and minimum time of synchronization are obtained by proposed method. Examples show the optimal synchronization of two different systems with two different cost functions. First, we use an ordinary integer cost function then we use a fractional‐order cost function and comparing the results. Finally, we suggest a cost function which has the optimal solution of this problem, and we can extend this solution to solve other synchronization problems. © 2016 Wiley Periodicals, Inc. Complexity 21: 401–416, 2016     

Linear programming approach to the control of discrete-time periodic systems with uncertain inputs

The problem is considered of finding a control strategy for a linear discrete-time periodic system with state and control bounds in the presence of unknown disturbances that are only known to belong to a given compact set. This kind of problem arises in practice in resource distribution systems where the demand has typically a periodic behavior, but cannot be estimated a priori without an uncertainty margin. An infinite-horizon keeping problem is formulated, which consists in confining the state within its constraint set using the allowable control, whatever the allowed disturbances may be. To face this problem, the concepts of periodically invariant set and sequence are introduced. They are used to formulate a solution strategy that solves the keeping problem. For the case of polyhedral state, control, and disturbance constraints, a computationally feasible procedure is proposed. In particular, it is shown that periodically invariant sequences may be computed off-line, and then they may be used to synthesize on-line a control strategy. Finally, an optimization criterion for the control law is discussed.     

Existence of solutions and periodic solutions for nonlinear evolution inclusions

In this paper we consider nonlinear-dependent systems with multivalued perturbations in the framework of an evolution triple of spaces. First we prove a surjectivity result for generalized pseudomonotone operators and then we establish two existence theorems: the first for a periodic problem and the second for a Cauchy problem. As applications we work out in detail a periodic nonlinear parabolic partial differential equation and an optimal control problem for a system driven by a nonlinear parabolic equation.     

A reducing of a chaotic movement to a periodic orbit,of a micro-electro-mechanical system,by using an optimal linear control design

This paper analyzes the non-linear dynamics, with a chaotic behavior of a particular micro-electro-mechanical system. We used a technique of the optimal linear control for reducing the irregular (chaotic) oscillatory movement of the non-linear systems to a periodic orbit. We use the mathematical model of a (MEMS) proposed by Luo and Wang.     

Hybrid event-triggered and impulsive control for time-delay systems

In this paper, we study the problem of hybrid event-triggered control for a class of nonlinear time-delay systems. Using a Razumikhin-type input-to-state stability result for time-delay systems, we design an event-triggered control algorithm to stabilize the given time-delay system. In order to exclude Zeno behavior, we combine the impulsive control mechanism with our event-triggered strategy. In this sense, the proposed algorithm is a hybrid impulsive and event-triggered strategy. Sufficient conditions for the stabilization of the nonlinear systems with time delay are obtained by using Lyapunov method and Razumikhin technique. Numerical simulations are provided to show the effectiveness of our theoretical results.     

Optimal feedback control of the forced van der Pol system

A simple feedback control strategy for chaotic systems is investigated using the forced van der Pol system as an example. The strategy regards chaos control as an optimization problem, where the maximum magnitude Floquet multiplier of a target unstable periodic orbit (UPO) is used as a cost function that needs to be minimized. Thus, the method obtains the optimal control gain in terms of the stability of the target UPO. This strategy was recently proposed for the proportional feedback control (PFC) method. Here, it is extended to the highly popular delayed feedback control (DFC) method. Since the DFC method treats the system as a delay-differential equation whose phase space is infinite-dimensional, the characteristic multipliers are found through a truncation in the number of delayed states. Control of a target UPO is achieved for several values of the forcing amplitude. We compare the DFC and PFC methods in terms of stability of the controlled orbit, steady state error and control effort.     

Relaxation methods for mixed-integer optimal control of partial differential equations

We consider integer-restricted optimal control of systems governed by abstract semilinear evolution equations. This includes the problem of optimal control design for certain distributed parameter systems endowed with multiple actuators, where the task is to minimize costs associated with the dynamics of the system by choosing, for each instant in time, one of the actuators together with ordinary controls. We consider relaxation techniques that are already used successfully for mixed-integer optimal control of ordinary differential equations. Our analysis yields sufficient conditions such that the optimal value and the optimal state of the relaxed problem can be approximated with arbitrary precision by a control satisfying the integer restrictions. The results are obtained by semigroup theory methods. The approach is constructive and gives rise to a numerical method. We supplement the analysis with numerical experiments.