Controlled random fields,von Neumann–Gale dynamics and multimarket hedging with risk

We develop a model of asset pricing and hedging for interconnected financial markets with frictions – transaction costs and portfolio constraints. The model is based on a control theory for random fields on a directed graph. Market dynamics are described by using von Neumann–Gale dynamical systems first considered in connection with the modelling of economic growth [13,24]. The main results are hedging criteria stated in terms of risk-acceptable portfolios and consistent price systems, extending the classical superreplication criteria formulated in terms of equivalent martingale measures.     

Fault tolerant saturated control for T–S fuzzy discrete-time systems with delays

In this paper, fault tolerant saturated control (FTSC) problem for discrete-time T–S fuzzy systems with delay is studied. Sufficient conditions for observer based control design with saturation constraints are developed. Based on a multiple Lyapunov functions and the slack variables, design conditions are formulated in LMI terms using the cone complementarity technique. The obtained results are illustrated on a numerical example showing fault detection, its localization and reconfiguration of the control law to maintain asymptotic stability even in the presence of both permanent sensor faults and saturation on the control.     

Sub-optimal constant-volume control for open channel networks

A state variable mathematical model for use in the synthesis of automatic control systems for open-channel networks is presented. The system considered here consists of n-cascaded reaches joined by control gates.The linear time invariant model consists of a controllable and observable representation where the state variables are the stored water volume variations in each reach and the control signals are the variations of the control gates opening sections. The model derives, through appropriate simplifications, from a more complex one in terms of transfer functions which was derived by linearizing the Saint-Venant equations.The problem of a linear quadratic optimal regulator is formulated in classical terms for the canal system and the constant-volume control laws obtained for the simplified model have been imposed on the complex one: such a control is therefore to be considered sub-optimal.The results of a digital simulation of the controlled system behaviour indicate that the system operates with practically constant stored water volumes in each reach and that such behaviour is fairly close to that of a pressure-water pipe system.     

Restricted non-linear approximation in sequence spaces and applications to wavelet bases and interpolation

Restricted non-linear approximation is a type of N-term approximation where a measure ν on the index set (rather than the counting measure) is used to control the number of terms in the approximation. We show that embeddings for restricted non-linear approximation spaces in terms of weighted Lorentz sequence spaces are equivalent to Jackson and Bernstein type inequalities, and also to the upper and lower Temlyakov property. As applications we obtain results for wavelet bases in Triebel–Lizorkin spaces by showing the Temlyakov property in this setting. Moreover, new interpolation results for Triebel–Lizorkin and Besov spaces are obtained.     

Existence of Solutions for a Class of Nonlinear Evolution Inclusions with Nonlocal Conditions

This paper deals with the nonlocal problems for a class of nonlinear first-order evolution inclusions. Some existence results are established for the cases of a convex and of a nonconvex valued perturbation terms. Also, the existence of extremal solutions and a strong relaxation theorem are obtained. Subsequently a nonlinear hyperbolic optimal control problem is considered and the existence theorems based on the proven results are obtained. Then the nonlinear version of “bang–bang” principle for control systems is given as well by utilizing the relaxation theorem.     

Analytical Solution for Power-Limited Optimal Rendezvous near an Elliptic Orbit

This paper presents an analytical solution to the problem of the optimal rendezvous using power-limited propulsion for a spacecraft in an elliptic orbit in a gravitational field. The derivation of the result assumes small relative distances, but does not make any assumption on the eccentricity of the orbit and does not require numerical integration. The results are generalized to include the possibility of different weights on the control effort for each axis (radial, along-track, and out-of-plane). When the weights on the control efforts are unequal, several integrals are used whose solutions may be represented by infinite series in a small parameter dependent on the eccentricity. A methodology is introduced where the series can be extended trivially to as many terms as desired. Furthermore, for a given numerical tolerance, an upper bound on the number of terms required to represent the series is also obtained. When the weights are equal for all the three axes, the series representations are no longer necessary. The results can be used easily to design optimal feedback controls for rendezvous maneuvers, or for generating initial guesses for two-point boundary-value problems for numerical solutions to the nonlinear rendezvous problem.     

Dissipativity and synchronization control of fractional‐order memristive neural networks with reaction‐diffusion terms

This paper deals with the dissipativity and synchronization control of fractional‐order memristive neural networks (FOMNNs) with reaction‐diffusion terms. By means of fractional Halanay inequality, Wirtinger inequality, and Lyapunov functional, some sufficient conditions are provided to ensure global dissipativity and exponential synchronization of FOMNNs with reaction‐diffusion terms, respectively. The underlying model and the obtained results are more general since the reaction‐diffusion terms are first introduced into FOMNNs. The given conditions are easy to be checked, and the correctness of the obtained results is confirmed by a living example.     

Problems of conflict control of high dimensionality functional systems

A minimax control problem with a performance index which is the sum of two terms is considered for a system with a delay. The first of these two terms in the Euclidean norm of the set of deviations of the motion of the system at specified instants of time from the stipulated objectives, while the second term is an integral-quadratic penalty which is imposed on the form of the control actions. The problem arises in a differential game. In this case, the history of the motion serves as the information for the strategies. A functional treatment of the control process in question is given which is based on an original prediction of the motion. A procedure for calculating the value of the game and for constructing minimax and maximun control strategies, which is convenient for numerical implementation, is obtained from this treatment and from the construction of hulls, convex upwards, of auxiliary functions from the method of stochastic program synthesis. The results of a numerical experiment are presented.     

Lyapunov Function for Nonuniform in Time Global Asymptotic Stability in Probability with Application to Feedback Stabilization

We extend the well-known Artstein-Sontag theorem by introducing the concept of control Lyapunov function for the notion of nonuniform in time global asymptotic stability in probability of stochastic differential system, when both the drift and diffusion terms are affine in the control. The main results of our work enable us to derive the necessary and sufficient conditions for feedback stabilization for affine in the control systems.     

Mixed strategy solutions for quadratic games

Mixed strategy solutions are given for two-person, zero-sum games with payoff functions consisting of quadratic, bilinear, and linear terms, and strategy spaces consisting of closed balls in a Hilbert space. The results are applied to linear-quadratic differential games with no information, and with quadratic integral constraints on the control functions.     

Two-Dimensional Chaos: The Baker Map Under Control

Some results on the stochastic control of a two-dimensional chaotic map, namely, the baker map, are presented. The approach is based on the probabilistic coupling of the controlled dynamics with a controlling system and the subsequent lifting of the coupled dynamics to a suitable functional space. The lifted dynamics is described in terms of probability densities and is governed by the linear Perron-Frobenius and Koopman operators. We obtain a sufficient condition for controllability and an estimation for the time to achieve control for a given accuracy in terms of the spectral decomposition of the Perron-Frobenius operator. Bibliography: 8 titles.__________Published in Zapiski Nauchnykh Seminarov POMI, Vol. 300, 2003, pp. 206–214     

A dynamic parameter estimator to control chaos with distinct feedback schemes

A dynamic strategy is proposed to estimate parameters of chaotic systems. The dynamic estimator of parameters can be used with diverse control functions; for example, those based on: (i) Lie algebra, (ii) backstepping, or (iii) variable feedback structure (sliding-mode). The proposal has adaptive structure because of interaction between dynamic estimation of parameters and a feedback control function. Without lost of generality, a class of dynamical systems with chaotic behavior is considered as benchmark. The proposed scheme is compared with a previous low-parameterized robust adaptive feedback in terms of execution and performance. The comparison is motivated to ask: What is the suitable adaptive scheme to suppress chaos in an specific implementation? Experimental results of proposed scheme are discussed in terms of control execution and performance and are relevant in specific implementations; for example, in order to induce synchrony in complex networks.     

参数激励Duffing-VanderPol振子的动力学响应及反馈控制

研究了Duffing-Van der Pol振子的主参数共振响应及其时滞反馈控制问题．依平均法和对时滞反馈控制项Taylor展开的截断得到的平均方程表明,除参数激励的幅值和频率外,零解的稳定性只与原方程中线性项的系数和线性反馈有关,但周期解的稳定性还与原方程中非线性项的系数和非线性反馈有关．通过调整反馈增益和时滞,可以使不稳定的零解变得稳定．非零周期解可能通过鞍结分岔和Hopf分岔失去稳定性,但选择合适的反馈增益和时滞,可以避免鞍结分岔和Hopf分岔的发生．数值仿真的结果验证了理论分析的正确性．     

On the exponential inequalities for negatively dependent random variables

A number of exponential inequalities for identically distributed negatively dependent and negatively associated random variables have been established by many authors. The proofs use the truncation technique together with the control of the bounded terms and unbounded terms. In this paper, we improve essentially the control of bounds for the unbounded terms and obtain exponential inequalities for negatively dependent random variables which include negatively associated random variables. Our results improve on the corresponding ones in the literature.     

New modelling approach concerning integrated disease control and cost-effectivity

Two new models for controlling diseases, incorporating the best features of different control measures, are proposed and analyzed. These models would draw from poultry, livestock and government expertise to quickly, cooperatively and cost-effectively stop disease outbreaks. The combination strategy of pulse vaccination and treatment (or isolation) is implemented in both models if the number of infectives reaches the risk level (RL). Firstly, for one time impulsive effect we compare three different control strategies for both models in terms of cost. The theoretical and numerical results show that there is an optimal vaccination and treatment proportion such that integrated pulse vaccination and treatment (or isolation) reaches its minimum in terms of cost. Moreover, this minimum cost of integrated strategy is less than any cost of single pulse vaccination or single treatment. Secondly, a more realistic case for the second model is investigated based on periodic impulsive control strategies. The existence and stability of periodic solution with the maximum value of the infectives no larger than RL is obtained. Further, the period T of the periodic solution is calculated, which can be used to estimate how long the infectious population will take to return back to its pre-control level (RL) once integrated control tactics cease. This implies that we can control the disease if we implement the integrated disease control tactics every period T. For periodic control strategy, if we aim to control the disease such that the maximum number of infectives is relatively small, our results show that the periodic pulse vaccination is optimal in terms of cost.     

Optimal Feedback Control for Linear Systems with Input Delays Revisited

The design problem of optimal feedback control for linear systems with input delays is very important in many engineering applications. Usually, the linear systems with input delays are firstly converted into linear systems without delays, and then all the design procedures are based on the delay-free linear systems. In this way, the feedback controllers are not designed in terms of the original states. This paper presents some new closed-form formula in terms of the original states for the delayed optimal feedback control of linear systems with input delays. We firstly reveal the essential role of the input delay in the optimal control design of the linear system with a single input delay: the input delay postpones the action of the optimal control only. Based on this fact, we calculate the delayed optimal control and find that the optimal state can be represented by a simple closed-form formula, so that the delayed optimal feedback control can be obtained in a simple way. We show that the delayed feedback gain matrix can be “smaller” than that for the controlled system with zero input delay, which implies that the input delay can be considered as a positive factor. In addition, we give a general formula for the delayed optimal feedback control of time-variant linear systems with multiple input delays. To show the effectiveness and advantages of the main results, we present five illustrative examples with detailed numerical simulation and comparison.     

Stability of linear time-varying delay systems and applications to control problems

This paper deals with the stability of a class of linear time-varying systems with multiple delays. Using the Lyapunov function method, we give sufficient delay-dependent conditions for the exponential stability with a given convergence rate, which are described in terms of linear matrix inequalities (LMI) and the solution of Riccati differential equations (RDE). The results are applied to the problem of stabilization of linear time-varying control systems with multiple delays. Numerical examples are given to illustrate the results.     

A choice of forcing terms in inexact Newton method

Inexact Newton method is one of the effective tools for solving systems of nonlinear equations. In each iteration step of the method, a forcing term, which is used to control the accuracy when solving the Newton equations, is required. The choice of the forcing terms is of great importance due to their strong influence on the behavior of the inexact Newton method, including its convergence, efficiency, and even robustness. To improve the efficiency and robustness of the inexact Newton method, a new strategy to determine the forcing terms is given in this paper. With the new forcing terms, the inexact Newton method is locally Q-superlinearly convergent. Numerical results are presented to support the effectiveness of the new forcing terms.     

Optimal solutions to differential inclusions

We treat a control problem given in terms of a differential inclusion $$\dot x(t) \in E(t,x(t))$$ and develop necessary conditions for a minimum in the problem. These conditions are given in terms of certain normals to arbitrary closed sets, and require no smoothness or convexity in the problem. The results subsume related works that incorporate convexity assumptions.     

Sufficient stability conditions and stabilizing control design for delay differential systems of a special type

This paper introduces some sufficient conditions for uniform and asymptotic global stability as well as the algorithms for design of stabilizing control for special systems like cascaded (triangular) systems and integrator chains. The results are presented in terms of semidefinite Lyapunov functions, and they hold for nonlinear nonautonomous systems. Application of the results proposed is illustrated by some classical examples.