On the Existence of a Lipschitz Feedback Control in a Control Problem with State Constraints

We consider a nonlinear control system with state constraints given as a solution set for a finite system of nonlinear inequalities. The problem of constructing a feedback control that ensures the viability of trajectories of the closed system in a small neighborhood of the boundary of the state constraints is studied. Under some assumptions, the existence of a feedback control in the form of a Lipschitz function of the state of the system is proved.     

基于极大极小方法的一类非线性系统的自适应控制

研究一类具有非线性不确定参数的非线性系统的自适应模型参考跟踪问题.假设系统的非线性项关于不确定参数是凸或凹的.去掉了在先前有关研究中要求参考模型矩阵有小于零的实特征值的条件.既考虑了状态反馈控制方式,也考虑了输出反馈控制方式.在采用输出反馈控制时,假设非线性项满足李普希兹条件,但李普希兹常数未知.基于一种极大极小方法,提出了一种自适应控制器的设计方法.控制器是连续的,能保证闭环系统的所有变量有界,并且渐近精确跟踪参考模型.举例说明了本结论的有用性.     

Dynamic Robust Output Min–Max Control for Discrete Uncertain Systems

Min–max control is a robust control, which guarantees stability in the presence of matched uncertainties. The basic min–max control is a static state feedback law. Recently, the applicability conditions of discrete static min–max control through the output have been derived. In this paper, the results for output static min–max control are further extended to a class of output dynamic min–max controllers, and a general parametrization of all such controllers is derived. The dynamic output min–max control is shown to exist in many circumstances under which the output static min–max control does not exist, and usually allows for broader bounds on uncertainties. Another family of robust output min–max controllers, constructed from an asymptotic observer which is insensitive to uncertainties and a state min–max control, is derived. The latter is shown to be a particular case of the dynamic min–max control when the nominal system has no zeros at the origin. In the case where the insensitive observer exists, it is shown that the observer-controller has the same stability properties as those of the full state feedback min–max control.     

Multi-Target Control Problems

We study control problems with several targets in the case of nonlinear dynamic systems. The map associating with every initial condition the minimal time to reach successively two given targets is characterized in the framework of differential inclusions through the notion of viability kernel. This approach allows one to treat the problem without assumptions of regularity and to build numerical schemes computing the minimal time. We also study the problem where an order of visit of the targets is required. The statements are also extended to the case of p targets under state constraints. Equivalent formulations in terms of Hamilton–Jacobi equations are also provided.     

Feedback boundary stabilization of the three-dimensional incompressible Navier–Stokes equations

We study the local stabilization of the three-dimensional Navier–Stokes equations around an unstable stationary solution w, by means of a feedback boundary control. We first determine a feedback law for the linearized system around w. Next, we show that this feedback provides a local stabilization of the Navier–Stokes equations. To deal with the nonlinear term, the solutions to the closed loop system must be in H^{3/2+ε,3/4+ε/2}(Q), with 0<ε. In [V. Barbu, I. Lasiecka, R. Triggiani, Boundary stabilization of Navier–Stokes equations, Mem. Amer. Math. Soc. 852 (2006); V. Barbu, I. Lasiecka, R. Triggiani, Abstract settings for tangential boundary stabilization of Navier–Stokes equations by high- and low-gain feedback controllers, Nonlinear Anal. 64 (2006) 2704–2746], such a regularity is achieved with a feedback obtained by minimizing a functional involving a norm of the state variable strong enough. In that case, the feedback controller cannot be determined by a well posed Riccati equation. Here, we choose a functional involving a very weak norm of the state variable. The compatibility condition between the initial state and the feedback controller at t=0, is achieved by choosing a time varying control operator in a neighbourhood of t=0.     

Regularity properties of optimal controls for problems with time-varying state and control constraints

In this paper we report new results on the regularity of optimal controls for dynamic optimization problems with functional inequality state constraints, a convex time-dependent control constraint and a coercive cost function. Recently, it has been shown that the linear independence condition on active state constraints, present in the earlier literature, can be replaced by a less restrictive, positive linear independence condition, that requires linear independence merely with respect to non-negative weighting parameters, provided the control constraint set is independent of the time variable. We show that, if the control constraint set, regarded as a time-dependent multifunction, is merely Lipschitz continuous with respect to the time variable, then optimal controls can fail to be Lipschitz continuous. In these circumstances, however, a weaker Hölder continuity-like regularity property can be established. On the other hand, Lipschitz continuity of optimal controls is guaranteed for time-varying control sets under a positive linear independence hypothesis, when the control constraint sets are described, at each time, by a finite collection of functional inequalities.     

Output Regulation in Differential Variational Inequalities using Internal Model Principle and Passivity-Based Approach

We consider the problem of designing state feedback control laws for output regulation in a class of dynamical systems where state trajectories are constrained to evolve within time-varying, closed, and convex sets. The first main result states sufficient conditions for existence and uniqueness of solutions in such systems. We then design a static state feedback control law using the internal model principle, which results in a well-posed closed-loop system and solves the regulation problem. As an application, we demonstrate how control input resulting from the solution of a variational inequality results in regulating the output of the system while maintaining polyhedral state constraints. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Properties of Attainable Sets of Evolution Inclusions and Control Systems of Subdifferential Type

In a separable Hilbert space we consider an evolution inclusion with a multivalued perturbation and the evolution operators that are the compositions of a linear operator and the subdifferentials of a time-dependent proper convex lower semicontinuous function. Alongside the initial inclusion, we consider a sequence of approximating evolution inclusions with the same perturbation and the evolution operators that are the compositions of the same linear operator and the subdifferentials of the Moreau–Yosida regularizations of the initial function. We demonstrate that the attainable set of the initial inclusion as a multivalued function of time is the time uniform limit of a sequence of the attainable sets of the approximating inclusions in the Hausdorff metric. We obtain similar results for evolution control systems of subdifferential type with mixed constraints on control. As application we consider an example of a control system with discontinuous nonlinearities containing some linear functions of the state variables of the system.     

Decentralized Adaptive Robust State Feedback for Uncertain Large-Scale Interconnected Systems with Time Delays

The problem of the decentralized robust control is considered for a class of large-scale time-varying systems withdelayed state perturbations and external disturbances in the interconnections. Here, the upper bounds of the delayed stateperturbations and external disturbances in the interconnections are assumed to be unknown. Adaptation laws areproposed to estimate such unknown bounds; by making use of the updated values of the unknown bounds, decentralized linear and nonlinear memoryless robust state feedback controllers are constructed. Based on Lyapunov stability theoryand Lyapunov–Krasovskii functionals, as well as employing the proposed decentralized nonlinear robust state feedback controllers, it is shown that the solutions of the resulting adaptive closed-loop large-scale time-delay system can be guaranteed to be uniformly bounded and that the states converge uniformly and asymptotically to zero. It is also shown that the proposed decentralized linear robust state feedback controllers can guarantee the uniform ultimate boundedness of the resulting adaptive closed-loop large-scale time-delay system. Finally, a numerical example is given to demonstrate the validity of the results.     

Bifurcation analysis and control of a class of hybrid biological economic models

This paper systematically studies a hybrid predator–prey economic model, which is formulated by differential-difference-algebraic equations. It shows that this model exhibits two bifurcation phenomena at the intersampling instants. One is saddle–node bifurcation, and the other is singular induced bifurcation which indicates that economic profit may bring impulse at some critical value, i.e., rapid expansion of biological population in terms of ecological implications. On the other hand, for the sampling instants, the system undergoes Neimark–Sacker bifurcation at a critical value of economic profit, i.e., the increase of economic profit destabilizes the system and generates a unique closed invariant curve. Moreover, the state feedback controller is designed so that singular induced bifurcation and Neimark–Sacker bifurcation can be eliminated and the population can be driven to steady states by adjusting harvesting costs and the economic profit. At the same time, by using Matlab software, numerical simulations illustrate the effectiveness of the results obtained here.     

Mathematical modeling and control of population systems: Applications in biological pest control

The aim of this paper is to apply methods from optimal control theory, and from the theory of dynamic systems to the mathematical modeling of biological pest control. The linear feedback control problem for nonlinear systems has been formulated in order to obtain the optimal pest control strategy only through the introduction of natural enemies. Asymptotic stability of the closed-loop nonlinear Kolmogorov system is guaranteed by means of a Lyapunov function which can clearly be seen to be the solution of the Hamilton–Jacobi–Bellman equation, thus guaranteeing both stability and optimality. Numerical simulations for three possible scenarios of biological pest control based on the Lotka–Volterra models are provided to show the effectiveness of this method.     

A finite dimensional extension of Lyusternik theorem with applications to multiobjective optimization

We consider a multiobjective program with inequality and equality constraints and a set constraint. The equality constraints are Fréchet differentiable and the objective function and the inequality constraints are locally Lipschitz. Within this context, a Lyusternik type theorem is extended, establishing afterwards both Fritz–John and Kuhn–Tucker necessary conditions for Pareto optimality.     

Optimal Control of State Constrained Integral Equations

We consider the optimal control problem of a class of integral equations with initial and final state constraints, as well as running state constraints. We prove Pontryagin’s principle, and study the continuity of the optimal control and of the measure associated with first order state constraints. We also establish the Lipschitz continuity of these two functions of time for problems with only first order state constraints.     

History Path Dependent Optimal Control and Portfolio Valuation and Management

Regarding the evolution of financial asset prices governed by an history dependent (path dependent) dynamical system as a prediction mechanism, we provide in this paper the dynamical valuation and management of a portfolio (replicating for instance European, American and other options) depending upon this prediction mechanism (instead of an uncertain evolution of prices, stochastic or tychastic). The problem is actually set in the format of a viability/capturability theory for history dependent control systems and some of their results are then transferred to the specific examples arising in mathematical finance or optimal control. They allow us to provide an explicit formula of the valuation function and to show that it is the solution of a ``Clio Hamilton–Jacobi–Bellman' equation. For that purpose, we introduce the concept of Clio derivatives of ``history functionals' in such a way we can give a meaning to such an equation. We then obtain the regulation law governing the evolution of optimal portfolios.     

Finding a Zero of The Sum of Two Maximal Monotone Operators

In this paper, the equivalence between variational inclusions and a generalized type of Weiner–Hopf equation is established. This equivalence is then used to suggest and analyze iterative methods in order to find a zero of the sum of two maximal monotone operators. Special attention is given to the case where one of the operators is Lipschitz continuous and either is strongly monotone or satisfies the Dunn property. Moreover, when the problem has a nonempty solution set, a fixed-point procedure is proposed and its convergence is established provided that the Brézis–Crandall–Pazy condition holds true. More precisely, it is shown that this allows reaching the element of minimal norm of the solution set.     

Scale of Viability and Minimal Time of Crisis

In this paper, we introduce and study the minimal time of a crisis map which measures the minimal time spent outside a given closed domain of constraints by trajectory solutions of a differential inclusion. The interest of such a notion is basically to tackle simultaneously viability and target issues. The main mathematical result characterizes the epigraph of the crisis map in terms of a viability kernel of an augmented problem. This allows the obtaining of the numerical schemes we specify and to derive an equivalent Hamilton–Jacobi formulation. A simple economic example illustrates the results.     

Normality of the maximum principle for nonconvex constrained Bolza problems

We consider a Bolza optimal control problem with state constraints. It is well known that under some technical assumptions every strong local minimizer of this problem satisfies first order necessary optimality conditions in the form of a constrained maximum principle. In general, the maximum principle may be abnormal or even degenerate and so does not provide a sufficient information about optimal controls. In the recent literature some sufficient conditions were proposed to guarantee that at least one maximum principle is nondegenerate, cf. [A.V. Arutyanov, S.M. Aseev, Investigation of the degeneracy phenomenon of the maximum principle for optimal control problems with state constraints, SIAM J. Control Optim. 35 (1997) 930–952; F. Rampazzo, R.B. Vinter, A theorem on existence of neighbouring trajectories satisfying a state constraint, with applications to optimal control, IMA 16 (4) (1999) 335–351; F. Rampazzo, R.B. Vinter, Degenerate optimal control problems with state constraints, SIAM J. Control Optim. 39 (4) (2000) 989–1007]. Our aim is to show that actually conditions of a similar nature guarantee normality of every nondegenerate maximum principle. In particular we allow the initial condition to be fixed and the state constraints to be nonsmooth. To prove normality we use J. Yorke type linearization of control systems and show the existence of a solution to a linearized control system satisfying new state constraints defined, in turn, by linearization of the original set of constraints along an extremal trajectory.     

Fuzzy Partial State Feedback Control of Discrete Nonlinear Systems with Unknown Time-Delay

A new discrete-time fuzzy partial state feedback control method for the nonlinear systems with unknown time-delay is proposed. Ma et al. proposed the design method of the fuzzy controller based on the fuzzy observer and Cao and Frank extend this result to be applicable to the case of the nonlinear systems with the time-delay. However, the time-delay is likely to be unknown in practical. In this paper, the sufficient condition for the asymptotic stability is derived with the assumption that the time-delay is unknown by applying Lyapunov–Krasovskii theorem and this condition is converted into the LMI problem.     

Cosmically Lipschitz Set-Valued Mappings

This paper introduces a kind of sub-Lipschitz continuity for set-valued mappings based on the cosmic metric. This type of Lipschitz behavior has applications with regards to necessary optimality conditions, the Hamilton–Jacobi equation, and invariance of unbounded differential inclusions. Cosmically Lipschitz assumptions allow for broader applications than previously allowed under Lipschitz assumptions. It is also shown that a cosmically Lipschitz mapping can be characterized by the normal cones to its graph using the coderivative, and various rules are presented in order to more easily identify such a mapping.     

New Approach to Linear Exact Model Matching for a Class of Nonlinear Systems

In this paper, a new approach to the linear exact model matching problem for a class of nonlinear systems, using static state feedback, is presented. This approach reduces the problem of determining the state feedback control law to that of solving a system of first-order partial differential equations. Based on these equations, two major issues are resolved: the necessary and sufficient conditions for the problem to have a solution and the general analytical expression for the feedback control law. Furthermore, the proposed approach is extended to solve the same problem via static output feedback.