Risk aggregation with dependence uncertainty

Risk aggregation with dependence uncertainty refers to the sum of individual risks with known marginal distributions and unspecified dependence structure. We introduce the admissible risk class to study risk aggregation with dependence uncertainty. The admissible risk class has some nice properties such as robustness, convexity, permutation invariance and affine invariance. We then derive a new convex ordering lower bound over this class and give a sufficient condition for this lower bound to be sharp in the case of identical marginal distributions. The results are used to identify extreme scenarios and calculate bounds on Value-at-Risk as well as on convex and coherent risk measures and other quantities of interest in finance and insurance. Numerical illustrations are provided for different settings and commonly-used distributions of risks.     

Robust State-Feedback Controllability of Linear Systems to a Hyperplane in a Class of Bounded Controls

The paper deals with a linear time-dependent dynamic system with scalar control and input uncertainty (disturbance). Two admissible classes of input uncertainty realizations are considered: the class of measurable bounded functions and the class of measurable quadratically integrable functions. The problem to be studied is the existence of a state feedback control with measurable bounded time realizations transferring the system to a given hyperplane (a target set) from any initial position in a prescribed time for any admissible input uncertainty realization. Necessary and sufficient conditions for the existence of such a control are derived, based on the explicit construction of this control by using an auxilary zero-sum linear-quadratic differential game with a cheap control for the minimizing player. Examples illustrting the theoritical results are presented.     

Global controllability and computation of controls in nonlinear systems

The authors consider the problem of constructing an admissible open-loop control of bounded energy steering a nonlinear system from a given initial state to a given final state under the condition that the first-approximation system is completely controllable. The convergent iterative procedure for computing an admissible control is verified. It is shown that a nonlinear system locally controllable with respect to the first approximation becomes globally completely controllable for any boundary conditions from the stability region if the initial nonlinear system is stabilizable up to the asymptotic stability in the large and in the whole and the nonlinear terms either satisfy the global Cauchy-Lipschitz condition or are polynomials of a certain degree in state coordinates with arbitrary coefficients. The nonlinear system of algebraic equations to computation of whose solutions the problem of constructing the admissible control reduces is indicated. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 26, Nonlinear Dynamics, 2005.     

Robust output feedback stabilization for two-dimensional continuous systems in roesser form

This paper considers the problem of robust stabilization via dynamic output feedbackcontrollers for uncertain two-dimensional continuous systems described by the Roesser's state space model. The parameter uncertainties are assumed to be norm-bounded appearing in all the matrices of the system model. A sufficient condition for the existence of dynamic output feedback controllers guaranteeing the asymptotic stability of the closed-loop system for all admissible uncertainties is proposed. A desired dynamic output feedback controller can be constructed by solving a set of linear matrix inequalities. Finally, an illustrative example is provided to demonstrate the applicability and effectiveness of the proposed method.     

Some problems of linear differential equations on abstract spaces and unbounded perturbations of linear operator semigroup

This paper is a survey for development of linear distributed parameter system. At first we point out some questions existing in current study of control theory for the L^p linear system with an unbounded control operator and an unbounded observation operator, such as stabilization problem and observer theory that are closely relevant to state feedback operator. After then we survey briefly some results on relevant problems that are related to solvability of linear differential equations in general Banach space and semigroup perturbations. As a principle, we propose a concept of admissible state feedback operator for system (A, B). Finally we give an existence result of admissible state feedback operators, including semigroup generation and the equivalent conditions of admissibility of state feedback operators, for an L^p well-posed system.     

Guaranteed Cost Control for Uncertain Neutral Stochastic Systems via Dynamic Output Feedback Controllers

This paper deals with the problem of guaranteed cost control for uncertain neutral stochastic systems. The parameter uncertainties are assumed to be time-varying but norm-bounded. Dynamic output feedback controllers are designed such that, for all admissible uncertainties, the resulting closed-loop system is mean-square asymptotically stable and an upper bound on the closed-loop value of the cost function is guaranteed. By employing a linear matrix inequality (LMI) approach, a sufficient condition for the solvability of the underlying problem is obtained. A numerical example is provided to demonstrate the potential of the proposed techniques.     

Optimal control problems for a class of nonlinear evolution equations

In this paper we study optimal control problems for systems governed by nonlinear evolution equations. First we develop an existence theory for systems with a priori feedback using the reduction technique and a convexity-type hypothesis involving property Q. In doing this we also establish the nonemptiness of the set of admissible state-control pairs, by solving a nonlinear evolution inclusion. Then we obtain necessary conditions for optimality for a class of problems with terminal cost criterion and initial condition which is not a priori given but is only required to belong to a given set (systems with insufficient data in the terminology of Lions). This revised version was published online in June 2006 with corrections to the Cover Date.     

Invariants for feedback equivalence and Cauchy characteristic multifoliations of nonlinear control systems

Linearization of a nonlinear feedback control system under nonlinear feedback is treated as a problem of equivalence-under the Lie pseudogroup of feedback transformations-of distributions on the product manifold of the state and control variables. The new feature of this paper is that it introduces the Cauchy characteristic sub-distributions of these distributions and their derived distributions. These Cauchy characteristic distributions are involutive and nested, hence define a Multifoliate Structure. A necessary condition for feedback equivalence of two nonlinear control systems is that these multifoliations be transformed under the feedback pseudogroup. For linear systems, this Cauchy characteristic multifoliate structuee is readily computed in terms of the (A, B)-matrix that defines the linear system. Assuming that the conditions for local feedback linearization are satisfied, the existence of a global feedback linearizing transformation is dependent on computing an element of the first cohomology group of the space with coefficients in the sheaf of groupoid of infinitesimal feedback automorphisms of the linear system. The theorem quoted above about the Cauchy characteristic multifoliations provides some information about this groupoid. It is computed explicitly and directly for control systems with one- or two-state dimensions. Finally, these Cauchy characteristic sub-distributions must inevitably play a role in the numerical or symbolic computational analysis of the Hunt-Su partial differential equations for the feedback-linearizing transformation.Senior Research Associate of the National Research Council at the Ames Research Center of NASA.     

Decentralized robust control for a class of uncertain large-scale interconnected nonlinear dynamical systems

The problem of the decentralized robust control for a class of large-scale interconnected nonlinear dynamical systems with input interconnection and external interconnection perturbations is considered. Based on the stabilizability of each nominal isolated subsystem (i.e., the isolated subsystem in the absence of interconnection perturbations), a class of decentralized local state feedback controllers is proposed, and some sufficient conditions are derived by making use of the Lyapunov stability criterion such that uncertain large-scale interconnected systems can be stabilized asymptotically by these decentralized state feedback controllers. For large-scale systems with only input interconnection perturbations, such decentralized controllers become a class of decentralized stabilizing state feedback controllers. That is, the decentralized stability of such large-scale systems can be guaranteed always by using the decentralized state feedback controllers proposed in the paper. Finally, a numerical example is given to demonstrate the validity of the results.     

Invariance relations in single server queues with LCFS service discipline

This paper is concerned with single server queues having LCFS service discipline. We give a condition to hold an invariance relation between time and customer average queue length distributions in the queues. The relation is a generalization of that in an ordinary GI/M/1 queue. We compare the queue length distributions for different single server queues with finite waiting space under the same arrival process and service requirement distribution of customer and derive invariance relations among them.This research was supported in part by a grant from the Tokyo Metropolitan Government. The latter part of this paper was written while the author resided at the University of California, Berkeley.     

Impulse free interconnection of dynamical systems

In this paper we address the problem that impulses might occur when a to-be-controlled plant is connected to a suitable controller. In the behavioral literature this issue is dealt when studying the so-called ‘regular feedback interconnection’ (RFI) of the plant and the controller behaviors. We address the question of when a given controlled behavior satisfying desired specifications can be obtained by RFI of the plant and some suitable controller. The paper presents an algorithm to construct a controller that yields the controlled behavior by RFI, if such a controller exists, and then parametrizes all such controllers. We extend the results to the more general case when all the to-be-controlled variables might not be available for interconnection. We show that any disturbances in the plant continue to remain free in the controlled behavior if and only if the interconnection is an RFI. This is also equivalent to the well-posedness of an interconnection. This paper also makes concrete the intuitive relation between absence of inadmissibility of initial conditions and RFI. A door closing mechanism is analysed in relation to the results in this paper.     

Exact self-similar solution of a certain equation of non-linear diffusion with dissipation

Using the renormalization-group method, an exact solution of the equation of non-linear diffusion with dissipation of the Cahn–Hilliard type is obtained. When finding an admissible form of the self-similar solution, only dimensionality arguments and the property of renormalization invariance are used. The form of the initial and boundary conditions, consistent with the non-linear equation cannot be given in advance, but is found from the solution of the renormalization-group equations.     

Asymptotical feedback controllability of continuous-time probabilistic logic control networks

In this article, we study the controllability of continuous-time probabilistic logic control networks (CT-PLCNs) under sampled-data feedback controls (SDFCs). First, we demonstrate that the concept of finite-time controllability with probability one for discrete-time probabilistic logic control networks cannot be generalized to CT-PLCNs. Then, we propose the concepts of asymptotical feedback reachability and asymptotical feedback controllability for CT-PLCNs. Based on the invariant subsets, we prove that a target state is asymptotically feedback reachable if and only if the target state is a control equilibrium point and any initial state has an admissible path to the target state. Moreover, we introduce the concept of reachability matrix and propose an easily verifiable criterion for asymptotical feedback reachability expressed in terms of the reachability matrix. Based on these, we prove that a CT-PLCN is asymptotically feedback controllable if and only if every state is a control equilibrium point and there is an admissible path between any pair of initial and target states. The relation between controllability and stabilizability is also discussed. We prove that a CT-PLCN is asymptotically feedback controllable if and only if every state is asymptotically feedback stabilizable. For a controllable CT-PLCN, we propose an algorithm of designing a stabilizing SDFC for any given target state. Additionally, we discuss the asymptotical feedback controllability of CT-PLCNs under time-varying nonuniform SDFCs. Finally, an illustrative example is presented to explain the proposed methods and verify the controllability criteria.     

Input-output linearization with simultaneous decoupling by restricted state feedback

The problem of input-output linearization with simultaneousdecoupling via restricted state feedback for nonsquare nonlinearsystems is investigated in this paper. The problem is treatedon the basis of a new algebraic approach whose main featureis that it reduces the determination of the admissible state-feedbackcontrol laws to the solution of an algebraic and a first-orderpartial differential system of equations. Verifiable necessaryand sufficient conditions of an algebraic nature are establishedfor the solvability of the aforementioned problem, which relyon the solvability of these systems of equations. Moreover,an explicit expression for a special admissible restricted state-feedbackcontroller is analytically derived.     

Entropy for semigroup actions on general topological spaces

This paper introduces both notions of topological entropy and invariance entropy for semigroup actions on general topological spaces. We use the concept of admissible family of open coverings to extending and studying the notions of Adler–Konheim–McAndrew topological entropy, Bowen topological entropy, and invariance entropy to the general theory of topological dynamics.     

Infinite horizon noncooperative differential games with nonsmooth costs

For a noncooperative differential game, the value functions of the various players satisfy a system of Hamilton-Jacobi equations. In the present paper, we study a class of infinite-horizon scalar games with either piecewise linear or piecewise smooth costs, exponentially discounted in time. By the analysis of the value functions, we find that results about existence and uniqueness of admissible solutions to the HJ system, and therefore of Nash equilibrium solutions in feedback form, can be recovered as in the smooth costs case, provided the costs are globally monotone. On the other hand, we present examples of costs such that the corresponding HJ system has infinitely many admissible solutions or no admissible solutions at all, suggesting that new concepts of equilibria may be needed to study games with general nonlinear costs.     

Pritchard-Salamon系统的可稳定化性和代数Riccati方程

在本文中,我们给出了光滑Pritchard－Salamon系统（简称PS系统）能以紧算子为可容反馈可稳定化的充分必要条件．应用此结果,我们给出了光滑PS系统中的代数Riccati方程的所有非负自伴解的参数化表示,把[1]中的主要结果推广到了光滑PS系统．     

Non‐linear initial boundary value problems of hyperbolic–parabolic type. A general investigation of admissible couplings between systems of higher order. Part 3: applications

This is the third part of an article that is devoted to the theory of non‐linear initial boundary value problems. We consider coupled systems where each system is of higher order and of hyperbolic or parabolic type. Our goal is to characterize systematically all admissible couplings between systems of higher order and different type. By an admissible coupling we mean a condition that guarantees the existence, uniqueness and regularity of solutions to the respective initial boundary value problem. In part 1, we develop the underlying theory of linear hyperbolic and parabolic initial boundary value problems. Testing the PDEs with suitable functions we obtain a priori estimates for the respective solutions. In particular, we make use of the regularity theory for linear elliptic boundary value problems that was previously developed by the author. In part 2, we prove the local in time existence, uniqueness and regularity of solutions to quasilinear initial boundary value problems using the so‐called energy method. In the above sense the regularity assumptions about the coefficients and right‐hand sides define the admissible couplings. In part 3 at hand, we extend the results of part 2 to the nonlinear initial boundary value problem (4.2). In particular, assumptions (B8) and (B9) about the respective parameters correspond to the previous regularity assumptions and hence define the admissible couplings now. Moreover, we exploit assumptions (B8) and (B9) for the case of two coupled systems. Copyright © 2002 John Wiley & Sons, Ltd.     

Non‐linear initial boundary value problemsof hyperbolic–parabolic type. A general investigationof admissible couplings between systems of higher order. Part 1: linear theory

In this article (which is divided in three parts) we investigate the non‐linear initial boundary value problems (1.2) and (1.3). In both cases we consider coupled systems where each system is of higher order and of hyperbolic or parabolic type. Our goal is to characterize systematically all admissible couplings between systems of higher order and different type. By an admissible coupling we mean a condition that guarantees the existence, uniqueness and regularity of solutions to the respective initial boundary value problem. In part 1 at hand, we develop the underlying theory of linear hyperbolic and parabolic initial boundary value problems. Testing the PDEs with suitable functions we obtain a priori estimates for the respective solutions. In particular, we make use of the regularity theory for linear elliptic boundary value problems that was previously developed by the author. In part 2, we prove the local in time existence, uniqueness and regularity of solutions to the quasilinear initial boundary value problem (1.2) using the so‐called energy method. In the above sense, the regularity assumptions about the coefficients and right‐hand sides define the admissible couplings. In part 3, we extend the results of part 2 to the non‐linear initial boundary value problem (1.3). In particular, the assumptions about the respective parameters correspond to the previous regularity assumptions and hence define the admissible couplings now. Moreover, we exploit the assumptions about the respective parameters for the case of two coupled systems. Copyright © 2002 John Wiley & Sons, Ltd.     

Robust Transferable Sets of Linear Transferring Strategies

The problem of transferring a controlled linear system with a disturbance to a prescribed target set (a hyperplane at a prescribed time instant) is considered. The control and disturbance are subject to geometrical constraints. The original transfer problem is reduced to a scalar one. Then, for a given transferring linear feedback strategy, the respective transferable set is studied, i.e. the set of all the initial positions in the time/state plane, from which the target set is reached, respecting the control constraints, against any admissible disturbance. Necessary and sufficient conditions for the existence of the transferable set are derived, its set-theoretical properties are established, the algorithm of its boundary construction is proposed, the boundary smoothness is studied. Illustrative examples are presented.