Topological methods for the global controllability of nonlinear systems

Sufficient conditions for the local and global controllability of general nonlinear systems, by means of controls belonging to a fixed finite-dimensional subspace of the space of all admissible controls, are established with the aid of topological methods, such as homotopy invariance principles. Some applications to certain classes of nonlinear control processes are given, and various known results on the controllability of perturbed linear systems are also derived as particular cases.     

非线性微分方程的正周期解

§ 1　 IntroductionIn[1 ] ,Saker and Agarwal studied the existence and uniqueness of positive periodicsolutions of the nonlinear differential equationN′(t) =-δ(t) N(t) + p(t) N(t) e- a N(t) ,(1 )whereδ(t) and p(t) are positive T-periodic functions.They proved that if p* >δ* ,then(1 ) has a unique T-periodic positive solution,wherep* =min0≤ t≤ Tp(t) ,δ* =max0≤ t≤ Tδ(t) .　　In view ofthe papermentioned above,whatcan be said aboutequation(1 ) when p* ≤δ* ?In this paper,we conside…     

Linear control systems: Controllability with constrained controls

We study the null controllability with bounded controls of an ordinary control process in ⁿ . To prove controllability conditions for the nonautonomous case, we need to generalize the definition of characteristic exponents of a linear system. Some theorems on characteristic exponents are stated.This work follows the program of the Gruppo Nazionale per l'Analisi Funzionale e le Sue Applicazioni, Consiglio Nazionale delle Ricerche, Rome, Italy.     

Characterizations of local upper Lipschitz property of perturbed solutions to nonlinear second-order cone programs

We characterize the local upper Lipschitz property of the stationary point mapping and the Karush–Kuhn–Tucker (KKT) mapping for a nonlinear second-order cone programming problem using the graphical derivative criterion. We demonstrate that the second-order sufficient condition and the strict constraint qualification are sufficient for the local upper Lipschitz property of the stationary point mapping and are both sufficient and necessary for the local upper Lipschitz property of the KKT mapping.     

Moduli of continuity of local times of strongly symmetric Markov processes via Gaussian processes

LetX be a strongly symmetric standard Markov process on a locally compact metric spaceS with 1-potential densityu ¹(x, y). Let {L _t ^y , (t, y)R ⁺×S} denote the local times ofX and letG={G(y), yS} be a mean zero Gaussian process with covarianceu ¹(x, y). In this paper results about the moduli of continuity ofG are carried over to give similar moduli of continuity results aboutL _t ^y considered as a function ofy. Several examples are given with particular attention paid to symmetric Lévy processes.The research of both authors was supported in part by a grant from the National Science Foundation. In addition the research of Professor Rosen was also supported in part by a PSC-CUNY research grant. Professor Rosen would like to thank the Israel Institute of Technology, where he spent the academic year 1989–90 and was supported, in part, by the United States-Israel Binational Science Foundation. Professor Marcus was a faculty member at Texas A&M University while some of this research was carried out.     

Global null controllability of nonlinear delay equations with controls in a compact set

The question of controlling some nonlinear retarded functional differential equations from an initial function to the zero function is considered. The control setsM() are square integrable functions with values in the unit closed sphereL ₂([t ₀, ),E ^m ) with center the origin. Assuming that the linear approximation of the nonlinear equation is null controllable with some integrable controls on some interval [t ₀,t ₁–2r], wheret ₀ is sufficiently large and wherer>0 is the delay, and assuming that the nonlinear system, withu=0, is uniformly globally asymptotically stable, we show that the nonlinear control process is globally null controllable with controlsuM(). The paper gives conditions which guarantee the stability assumptions, and also indicates conditions which yield the null controllability assumptions of the linear approximation. Our research extends known results on ordinary differential processes.     

Existence of periodic solutions of higher order nonlinear functional difference equations

Sufficient conditions for the existence of at least one periodic solution of two classes of functional difference equations are established, respectively.     

Existence of solutions for a class of third-order nonlinear boundary value problems

In this paper, we are concerned with the following third-order ordinary differential equation:

     

Small-time local controllability and continuity of the optimal time function for linear systems

For a finite-dimensional linear system, in which the control is restricted to belong to a completely arbitrary setJ, we give a simple necessary and sufficient condition for small-time local controllability from a pointp. The condition is equivalent to a characterization of the property that the Bellman function for the corresponding minimum-time optimal control problem is continuous atp.This work was partially supported by the National Science Foundation, Grant No. MCS-78-02442.     

A framework and a mean-field algorithm for the local control of spatial processes

The Markov Decision Process (MDP) framework is a tool for the efficient modelling and solving of sequential decision-making problems under uncertainty. However, it reaches its limits when state and action spaces are large, as can happen for spatially explicit decision problems. Factored MDPs and dedicated solution algorithms have been introduced to deal with large factored state spaces. But the case of large action spaces remains an issue. In this article, we define graph-based Markov Decision Processes (GMDPs), a particular Factored MDP framework which exploits the factorization of the state space and the action space of a decision problem. Both spaces are assumed to have the same dimension. Transition probabilities and rewards are factored according to a single graph structure, where nodes represent pairs of state/decision variables of the problem. The complexity of this representation grows only linearly with the size of the graph, whereas the complexity of exact resolution grows exponentially. We propose an approximate solution algorithm exploiting the structure of a GMDP and whose complexity only grows quadratically with the size of the graph and exponentially with the maximum number of neighbours of any node. This algorithm, referred to as MF-API, belongs to the family of Approximate Policy Iteration (API) algorithms. It relies on a mean-field approximation of the value function of a policy and on a search limited to the suboptimal set of local policies. We compare it, in terms of performance, with two state-of-the-art algorithms for Factored MDPs: SPUDD and Approximate Linear Programming (ALP). Our experiments show that SPUDD is not generally applicable to solving GMDPs, due to the size of the action space we want to tackle. On the other hand, ALP can be adapted to solve GMDPs. We show that ALP is faster than MF-API and provides solutions of similar quality for most problems. However, for some problems MF-API provides significantly better policies, and in all cases provides a better approximation of the value function of approximate policies. These promising results show that the GMDP model offers a convenient framework for modelling and solving a large range of spatial and structured planning problems, that can arise in many different domains where processes are managed over networks: natural resources, agriculture, computer networks, etc.     

Optimal control of one dimensional non-conservative quasi-diffusion processes

An extension of the work of P. Mandl concerning the optimal control of time-homogeneous diffusion processes in one dimension is given. Instead of a classical second order differential operator as infinitesimal generator, Feller's generalized differential operator D_mD⁺_p with a possibly nondecreasing weight function m is used. In this manner an optimal control of a wider class of one dimensional Marcov processes-including diffusions as well as birth and death processes-is realized.     

Multiple positive periodic solutions of nonlinear functional differential system with feedback control

In this paper, we employ Avery-Henderson fixed point theorem to study the existence of positive periodic solutions to the following nonlinear nonautonomous functional differential system with feedback control:

     

Local controllability for autonomous nonlinear systems

This paper is concerned with the study of local controllability for nonlinear, analytic, autonomous, control systems. To this aim, we consider the control system as a set-valued differential system, and we show that the convexification of the set of velocities does not change the local controllability properties. Given a finite family of vector fields, we also give a necessary condition for local controllability at timet, for everyt.This work was performed under the auspices of the National Research Council of Italy (CNR).     

Limit laws for the intersection local time of stable processes in u2

We prove renormalization results for self‐intersections of a stable process of index β in the plane. If 1 < β ≤4÷3 we show that a suitably renormalized version of the intersection local time converges in law to a Brownian motion     

Switching control of piecewise-deterministic processes

A finite collection of piecewise-deterministic processes are controlled in order to minimize the expected value of a performance functional with continuous operating cost and discrete switching control costs. The solution of the associated dynamic programming equation is obtained by an iterative approximation using optimal stopping time problems.This research was supported in part by NSF Grant No. DMS-8508651 and by University of Tennessee Science Alliance Research Incentive Award.     

Multiple positive solutions for a class of nonlinear elliptic equations

Via delicate estimates, we characterize an exact growth order near zero for positive solutions of a class of nonlinear elliptic equations. Using this characterization, we obtain multiple positive solutions for equations involving critical nonlinearity.     

Index theory, nontrivial solutions, and asymptotically linear second-order Hamiltonian systems

In this paper, we consider the existence and multiplicity of solutions of second-order Hamiltonian systems. We propose a generalized asymptotically linear condition on the gradient of Hamiltonian function, classify the linear Hamiltonian systems, prove the monotonicity of the index function, and obtain some new conditions on the existence and multiplicity for generalized asymptotically linear Hamiltonian systems by global analysis methods such as the Leray-Schauder degree theory, the Morse theory, the Ljusternik-Schnirelman theory, etc.