On a team control problem under obstacles

The paper deals with the problem of coordinated goal-oriented target control for a group of control systems that are to realize a joint movement towards a given target set under collision avoidance. The members of the group are obliged to lie inside a virtual ellipsoidal container, which realizes a reference motion while also avoiding external obstacles specified in advance. We describe a general solution scheme based on decomposing the main problem into auxiliary subproblems, for which we indicate solution methods as well as the necessity of coordinating these solutions at the final stage.     

Local Optima Avoidance in Depot Location

The locating of a set of depots to serve a given set of customers is complicated by the existence of local optima of the cost function. It is shown that the strategic siting of hazards (conventionally used to model natural obstacles) can lead to the avoidance of such local optima. A set of heuristics for hazard-siting is given and some examples are considered.     

Characterization and computation of control invariant sets for linear impulsive control systems

Impulsive control systems are suitable to describe and control a venue of real-life problems, going from disease treatment to aerospace guidance. The main characteristic of such systems is that they evolve freely in-between impulsive actions, which makes it difficult to guarantee its permanence in a given state-space region. In this work, we develop a method for characterizing and computing approximations to the maximal control invariant sets for linear impulsive control systems, which can be explicitly used to formulate a set-based model predictive controller. We approach this task using a tractable and non-conservative characterization of the admissible state sets, namely the states whose free response remains within given constraints, emerging from a spectrahedron representation of such sets for systems with rational eigenvalues. The so-obtained impulsive control invariant set is then explicitly used as a terminal set of a predictive controller, which guarantees the feasibly asymptotic convergence to a target set containing the invariant set. Necessary conditions under which an arbitrary target set contains an impulsive control invariant set (and moreover, an impulsive control equilibrium set) are also provided, while the controller performance are tested by means of two simulation examples.     

Avoidance Control on Time Scales

We consider dynamic systems on time scales under the control of two agents. One of the agents desires to keep the state of the system out of a given set regardless of the other agent’s actions. Leitmann’s avoidance conditions are proved to be valid for dynamic systems evolving on an arbitrary time scale.     

On the closure of sets of attainability in ℝ2

We display a class of control systems in the plane for which it is generically true that the set attainable from any point in the plane is a closed Whitney prestratified set. That is, even if the attainable set of some given system is not closed, we can approximate the system as closely as we wish by another system whose attainable sets are closed.     

Control synthesis for discrete time systems with control and state bounds in the presence of disturbances

The problem of control in the presence of unknown but limited disturbance for a discrete-time linear system with polyhedral input and state bounds is investigated. Two problems are considered: that of reaching an assigned target set in the state space; and that of keeping the state in a given region using the available controls. In both cases, a solution is given via linear programming. A computational procedure for the control synthesis is proposed which can be implemented to obtain a feedback control.The author thanks Professor G. Leitmann for his helpful suggestions.     

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.     

Asymptotics of pattern avoidance in the Klazar set partition and permutation-tuple settings

We consider asymptotics of set partition pattern avoidance in the sense of Klazar. Our main result derives the asymptotics of the number of set partitions avoiding a given set partition within an exponential factor, which leads to a classification of possible growth rates of set partition pattern classes. We further define a notion of permutation-tuple avoidance, which generalizes notions of Aldred et al. and the usual permutation pattern setting, and similarly determine the number of permutation-tuples avoiding a given tuple to within an exponential factor. We note a seeming connection to previous results on hereditary properties of labelled graphs, prompting the question of if there is a generalization to ordered graphs.     

Additional necessary conditions for optimal control with state-variable inequality constraints

It is shown that, when the set of necessary conditions for an optimal control problem with state-variable inequality constraints given by Bryson, Denham, and Dreyfus is appropriately augmented, it is equivalent to the (different) set of conditions given by Jacobson, Lele, and Speyer. Relationships among the various multipliers are given.This work was done at NASA Ames Research Center, Moffett Field, California, under a National Research Council Associateship.     

Existence of optimal controls for the diffusion equation

The existence is considered of a boundary control which drives a system governed by the one-dimensional diffusion equation from the zero state to a given final state, and at the same time minimizes a given functional. The problem is first modified to one in which the minimum is sought of a functional defined on a set of Radon measures. The existence of a minimizing measure is demonstrated, and it is shown that this measure may be approximated by a piecewise constant control. Finally, conditions are given under which a minimizing measurable control exists for the unmodified problem.     

Coordination and collision avoidance for Lagrangian systems with disturbances

In this paper we address the problem of cooperation and collision avoidance for Lagrangian systems with input disturbances. We design control laws that guarantee cooperation as well as collision-free maneuvers. We show, using a two-step proof, that the avoidance part of the control laws guarantees safety of the agents independently of the coordinating part. Then, we establish an ultimate bound on the region to which all the agents converge to. The obtained theoretical results are illustrated through numerical examples.     

Continuity in the parameter of the minimum value of an integral functional over the solutions of an evolution control system

The problem of minimization of an integral functional with an integrand that is nonconvex with respect to the control is considered. We minimize our functional over the solution set of a nonlinear evolution control system with a time-dependent subdifferential operator in a Hilbert space. The control constraint is given by a nonconvex closed bounded set. The integrand, the control constraint, the initial conditions and the operators in the equation describing the control system all depend on a parameter. We consider, along with the original problem, the problem of minimizing an integral functional with an integrand convexified with respect to the control over the solution set of the same system, but now subject to the convexified control constraint. By a solution of the control system we mean a “trajectory–control” pair. We prove that for each value of the parameter the convexified problem has a solution, which is the limit of a minimizing sequence of the original problem, and the minimum value of the functional of the convexified problem is a continuous function of the parameter.     

Guaranteed avoidance strategies

We consider a dynamical system under the control of two agents, one of whom desires to keep the system's state out of a given set regardless of the other agent's actions. The guaranteed avoidance conditions of Ref. 1 are generalized to admit piecewiseC ¹ test functions; this generalization permits application to a wider class of problems. The results are illustrated by application to a version of the homicidal chauffeur game.Dedicated to R. BellmanThis paper is based on research supported by NSF under Grant No. ENG-78-13931.     

Efficient robust optimization for robust control with constraints

This paper proposes an efficient computational technique for the optimal control of linear discrete-time systems subject to bounded disturbances with mixed linear constraints on the states and inputs. The problem of computing an optimal state feedback control policy, given the current state, is non-convex. A recent breakthrough has been the application of robust optimization techniques to reparameterize this problem as a convex program. While the reparameterized problem is theoretically tractable, the number of variables is quadratic in the number of stages or horizon length N and has no apparent exploitable structure, leading to computational time of per iteration of an interior-point method. We focus on the case when the disturbance set is ∞-norm bounded or the linear map of a hypercube, and the cost function involves the minimization of a quadratic cost. Here we make use of state variables to regain a sparse problem structure that is related to the structure of the original problem, that is, the policy optimization problem may be decomposed into a set of coupled finite horizon control problems. This decomposition can then be formulated as a highly structured quadratic program, solvable by primal-dual interior-point methods in which each iteration requires time. This cubic iteration time can be guaranteed using a Riccati-based block factorization technique, which is standard in discrete-time optimal control. Numerical results are presented, using a standard sparse primal-dual interior point solver, that illustrate the efficiency of this approach.     

Set stabilizability of switched Boolean control networks via Ledley antecedence solution

This paper studies two kinds of set stabilizability issues of switched Boolean control networks (SBCNs) by Ledley antecedence solution, that is, pointwise set stabilizability and set stabilizability under arbitrary switching signals. Firstly, based on the state transition matrix of SBCNs, the mode-dependent truth matrix is defined. Secondly, using the mode-dependent truth matrix in every step, a switching signal and the corresponding Ledley antecedence solutions are determined. Furthermore, a state feedback switching signal and a state feedback control are obtained for the pointwise set stabilizability. Thirdly, with the help of all mode-dependent truth matrices, the Ledley antecedence solutions are derived for a set of Boolean inclusions, which admits a state feedback control for the set stabilizability under arbitrary switching signals. Finally, an example is given to show the effectiveness of the proposed results.     

Conditions for swappability of records in a microdata set when some marginals are fixed

We consider swapping of two records in a microdata set for the purpose of disclosure control. We give some necessary and sufficient conditions that some observations can be swapped between two records under the restriction that a given set of marginals are fixed. We also give an algorithm to find another record for swapping if one wants to swap out some observations from a particular record.     

ON AVOIDANCE ACTIVITIES IN FISHERY ENFORCEMENT MODELS

Compliance and enforcement in fisheries are important issues from an economic point of view since management measures are useless without a certain level of enforcement. These conclusions come from the well‐established theoretical literature on compliance and enforcement problems within fisheries and a common result is that, it is efficient to set fines as high as possible and monitoring as low as possible, when fines are costless and offenders are risk neutral. However, this result is sensitive to the assumption that fishermen cannot engage in avoidance activities, e.g., activities to reduce the likelihood of being detected when noncomplying. The paper presents a model of fisheries that allows the fishermen to engage in avoidance activities. The conclusions from the model are that, under certain circumstances, fines are costly transfers to society since they not only have a direct positive effect on the level of deterrence, but also an indirect negative effect in the form of increased avoidance activities to reduce the probability of detection. The paper contributes to the literature on avoidance activities by introducing the externality from the illegal behavior as an endogenous effect on other offenders. For an externality, that has an exogenous effect on other actors, Malik shows that fines are only costly transfers for conditional deterrence (when one actor is deterred while another actor is not). For fisheries, we show that fines are also costly transfers under no deterrence (when no agents are deterred).     

Control with conic control constraint set

A sufficient condition is given for controllability with conic control constraint set, and a control is computed which performs the transfer to the origin.     

随机递归最优控制和混合最优控制问题

对随机递归最优控制问题即代价函数由特定倒向随机微分方程解来描述和递归混合最优控制问题即控制者还需 决定最优停止时刻, 得到了最优控制的存在性结果. 在一类等价概率测度集中,还给出了递归最优值函数的最小和最大数学期望.     

Another Jacobi sufficiency criterion for optimal control with smooth constraints

In order to tighten the gap between necessary and sufficient conditions, new second-order sufficient conditions are developed for optimal control problems, where the control set is given by smooth functions. When the control set is polyhedral, our criterion generalizes prior results of the same kind, namely, the Jacobi criterion in Hamiltonian form and that in Lagrangian form (Refs. 1–3).The research of V. Zeidan was supported by NSERC Grant A-8570, which is gratefully acknowledged.