On a minimax control problem for a positional functional under geometric and integral constraints on control actions

Within the game-theoretical approach, we consider a minimax feedback control problem for a linear dynamical system with a positional quality index, which is the norm of the deviation of the motion from given target points at given times. Control actions are subject to both geometric and integral constraints. A procedure for the approximate calculation of the optimal guaranteed result and for the construction of a control law that ensures the result is developed. The procedure is based on the recursive construction of upper convex hulls of auxiliary program functions. Results of numerical simulations are presented.     

On the numerical solution of a minimax control problem with a positional functional

The probabilistic analysis of an approximation algorithm for the minimum-weight m-peripatetic salesman problem with different weight functions of their routes (Hamiltonian cycles) is presented. The time complexity of the algorithm is O(mn ²). It is assumed that the elements of the distance matrix are independent equally distributed random variables with values in an upper unbounded domain [a _n , ∞), where a _n > 0. The analysis is carried out for the example of truncated normal and exponential distributions. Estimates for the relative error and failure probability, as well as conditions for the asymptotic exactness of the algorithm, are found.     

Minimax risk (regret) strategy for one class of control problems under dynamic disturbances

A controlled system under dynamic disturbances is considered. We formulate the problem of finding a strategy that is optimal in the sense of Savage’s minimax risk (regret) criterion, list basic properties of such problems, and describe a construction of a strategy optimal in the above sense for one class of systems containing linear systems.     

On some properties of optimal schedules in the job shop problem with preemption and an arbitrary regular criterion

Optimal schedules in the job shop problem with preemption and with the objective of minimizing an arbitrary regular function of operation completion times are studied. It is shown that for any instance of the problem there always exists an optimal schedule that meets several remarkable properties. Firstly, each changeover date coincides with the completion time of some operation, and so, the number of changeover dates is not greater than the total number of operations, while the total number of interruptions of the operations is no more than the number of operations minus the number of jobs. Secondly, every changeover date is “super-integral”, which means that it is equal to the total processing time of some subset of operations. And thirdly, the optimal schedule with these properties can be found by a simple greedy algorithm under properly defined priorities of operations on machines. It is also shown that for any instance of the job shop problem with preemption allowed there exists a finite set of its feasible schedules which contains at least one optimal schedule for any regular objective function (from the continuum set of regular functions).     

On the asymptotic properties of a priori minimax estimates

Properties are studied of the a priori minimax estimates /1,2/ of an unknown initial state of a linear dynamic object under the condition that the measurements are taken at discrete instants, while noise exists only in the measurement channel and is modelled as an independent random variable. The paper continues the investigations begun in /3/.     

On a problem of impulse control under interference

This paper contains several results about the structure of the congruence kernel C^(S)(G) of an absolutely almost simple simply connected algebraic group G over a global field K with respect to a set of places S of K. In particular, we show that C^(S)(G)) is always trivial if S contains a generalized arithmetic progression. We also give a criterion for the centrality of C^(S)(G) in the general situation in terms of the existence of commuting lifts of the groups G(K_v) for v ? S in the S-arithmetic completion ?^(S). This result enables one to give simple proofs of the centrality in a number of cases. Finally, we show that if K is a number field and G is K-isotropic, then C^(S)(G) as a normal subgroup of ?^(S) is almost generated by a single element.     

A minmax regret median problem on a tree under uncertain locations of the demand points

This paper deals with the location of a new facility on a tree according to the minimization of the weighted distance to the customers. The weights represent demands of the set of nodes. The exact location of each customer will be assumed unknown but close   to its corresponding node. In this context, an algorithm to find a minmax regret median is proposed and its complexity is shown to be O(nlog(n))$O(nlog\left(n\right))$, where n$n$ is the number of nodes of the tree     

Reliability of disturbance rejection control based on the geometrical disturbance decoupling

In this note, a reliable disturbance decoupled control system based on the geometrical technique is proposed. For this purpose, first, the conditions for nominal disturbance detection and rejection is introduced by using invariant subspaces. Then, the fragility issue of the controller/observer is addressed with appropriate H_∞ constraints in the form of some linear matrix inequalities (LMIs). In contrast to conventional pole placement method suitable for systems by neglecting sources of uncertainty, in the presented method, the disturbance rejection problem reduces to a quadratic stability problem. It is observed that the proposed method has a better attenuation performance than classical observer-based non-fragile controller. © 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On some properties of the groupsG(n,l)

Sunto In [1]è stata costruita una famiglia di 3-varietà chiuse e connesse mediante l'uso di grafi 4-colorati sugli spigoli. Nella presente nota si studiano alcune proprietà algebriche dei gruppi fondamentali delle suddette 3-varietà, per i quali una presentazione è stata ottenuta in [2].Il lavoro ha soprattutto un interesse algebrico, più che topologico, per le tecniche usate nelle dimostrazioni.

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Work performed under the auspicies of the G.N.S.A.G.A. of the C.N.R. (National Research Council) of Italy and within the project « Geometria delle Varietà Differenziabili » (40%) of the M.P.I. (Italy).     

On a problem of (0,2)-interpolation

The object in this paper is to consider the problem of existence, uniqueness, explicit representation of (0,2)-interpolation on the zeros of (1−x²)P_n−1(x)/x when n is odd, where P_n−1 denotes Legendre polynomial of degreen−1, and the problem of convergence of interpolatory polynomials.     

Search for maximum points of a vector criterion based on decomposition properties

We address the problem of optimal reconstruction of the values of a linear operator on ℝ ^d or ℤ ^d from approximate values of other operators. Each operator acts as the multiplication of the Fourier transform by a certain function. As an application, we present explicit expressions for optimal methods of reconstructing the solution of the heat equation (for continuous and difference models) at a given instant of time from inaccurate measurements of this solution at other time instants.     

On some properties of (fc)-sequences of ideals in local rings

The paper characterizes the length of maximal sequences satisfying conditions (i) and (ii) of (FC)-sequences, and proves some properties of (FC)-sequences, such as a bound on their lengths. As a consequence we get some results for mixed multiplicities and multiplicities of Rees rings of equimultiple ideals. We also prove that if is an ideal of positive height and is an arbitrary maximal sequence in satisfying conditions (i) and (ii) of (FC)-sequences, then is a reduction of

     

Abort landing in the presence of windshear as a minimax optimal control problem,part 1: Necessary conditions

The landing of a passenger aircraft in the presence of windshear is a threat to aviation safety. The present paper is concerned with the abort landing of an aircraft in such a serious situation. Mathematically, the flight maneuver can be described by a minimax optimal control problem. By transforming this minimax problem into an optimal control problem of standard form, a state constraint has to be taken into account which is of order three. Moreover, two additional constraints, a first-order state constraint and a control variable constraint, are imposed upon the model. Since the only control variable appears linearly, the Hamiltonian is not regular. Thus, well-known existence theorems about the occurrence of boundary arcs and boundary points cannot be applied. Numerically, this optimal control problem is solved by means of the multiple shooting method in connection with an appropriate homotopy strategy. The solution obtained here satisfies all the sharp necessary conditions including those depending on the sign of certain multipliers. The trajectory consists of bang-bang and singular subarcs, as well as boundary subarcs induced by the two state constraints. The occurrence of boundary arcs is known to be impossible for regular Hamiltonians and odd-ordered state constraints if the order exceeds two. Additionally, a boundary point also occurs where the third-order state constraint is active. Such a situation is known to be the only possibility for odd-ordered state constraints to be active if the order exceeds two and if the Hamiltonian is regular. Because of the complexity of the optimal control, this single problem combines many of the features that make this kind of optimal control problems extremely hard to solve. Moreover, the problem contains nonsmooth data arising from the approximations of the aerodynamic forces and the distribution of the wind velocity components. Therefore, the paper can serve as some sort of user's guide to solve inequality constrained real-life optimal control problems by multiple shooting.An extended abstract of this paper was presented at the 8th IFAC Workshop on Control Applications of Nonlinear Programming and Optimization, Paris, France, 1989 (see Ref. 1).This paper is dedicated to Professor Hans J. Stetter on the occasion of his 60th birthday.     

On a control problem for a linear system with delay in the control

The paper presents a realization of an orientation-preserving Morse–Smale 3-diffeomorphism in each class of the topological conjugacy by means of an abstract scheme.