Optimal control of the delay linear systems with allowance for the terminal state constraints

Consideration was given to the linear problem of optimal control of one type of the delay systems where the delay appears in one equation of the system mathematical model. The terminal states of the system are bounded, the optimal control is realized by the discrete control actions obeying the geometrical constraints. Consideration was given to two types of solutions—program and positional. A dual method of calculation of the optimal programs was presented. Described was an algorithm of the optimal controller generating in real time the current values of the positional solution (optimal feedback). The results obtained were illustrated by the example of control of a system with the fourth-order delay.     

A cascadic monotonic time-discretized algorithm for finite-level quantum control computation

A computer package (CNMS) is presented aimed at the solution of finite-level quantum optimal control problems. This package is based on a recently developed computational strategy known as monotonic schemes.Quantum optimal control problems arise in particular in quantum optics where the optimization of a control representing laser pulses is required. The purpose of the external control field is to channel the system's wavefunction between given states in its most efficient way. Physically motivated constraints, such as limited laser resources, are accommodated through appropriately chosen cost functionals.

Program summary

Program title: CNMSCatalogue identifier: ADEB_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADEB_v1_0.htmlProgram obtainable from: CPC Program Library, Queen's University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 770No. of bytes in distributed program, including test data, etc.: 7098Distribution format: tar.gzProgramming language: MATLAB 6Computer: AMD Athlon 64 × 2 Dual, 2:21 GHz, 1:5 GB RAMOperating system: Microsoft Windows XPWord size: 32Classification: 4.9Nature of problem: Quantum controlSolution method: IterativeRunning time: 60-600 sec     

Discrete optimization by optimal control methods. II. The static traveling salesman problem

The static minisum traveling salesman problem is formulated as an optimal control problem. Two-sided algorithms based on the sufficient conditions for global optimality for solving this problem and a new algorithm for approximating the quality criterion from above to its optimal value are designed.     

A class of µ-commutative bilinear control systems

A class of μ-commutative bilinear control systems is defined and the properties of coefficients of a μ-commutative system are studied. A μ-commutative scalar control system is studied and an explicit representation for its input-output mapping is derived. This mapping is used to study the properties of a system, particularly the geometry of its reachable set. Original Russian Text ? M.V. Topunov, 2006, published in Avtomatika i Telemekhanika, 2006, No. 6, pp. 113–125. This paper was recommended for publication by B.T. Polyak, a member of the Editorial Board     

Discrete optimization by optimal control methods I. Separable problems

Two general solution schemes are designed for separable discrete optimization problems. Approximations from below and from above to the optimal value of the quality criterion are determined. These schemes are based on a unified theoretical base—sufficient conditions for the global optimal known in optimal control theory. Known and new methods for defining a resolving function, which is essential for applying these conditions, are described.     

External bounds and step controllability of the linear interval system

The differential equations describing double-sided bounds of the fundamental matrix of solutions for the linear control system with variable interval coefficients are obtained. Based on those bounds, the controllability problem, i.e., the possibility to transfer the system from a given parallelepiped to another in finite time in the class of step (piecewise constant) controls, is examined. Sufficient conditions of controllability in the form of solvability of an auxilliary linear programming problem are established. This problem results in step control transferring a bundle of system trajectories from an initial parallelepiped to a minimal neighborhood of a final parallelepiped.     

On the existence of periodic motion and the maximum sector bounds for absolute stability of nonlinear nonstationary systems

In this contribution we consider the absolute stability problem. We derive necessary and sufficient conditions for the existence of periodic motion using an operator approach. The results yield an efficient algorithm to approximate the maximum sector bounds for absolute stability numerically.     

Turnpike solutions of optimal control problems

Minimizing sequences for degenerate optimal control problems are constructed from turnpike solutions. Two variational approximation schemes for the turnpike solution are described: the first is a direct improvement of the simple approximation of piecewise-continuous turnpikes by the solutions of the initial differential system and the second consists of constructing an approximate optimal control in the neighborhood of a turnpike by a parametric curve in the state space. Since a sequence of state-linear feedback controls with variable coefficients is generated in both variants, turnpikes can be easily realized in practice.     

Comparison equations in problems of motion stability

New results of the investigation of stability of solutions to differential equations describing the behavior of different nonlinear nonstationary systems are represented; the results are based on the joint use of the method of logarithmic matrix norms and the method of limit equations.     

Methods of perturbations in quadratic problems of optimal control

The suggestion is made to use the methods of perturbations for the implementation of the conditions of nonlocal improvement in the class of optimal control problems that are quadratic in the state and linear in control. The methods are specified by the absence of the operation of the weak or the acicular variation and by the basic possibility of the nonlocal improvement of controls that satisfy the maximum principle.     

On some models and methods of the solution of scheduling problems in discrete enterprises

Scheduling problems of operation of machine-building enterprises are considered, the composition of which includes a few manufacturing systems and sections. For their solution, methods for construction of consistent plans of operation of enterprises are suggested, which can be detailed if necessary before the schedule of processing of individual parts.     

Generalized synthesis for singular nonlinear optimal control problems

We consider the problem of minimizing a quadratic noncoercive functional along the trajectories of a control-affine system. Due to lack of coercivity, existence of “classical” minimizers cannot, in general, be guaranteed. Under appropriate commutativity assumptions the problem can be extended into the space of generalized controls of class W _?1,∞ and reduced into a new problem which is generically coercive but nonconvex. We show how to extend further the problem in order to include generalized controls which are “generalized derivatives of one-parameter families of regular probability measures,” thus achieving convexification. Generalized trajectories for this type of controls exist only in a weak sense. We discuss a version of the maximum principle suitable to this class of problems and show how a generalized synthesis can be obtained.     

Pulse control problems under ellipsoidal constraints: Constraint parameters sensitivity

Consideration is given to pulse control problems for the tube of trajectories of the linear differential system under incomplete information on the initial data. Apart from the ordinary requirement for boundedness of variation, a special constraint of the ellipsoidal type is imposed on control actions; at this constraint, vectors of jumps of generalized controls belong to the specified ellipsoid of the corresponding finite-dimensional space. The dependence of the variation of optimal pulse control that solves the control problem for the tube of trajectories on parameters of the ellipsoidal constraint on admissible controls is studied.     

Realization of sliding modes as generalized solutions to optimal control problems

Consideration is given to the problem of realization of generalized solutions of different nature in classes of admissible models traditional for applications for optimal control problems. The main attention is focused on sliding modes in systems with a bounded set of velocities (velocity hodograph). Constructive correction schemes for elements of the sequence converging to the sliding mode are specified so that they are strictly admissible. This defines the procedure for constructing a minimizing sequence the elements of which are approximate optimal practically realizable solutions to estimates of accuracy.     

On the Newton-type method with admissible trajectories for mixed complementatiry problems

A new strategy is suggested for globalization of the convergence of the Newton method, which we proposed earlier, of the active set for the solution of mixed complementarity problems. The most important feature of the new algorithm is the admissibility of the trajectories generated by it.     

Stability against small noises in control problems with non-Lipschitz right-hand side of the dynamic equation

Control problems in systems with non-Lipschitz right-hand side are studied for the performance functional continuously dependent on the path. Are considered two variants of the optimization problem depending on the fact whether the ally controls the realized path from the set generated by a useful control. Relaxation of original optimization problems, namely, a sequence of perturbed problems with vanishing perturbations (the right-hand side of the equation and initial conditions) is proposed. An asymptotically optimal solution to the relaxation problem is obtained by N.N. Krasovskii and A.I. Subbotin’s extreme shift method. As is shown, the value achieved at this can be considerably better than the optimal result of the original problem.     

Exponential lag synchronization for delayed fuzzy cellular neural networks via periodically intermittent control

In this paper, lag synchronization for a class of delayed fuzzy cellular networks is investigated. By utilizing inequality technique, Lyapunov functional theory and the analysis method, some new and useful criteria of lag synchronization for the addressed networks are derived in terms of p-norm under a periodically intermittent controller. Finally, an example with simulation is given to show the effectiveness of the obtained results.     

Control of the medical treatment of AIDS

This work deals with the statement and the solution of problems of optimal control of the human immune system affected by a human immunodeficiency virus (HIV). The solution is performed by the numerical methods of optimal control on the basis of a dynamic model of the development of an infection in the organism of a human being. The results illustrate the possibilities for the use of mathematical methods in the problem of extending the life of HIV-infected patients.     

Boundary control of a laminar flow of a viscous incompressible fluid in the generalized Couette cell. The asymptotic approach

In this paper, a limiting problem for an optimal boundary control problem of a laminar flow of a viscous incompressible fluid in the generalized Couette cell, when the number of inner cylinders unrestrictedly increases, is obtained.     

The Fenchel duality, S-procedure, and the Yakubovich-Kalman lemma

The role of the Fenchel duality in the theorem on the losslessness of the S-procedure and in Yakubovich-Kalman lemma is studied. The Fenchel duality theorem implying the well-known results on the losslessness of the S-procedure is formulated. A relation between the Yakubovich-Kalman lemma and a special extremal problem defined on a set of positive-demidefinite solutions of the generalized Lyapunov inclusion is derived. Every assertion of the lemma is shown to be necessary and sufficient for the value of the extremal problem to be bounded.