Degenerate problems of optimal control for discrete-continuous (hybrid) systems

The notion of the degenerate problem of optimal control for the discrete-continuous systems was formulated. The main approaches to the problems of this class that were developed for the uniform continuous and discrete systems such as the transformations to the derivative systems and the method of multiple maxima, a special technique to define the Krotov functions under the like sufficient conditions, were extended to the discrete-continuous systems. The fields of possible efficient applications were indicated, and an example was presented.     

Turnpike solutions in optimal control problems for quantum-mechanical systems

We consider the problem of optimal impulse control for a quantum-mechanical spin sequence as a system of oscillators. We study this problem with a double transformation to a derivative problem in which phase variables are oscillator amplitudes. As a result, we get the turnpike solutions. For special (turnpike) sets of boundary conditions they correspond exactly to generalized solutions of the original problem, while for other conditions they can be used as initial approximations for iterative procedures. This problem is a generalization of the special case of two oscillators which we study exhaustively and use as an illustrative example.     

Optimization of state-linear discrete-continuous systems

An iterative method for process optimization in the state-linear discrete-continuous systems was proposed on the basis of the sufficiency conditions and the minimax scheme of nonlocal improvement. For the discrete systems, in particular, it was reduced to an essentially new method. Consideration was given to the possibility of applying it to the general nonlinear systems through approximation by systems of the class under consideration. Examples were presented.     

An optimization method for solving mixed discrete-continuous programming problems

This paper presents a heuristic approach for minimizing nonlinear mixed discrete-continuous problems with nonlinear mixed discrete-continuous constraints. The approach is an extension of the boundary tracking optimization that was developed by the authors to solve the minimum of nonlinear pure discrete programming problems with pure discrete constraints. The efficacy of the proposed approach is demonstrated by solving a number of test problems of the same class published in recent literature. Among these examples is the complex problem of minimizing the cost of a series–parallel structure with redundancies subject to reliability constraint. All tests conducted so far show that the proposed approach obtains the published minima of the respective test problems or finds a better minimum. While it is not possible to compare computation time due to the lack of data on the test problems, for all the tests the minimum is found in a reasonable time.     

Structure of differential-algebraic equations for control problems in mechanics

A problem of control of a mechanical system in a partly specified motion is studied. It is shown that the structure of differential-algebraic equations (DAEs) describing the problem may essentially differ from the structure of governing equations for a constrained mechanical system. The control forces chosen to realize the program constraints may have arbitrary directions with respect to the constraint manifold, and, in the extreme, they may be tangent. Thus, the index of the original DAEs of the problem may exceed three and a special approach to the problem solution has to be undertaken. Such an approach is proposed; a systematic formulation is given for the transformation of the DAEs into index-one DAEs or ODEs. Criteria for solvability of the problem are included.     

Double solutions of boundary value problems for 2mth-order differential equations and difference equations

A double fixed-point theorem is applied to obtain the existence of at least two positive solutions for the boundary value problem, (−1)^my^(2m)(t) = f(y(t)), t ϵ [0, 1], y⁽²ⁱ⁾(0) = y⁽²ⁱ⁺¹⁾(1) = 0, 0 ≤ i ≤ m−1. It is later applied to obtain the existence of at least two positive solutions for the analogous discrete boundary value problem, (−1)^mΔ^2mu(k) = g(u(k)), k ϵ {0, …, N}, Δ²ⁱu(0) = Δ²ⁱ⁺¹u(N + 1) = 0, 0 ⩽ m − 1.     

Minimum time control problems for non-autonomous differential equations

In this paper, we investigate a minimum time problem for controlled non-autonomous differential systems, with dynamics depending on the final time. The minimal time function associated to this problem does not satisfy the dynamic programming principle. However, we will prove that it is related to a standard front propagation problem via the reachability function. Two simple numerical examples are given to illustrate our approach.     

Efficient methods for enclosing solutions of systems of nonlinear equations

We consider modifications of the interval Newton method which combine two ideas: Reusing the same evaluation of the Jacobian several (says) times and approximately solving the Newton equation by some ‘linear’ iterative process. We show in particular that theR-order of these methods may becomes+1. We illustrate our results by a numerical example.     

Minimax filtering in linear stochastic uncertain discrete-continuous systems

Filtering of the states of a system, whose dynamics is defined by an Ito stochastic differential equation, by discrete and discrete-continuous observations is studied under the assumption that the intensities of continuous noises and covariance matrices of discrete noises are known only within to membership of certain uncertainty sets. A minimax approach is used to solve the problem. The filter is optimized with an integral quality criterion. Minimax filtering equations are derived from the solution of the dual optimization problem. A numerical solution algorithm for the problem is designed. Results of numerical experiments are presented.     

Approximating solutions for the systems of strongly accretive operator equations

The purpose of this paper is to study a strong convergence of multi-step iterative scheme to a common fixed point for a finite family of strongly pseudo-contractive mappings. As a consequence, the strong convergence theorems for the modified multi-step iterative sequence to a solution of systems of strongly accretive operator equations are obtained in real uniformly smooth Banach spaces. The results presented in this paper improve and extend the corresponding Xu [Y. Xu, Ishikawa and Mann iterative processes with errors for nonlinear strongly accretive operator equations, J. Math. Anal. Appl. 224 (1998) 91–101] and Noor, Rassias and Huang [M.A. Noor, T.M. Rassias, Z. Huang, Three-step iterations for nonlinear accretive operator equations, J. Math. Anal. Appl. 274 (2002) 59–68], and many others.     

Parallel Solutions for Large-Scale General Sparse Nonlinear Systems of Equations

In solving application problems,many large-scale nonlinear systems of equaions result in sparse Jacobian matrices.Such nonlinear systems are called sparse nonlinear systems.The irregularity of the locations of nonzrero elements of a general sparse matrix makes it very difficult to generally map sparse matrix computations to multiprocessors for parallel processing in a well balanced manner.To overcome this difficulty,we define a new storage scheme for general sparse matrices in this paper,With the new storage scheme,we develop parallel algorithms to solve large-scale general sparse systems of equations by interval Newton/Generalized bisection methods which reliably find all numerical solutions within a given domain.I n Section 1,we provide an introduction to the addressed problem and the interval Newton‘s methods.In Section 2,some currently used storage schemes for sparse systems are reviewed.In Section 3,new index schemes to store general sparse matrices are reported.In Section 4,we present a parallel algorithm to evaluate a general sparse Jacobian matrix.In Section 5,we present a parallel algorithm to solve the corresponding interval linear system by the all-row preconditioned scheme.Conclusions and future work are discussed in Section 6.     

Singular solutions in problems of optimal control

A particular class of optimization problems, in which the system equations and index of performance are linear in the control variable, is examined in detail. Pontryagin's Maximum Principle seems to indicate an optimal control of the bang-bang type for this class of problems. However, it is shown that the optimal control may actually consist of intervals of variable control effort (called "singular control") combined with intervals of bang-bang control. The conditions which characterize singular control are derived. Some techniques are given which may be helpful in detecting and calculating singular controls in this class of problems. In general, it cannot be stated a priori that singular control will necessarily constitute part of the optimal control. Two examples are worked out in detail to illustrate application of the techniques given in the paper.     

Numerical solutions for constrained time-delayed optimal control problems

In this paper, time-delayed optimal control problems governed by delayed differential equation are solved. Two different techniques based on integration and differentiation matrices are considered. The time-delayed term of the problem has been approximated by Chebyshev interpolating polynomials. On this basis, the optimal control problem can be solved as a mathematical programming problem. The example illustrates the robustness, accuracy and efficiency of the proposed numerical techniques.     

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.