Degenerate problems of optimal control. I

Considered is the practically important and theoretically challenging class of optimal control problems which can be integrated within the common notion of “degenerate problems.” The general definition of such problems is given, which arises from the connection between the degeneracy and the presence of hidden passive differential constraints or discrete chains in the problem. This definition is analyzed with the focus on its relation with the classical notion of degeneracy in the variational calculus and the notion of singular and sliding modes well known in the control theory. This paper is the first one in the series of three, which are aimed at presenting a survey of the main facts and applications of the special theory of such problems, which is essentially based on finding and eliminating passive constraints. New results and generalizations are also reported.     

Degenerate problems of optimal control. II

The paper is the second part of the review of the results obtained by the special theory of the degenerate problems of optimal control and its applications, including the newest ones. Consideration was given to the methods based on the rearrangement of the degenerate problems in the smaller-order regular derivative problems, the main types of solutions of the derivative problems, and the issues of their realization as the generalized solutions of the original problems. The generalized conditions for their optimality were derived. Examples, both methodological and applied, were presented to demonstrate the efficiency of these methods.     

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.     

Perturbed optimal control problems

We examine a family of problems obtained by perturbing infinite-dimensionally the dynamics of an optimal control problem. A formula is derived for the generalized gradient of the associated value function, one which specializes to yield, for instance, information about ordinary directional derivatives. Several examples are discussed.     

Hybridization of optimal control problems

Optimization problems for hybrid systems have attracted a lot of attention in recent years. This interest stimulated numerous scientific works and the development of tools to study optimal trajectories, such as necessary conditions. The idea of the present contribution is to use such results for hybrid systems to construct a hybridization of an optimal control problem. The approximation via a hybrid system was already considered in the literature, but using different methods to obtain a reduction to minimal path problems on a graph. Our procedure gives an efficient approximation, i.e. more efficient than standard discretization.     

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.     

On dual problems of optimal control

The common structure of duality of optimal control problems is suggested, which is based on the extension principle. The iterative computational algorithm is described, in which the primal and the dual problem are solved in parallel.     

Stability of elliptic optimal control problems

We consider optimal problems governed by elliptical equations on parameters and give sufficient conditions for the continuous dependence of the solutions on the parameters. The techniques are based on variational methods.     

Equivalent transformation of optimal control problems

The equivalent transformation of the optimal control problems is proposed. The equivalent transformation is the expanded canonical transformation involving the transformation of the control. The generating function plays an important role in the equivalent transformation. The quadratic generating function is discussed. In this case the equivalent transformation is linear.     

Butterworth最优控制的逆问题

为了认识Butterworth最优控制的本质,揭开Butterworth最优传递函数与加权矩阵Q,R的相互关系,本文研究Butterworth最优控制的逆问题.首先用Butterworth最优控制确定状态反馈增益阵K,然后给出计算加权矩阵Q的参数化公式,最后用一个例子说明这种确定加权矩阵Q,R的方法的有效性和简便性.     

Explicitly solvable non-linear optimal control problems

The method of collective hamiltonians is applied to the study of optimal control problems on manifolds. This approach enables us to single out exactly solvable situations. Various examples are considered. Necessary results from hamiltonian formalism are given, including an ‘industrial’ technique for the construction of completely integrable systems.     

Numerical treatment of multiobjective optimal control problems

We present a numerical procedure for solving optimal control problems with both linear terminal constraints and multiple criteria. Using a Chebyshev spectral procedure, the problem reduces to a constrained optimization problem which can be solved using hybrid penalty partial quadratic interpolation (HPPQI) technique. The proposed procedure compares quite favorably with other methods on a sample of well-known examples.     

Trigonometric approximation of optimal periodic control problems

Trigonometric polynomials are used to approximate the state trajectory, the control and the functions in differential equations and in the criterion of the constrained optimal periodic control problem. Several discretized problems using such polynomials are proposed to approximate the problem considered. Sufficient conditions for the convergence of solutions of approximating problems to the optimal solution of the basic problem are given. The application to a class of optimal periodic control problems in chemical engineering is discussed.     

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.     

On transformations of degenerate optimal control problems

We consider practical algorithms for transforming a system with unbounded controls to a new model regular from the point of view of known general optimal control methods. In addition to a transformation to a certain derivative system on the entire time interval, which is traditional for degenerate problems, we consider a family of such transformations, choosing a specific one at every moment of time. As a result we get a controllable system which is in a certain sense equivalent to the original one, and one can operate with it like a regular system. Its solution is realized in the original class as an impulse sliding mode.     

Difference approximation of optimal periodic control problems

The constrained optimal periodic control problem is approximated by a sequence of discretized problems in which the system of differential equations of the basic continuous problem is replaced by a system of one–step difference equations. Two kinds of approximate optimal controls are derived from the optimal solutions of discretized problems: the first in the form of a step function and the second in the form of a special trigonometric polynomial generated by a positive kernel. Sufficient conditions for approximate solutions to be weakly convergent to the optimal solution of the basic problem are given. Certain improvements in the difference approximation considered are discussed and potential applications given.     

Algebraic approximation of optimal cyclic control problems

Algebraic polynomials are used to approximate the state trajectory, the control and the functions in differential equations and in the criterion for the constrained optimal cyclic control problem. Several discretized problems using algebraic polynomials are proposed to approximate the basic control problem. Sufficient conditions for the convergence of solutions of approximating problems to the optimal solution of the basic problem are given and the convergence rate is estimated. The application to a class of industrial optimal cyclic control problems is discussed.     

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.