On conditions of optimality for singular control problems

Satisfaction of the generalized Legendre-Clebsch condition is known to be a necessary condition of optimality in singular control problems. Recently, a new additional necessary condition of optimality was discovered. In this note, we demonstrate by means of an example that together these two necessary conditions are, in general, insufficient for optimality in singular control problems.     

Global optimal feedback control for general nonlinear systems with nonquadratic performance criteria

Optimal control of general nonlinear nonaffine controlled systems with nonquadratic performance criteria (that permit state- and control-dependent time-varying weighting parameters), is solved classically using a sequence of linear- quadratic and time-varying problems. The proposed method introduces an “approximating sequence of Riccati equations” (ASRE) to explicitly construct nonlinear time-varying optimal state-feedback controllers for such nonlinear systems. Under very mild conditions of local Lipschitz continuity, the sequences converge (globally) to nonlinear optimal stabilizing feedback controls. The computational simplicity and effectiveness of the ASRE algorithm is an appealing alternative to the tedious and laborious task of solving the Hamilton–Jacobi–Bellman partial differential equation. So the optimality of the ASRE control is studied by considering the original nonlinear-nonquadratic optimization problem and the corresponding necessary conditions for optimality, derived from Pontryagin's maximum principle. Global optimal stabilizing state-feedback control laws are then constructed. This is compared with the optimality of the ASRE control by considering a nonlinear fighter aircraft control system, which is nonaffine in the control. Numerical simulations are used to illustrate the application of the ASRE methodology, which demonstrate its superior performance and optimality.     

Conditions for optimality and transition surfaces using the reprisal concept†

The concept of optimality in zero-sum, two-player differential games introduced by the authors in a previous paper is studied. Furthermore, a number of tools to aid in the study of differential games in which there exist ‘ non-terminating ’ strategies are developed.

In the all-terminating case it is shown that for C^(z)) strategies or for strategies containing a specified class of discontinuities, satisfying Isaacs' equation is both necessary and sufficient for optimality. In linear, all-terminating games, easily verifiable necessary and sufficient conditions are given and it is shown that, subject to one condition, points of transition surfaces must necessarily be in the zero-set of one of two switching functions.

In ‘ non-terminating ’ games, satisfying Isaacs' equation is necessary for optimality but no longer sufficient. Sufficiency conditions for optimality in these games are then proven along with three alternative necessary conditions which points of transition surfaces must satisfy. An example illustrating the application of these results is included together with a heuristic discussion of the ‘ double transition surface ’ phenomenon.     

Sulla ottimizzazione dei sistemi discreti

As an extension of the results of the first part of this work, necessary conditions are derived for the optimality of discrete nonlinear systems with constraint sets on the state and control variables which are «locally approximable by convex cones». The conditions are deduced geometrically from general properties of such cones and their dual cones in relation to non linear mappings an intersections of sets.     

Generalized maximum principle for optimal control of generalized state-space systems

This paper is mainly concerned with the derivation of the necessary conditions, called the ‘generalized maximum principle’, for the optimal control of generalized state-space systems with a more general form. By making use of a method of the modern calculus of variations which has been used for the proof of Pontryagin's maximum principle, the generalized maximum principle is derived, and some problems related to this principle are discussed in detail. In addition, an illustrative example is given in the light of this principle.     

First-order and second-order necessary conditions for optimality of singular controls

The derivation of necessary conditions for the optimality of singular controls (in the sense of Pontryagin's maximum principle) in optimal control problems is inves- tigated. A new class of the necessary conditions is developed. This class of necessary conditions is composed of two parts, the first-order and second-order necessary conditions. As an application of these necessary conditions, an illustrative example is given, which shows that the necessary conditions developed are stronger than the generalized Legendre-Clebsch condition. That is, the singular control in the example satisfies the generalized Legendre-Clebsch condition, but does not satisfy the necessary conditions developed in this paper.     

Discrete‐time low‐gain control of linear systems with input/output nonlinearities

Discrete‐time low‐gain control strategies are presented for tracking of constant reference signals for finite‐dimensional, discrete‐time, power‐stable, single‐input, single‐output, linear systems subject to a globally Lipschitz, non‐decreasing input nonlinearity and a locally Lipschitz, non‐decreasing, affinely sector‐bounded output nonlinearity (the conditions on the output nonlinearities may be relaxed if the input nonlinearity is bounded). Both non‐adaptive and adaptive gain sequences are considered. In particular, it is shown that applying error feedback using a discrete‐time ‘integral’ controller ensures asymptotic tracking of constant reference signals, provided that (a) the steady‐state gain of the linear part of the plant is positive, (b) the positive gain sequence is ultimately sufficiently small and (c) the reference value is feasible in a very natural sense. The classes of input and output nonlinearities under consideration contain standard nonlinearities important in control engineering such as saturation and deadzone. The discrete‐time results are applied in the development of sampled‐data low‐gain control strategies for finite‐dimensional, continuous‐ time, exponentially stable, linear systems with input and output nonlinearities. Copyright © 2001 John Wiley & Sons, Ltd.     

Slope Intervals,Generalized Gradients,Semigradients, Slant Derivatives,and Csets

Many practical optimization problems are nonsmooth, and derivative-type methods cannot be applied. To overcome this difficulty, there are different concepts to replace the derivative of a function f : : interval slopes, semigradients, generalized gradients, and slant derivatives are some examples. These approaches generalize the success of convex analysis, and are effective in optimization. However, with the exception of interval slopes, it is not clear how to automatically compute these; having a general analogue to the chain rule, interval slopes can be computed with automatic differentiation techniques. In this paper we study the relationships among these approaches for nonsmooth Lipschitz optimization problems in finite dimensional Euclidean spaces. Inclusion theorems concerning the equivalence of these concepts when there exist one sided derivatives in one dimension and in multidimensional cases are proved separately. Valid enclosures are produced. Under containment set (cset) theory, for instance, the cset of the gradient of a locally Lipschitz function f near x is included in its generalized gradient.     

On an Optimization Problem for a Class of Impulsive Hybrid Systems

This contribution addresses the problem of optimal control for a class of hybrid systems, where discrete transitions are accompanied by instantaneous changes in the continuous state variables, and where these changes can be considered as control variables. Based on a variational approach, necessary conditions of optimality are first established. The problem is then cast as a parametric optimization problem for which gradient information is derived. Finally, we discuss assumptions that guarantee convergence of a conceptual algorithm to a stationary solution. A brief discussion on the main implementation issues is also included.     

Necessary and sufficient optimality conditions for discrete time-invariant automaton-type systems

The optimal control of deterministic discrete time-invariant automaton-type systems is considered. Changes in the system’s state are governed by a recurrence equation. The switching times and their order are not specified in advance. They are found by optimizing a functional that takes into account the cost of each switching. This problem is a generalization of the classical optimal control problem for discrete time-invariant systems. It is proved that, in the time-invariant case, switchings of the optimal trajectory (may be multiple instantaneous switchings) are possible only at the initial and (or) terminal points in time. This fact is used in the derivation of equations for finding the value (Hamilton–Jacobi–Bellman) function and its generators. The necessary and sufficient optimality conditions are proved. It is shown that the generators of the value function in linear–quadratic problems are quadratic, and the value function itself is piecewise quadratic. Algorithms for the synthesis of the optimal closed-loop control are developed. The application of the optimality conditions is demonstrated by examples.     

An approach to a singular optimal control problem for non-linear systems using differential form theory

By applying differential form theory, we consider the singular control problem for non-linear systems with control variables appearing linearly in both the system dynamics and the performance index. First, we derive necessary conditions of singular optimality for a single-input system, including the relation to the Euler-Poisson equation and to the generalized Legendre-Clebsch condition. Defining the degree of singularity, we develop necessary conditions satisfied by the singular trajectory embedded in a reduced space. For a time-invariant system, we clarify the relation between the dynamic and the related static optimality. Second, we derive necessary conditions for singular optimality for a multi-input system where the dimension of the control vector is equal to that of the state space. We show that the Shima-Sawaragi condition for the optimality of boundary controls and the generalized Legendre-Clebsch condition are obtained from these conditions. The results are also applied to the analysis of a time-invariant system.     

Classical necessary optimality conditions in discrete optimal control problems with nonlocal conditions

We consider an optimal control problem in which the system’s state is described by a system of difference equations with nonlocal (two-point) conditions; this problem includes, as particular cases, the initial value problem (Cauchy problem) and different types of boundary value problems. It is assumed that the admissible controls take values from an open set. The first and second functional variations are calculated; these variations are used to express first and second order necessary optimality conditions in the classical sense for discrete optimal control problems.     

一类脉冲切换系统的最优控制

研究一类脉冲依赖于状态的脉冲切换系统的最优控制问题. 考虑了目标函数的两种情况: 当目标函数光滑时, 通过将跳跃瞬间转化为一个新的待优化参数, 得到了该脉冲切换系统的必要最优性条件; 当目标函数不光滑时, 利用非光滑分析的知识, 得到了广义微分形式的必要最优性条件. 算例分析验证了所提出方法的有效性.     

A sufficiency condition for discrete time optimal processes†

Sufficient conditions for optimality of a class of linear discrete time processes are presented. In distinction to previously obtained necessary conditions for optimality, these sufficient conditions hold under very weak hypotheses. An example, which might arise in a sampled data system, is presented. For the example the sufficient conditions yield the optimal control, but the necessary conditions are not applicable since the required hypotheses are not satisfied.     

Nonsmooth continuous-time optimization problems: necessary conditions

We consider Lipschitz continuous-time nonlinear optimization problems and provide first-order necessary optimality conditions of both Fritz John and Karush-Kuhn-Tucker types.     

A heuristic approach to necessary conditions of optimality

A simple, direct method is presented for deriving necessary conditions of optimality covering most of the situations encountered in practice. The method is based on the observation that, if generalized functions are admitted, a broad class of optimal control problems can be transcribed into a canonical form called the Canonical Variational Problem (CVP). The necessary optimality condition for CVP is phrased in the form of a Lagrange multiplier rule. This condition contains many of the results relating to minimum principles in the optimal control literature.     

Solution of Nonconvex Nonsmooth Stochastic Optimization Problems

Different classes of nonconvex nonsmooth stochastic optimization problems are analyzed, their generalized differentiability properties and necessary optimality conditions are studied, and a technique for calculating stochastic gradients is developed. For each class of the problems, corresponding solution methods are proposed, in particular, generalizations of the stochastic quasigradient method.     

Multiple eigenvalues in structural optimization problems

This paper discusses characteristic features and inherent difficulties pertaining to the lack of usual differentiability properties in problems of sensitivity analysis and optimum structural design with respect to multiple eigenvalues. Computational aspects are illustrated via a number of examples.Based on a mathematical perturbation technique, a general multiparameter framework is developed for computation of design sensitivities of simple as well as multiple eigenvalues of complex structures. The method is exemplified by computation of changes of simple and multiple natural transverse vibration frequencies subject to changes of different design parameters of finite element modelled, stiffener reinforced thin elastic plates.Problems of optimization are formulated as the maximization of the smallest (simple or multiple) eigenvalue subject to a global constraint of e.g. given total volume of material of the structure, and necessary optimality conditions are derived for an arbitrary degree of multiplicity of the smallest eigenvalue. The necessary optimality conditions express (i) linear dependence of a set of generalized gradient vectors of the multiple eigenvalue and the gradient vector of the constraint, and (ii) positive semi-definiteness of a matrix of the coefficients of the linear combination.It is shown in the paper that the optimality condition (i) can be directly applied for the development of an efficient, iterative numerical method for the optimization of structural eigenvalues of arbitrary multiplicity, and that the satisfaction of the necessary optimality condition (ii) can be readily checked when the method has converged. Application of the method is illustrated by simple, multiparameter examples of optimizing single and bimodal buckling loads of columns on elastic foundations.Dedicated to the memory of Ernest F. MasurGuest professor during the period 16 November to 11 December, 1992 and 15 November to 12 December, 1993.     

Free lunches on the discrete Lipschitz class

The No-Free-Lunch theorem states that there does not exist a genuine general-purpose optimizer because all algorithms have the identical performance on average over all functions. However, such a result does not imply that search heuristics or optimization algorithms are futile if we are more cautious with the applicability of these methods and the search space. In this paper, within the No-Free-Lunch framework, we firstly introduce the discrete Lipschitz class by transferring the Lipschitz functions, i.e., functions with bounded slope, as a measure to fulfill the notion of continuity in discrete functions. We then investigate the properties of the discrete Lipschitz class, generalize an algorithm called subthreshold-seeker for optimization, and show that the generalized subthreshold-seeker outperforms random search on this class. Finally, we propose a tractable sampling-test scheme to empirically demonstrate the superiority of the generalized subthreshold-seeker under practical configurations. This study concludes that there exist algorithms outperforming random search on the discrete Lipschitz class in both theoretical and practical aspects and indicates that the effectiveness of search heuristics may not be universal but still general in some broad sense.