Sufficient optimality conditions for controlling switched systems

The optimal control problem for a switched system whose state vector includes both continuous and discrete components is considered. The continuous part of the system is governed by differential equations, while its discrete part, which models the operation of a switching device with memory, is governed by recurrence inclusions. The discrete part switches the operation modes of the continuous part of the system, and is itself affected by the continuous part. The switching times and their number are not specified in advance. They are found as a result of optimizing a performance index. This problem is a generalization of a classical optimal control problem for continuous–discrete systems. It is shown that the cost function is constructed from certain auxiliary functions—the so-called conditional cost functions. Equations for the conditional cost functions are derived and sufficient optimality conditions are proved. The application of these conditions is demonstrated by the example of designing a time optimal switched system that is similar to A.A. Fel’dbaum’s classical example of the time optimal continuous system.     

Sufficient optimality conditions for discrete automaton-type systems

A discrete system that models the operation of a dynamic automaton with memory is considered. In distinction from the ordinary models of discrete systems, in which the states are changed (switched) at prescribed instants of time, automaton-type systems may change their states at arbitrary instants. Furthermore, the choice of the instants when the automaton “fires” is a control resource and it is subject to optimization. Sufficient optimality conditions for such systems under a limited or unlimited number of switchings are proved. Equations for the synthesis of the optimal closed-loop control are derived. Application of the optimality conditions is illustrated by examples.     

Sufficient conditions of optimality for stochastic systems withcontrollable diffusions

This paper studies optimal controls for systems governed by Ito's stochastic differential equations. Both the drift and diffusion terms of the equations are allowed to depend on controls, and the systems are allowed to be degenerate. It is shown that the necessary conditions of optimality, namely, the maximum conditions in terms of the “ℋ-function” (which is a generalization of the usual Hamiltonian and is quadratic with respect to the diffusion coefficients), along with some convexity conditions, constitute sufficient conditions of optimality for such controlled systems     

Sufficient optimality conditions for Pontryagin extremals

L₁-local optimality of a given control (·) in an optimal control problem for an affine control system with bounded controls is investigated. Starting from the Pontryagin Maximum Principle, which is a first-order necessary optimality condition, we develop it in two directions: (1) extending the notions of 1st and 2nd variations of the system along (·), we obtain 1st and 2nd-order sufficient optimality conditions for bang-bang Pontryagin extremals; (2) developing Legendre-Jacobi-Morse-type results for the extended second variation we obtain 2nd-order sufficient optimality conditions for general (bang-bang-singular) type of Pontryagin extremals.     

Sufficient optimality conditions for continuous-discrete systems with multiple instantaneous switchings of the discrete part

A continuous-discrete system in which the motion of the continuous part is described by differential equations and the behavior of the discrete part is described by recurrences is considered; multiple instantaneous switchings are allowed. Such systems are called hybrid, and they occupy an intermediate position between continuous-discrete systems with single switchings and logical-dynamical systems. Sufficient optimality conditions under constraints on the number of switchings of the discrete part are obtained. Equations for the synthesis of the optimal closed-loop control are derived. Application of the optimality conditions is illustrated by examples.     

Sufficient optimality conditions for the control of discrete determinate systems with bounded composition of precise measurements

Sufficient optimality conditions for control of discrete determinate systems with incomplete state-vector feedback are formulated and demonstrated, and also relations for desired control determination are obtained. Application of the obtained relations is demonstrated by examples.     

Necessary optimality conditions for switched systems

A dynamic system that models the operation of a switching device (switch) is considered. During the operation, the system changes its state a finite number of times. The change of state (switching) is described by a recurrent inclusion, which corresponds to the representation of the switch by a dynamic finite state machine with memory; instantaneous multiple switchings are admitted. The instants of time at which switchings are made and the number of switchings are not given in advance. They are found by optimizing a functional in which the number of switchings and the cost of each of them are taken into account. Necessary optimality conditions for such systems are proved. Different versions of the optimality conditions for different types of constraints are given. In particular, under additional convexity conditions, conditions that are similar to the maximum principle for discrete systems are obtained. The application of the optimality conditions is illustrated by examples.     

Sufficient conditions of optimality of the automaton part of the logical-dynamical system

Deterministic logical-dynamical systems are considered with the dynamical part described by differential equations. The logical part modeling the work of an automaton with memory is described by recursive inclusions. Sufficient optimality conditions are obtained for the construction of the logical (automaton) part of the logical-dynamical system.     

Irregular optimization models and p-order Kuhn-Tucker optimality conditions

A problem with irregular constraints in the calculus of variations is investigated. Optimality conditions for a degenerate case are obtained that make it possible to reject false solutions. The reasoning is conducted for the finite dimensional case; however, the main results are valid for Banach spaces.     

Necessary optimality conditions for control of logical-dynamical systems

The deterministic logical-dynamical systems with the dynamical part described by differential equations and the logical part, simulating the operation of automatons with memory, by recurrent inclusions are considered. The necessary conditions of optimal control are obtained. The equations for variations of functionals determined on trajectories of logical-dynamical systems and the equations for finding the optimal program control are obtained. The application of the optimality conditions is exemplified.     

Sufficient conditions for parameter convergence in linearizable systems

The major drawback of the adaptive compensation design in the previous paper (2002) in terms of parameter identification is that the criterion for parameter convergence cannot be checked in advance. The purpose of this paper aims at circumventing such difficulties by providing a priori verifiable sufficient conditions that guarantee the asymptotic stability of the parametric equilibrium.     

Sufficient conditions for instability of delay differential systems

The paper gives sufficient conditions for instability of linear systems of both the neutral and retarded types with commensurate and non-commensurate time delays. The conditions are given in terms of time delays and coefficients of the characteristic quasi-polynomials of these systems. Illustrative examples are presented.     

Sufficient conditions for stability of linear time-varying systems

In this paper we consider sufficient conditions for the exponential stability of linear time-varying systems of the form . Stability guaranteeing upper bounds for different measures of parameter variations are derived.     

High-order necessary conditions of optimality for nonlinear control systems

For nonlinear affine control systems with unbounded controls, necessary high-order optimality conditions are derived. These conditions are stated in the form of a high-order maximum principle and are expressed in terms of Lie brackets and Newton diagrams.     

On some optimality conditions and their equivalence in time-dependent inventory models

The paper considers a deterministic, time-and-review-continuous, finite horizon inventory model with general time-dependent demand, holding cost, shortage cost and replenishment rate functions. It gives conditions for the optimal ordering schedule, for a fixed number of replenishments, which reduce the effort for its finding to a simple one-dimensional search. The paper also proves equivalence relationships between the derived optimality conditions and the corresponding ones for special cases of the general model where some of its component functions tend to infinity. It concludes with a discussion concerning the uniqueness of the optimal number of replenishments, and the convergence of the optimal ordering schedule for an infinite time horizon.     

Multi-agent systems with reinforcement hierarchical neuro-fuzzy models

This paper introduces a new multi-agent model for intelligent agents, called reinforcement learning hierarchical neuro-fuzzy multi-agent system. This class of model uses a hierarchical partitioning of the input space with a reinforcement learning algorithm to overcome limitations of previous RL methods. The main contribution of the new system is to provide a flexible and generic model for multi-agent environments. The proposed generic model can be used in several applications, including competitive and cooperative problems, with the autonomous capacity to create fuzzy rules and expand their own rule structures, extracting knowledge from the direct interaction between the agents and the environment, without any use of supervised algorithms. The proposed model was tested in three different case studies, with promising results. The tests demonstrated that the developed system attained good capacity of convergence and coordination among the autonomous intelligent agents.     

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.     

Sufficient conditions for hyperstability of a class of nonlinear systems

The concept of hyperstability is applied to obtain a new stability criterion for a class of nonlinear systems. The physical system considered is the chemical reactor which gives rise to a very interesting aspect of stability study.     

Synthesis of logical-dynamical systems on the basis of sufficient optimality conditions

A dynamical system controlled by an automaton with memory is considered. A continuous part of the systems is described by differential equations, and a logical (automaton) part, by recurrence equations. Moments when the states of the automaton part are changed are a priori not known and determined in the course of optimization. Modes with multiple instantaneous switchings of the automaton part are assumed to be possible. Based on sufficient optimality conditions, a technique for synthesizing suboptimal positional control is developed. The use of this technique is illustrated by an example.