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.

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

Optimal control problem for logic-dynamic systems is considered. Based on sufficient optimality conditions, similar to Krotov's conditions, a method for successive control improvement is developed. First- and second-order computational algorithms are constructed.

Iterative methods for solution of problems of optimal control of logic-dynamic systems

The problem of optimal control of automatic deterministic discrete systems in conditions of parametric uncertainty is considered. The state change (switchover) of a system is described by a recurrent equation in which instant multiple switchovers are admitted. The times and the number of switchovers are not specified in advance--they are found as a result of optimization of a functional. The initial state of the system is not known exactly, but the set of possible initial states is known; therefore, problems of average optimal control and guaranteed optimal control of bunches of trajectories are formulated. For solving these problems, it is suggested to use the principle of separation: namely, to control the bunch of trajectories by applying a control optimal for one specially chosen trajectory of the bunch. The resulting control of the bunch is suboptimal, but it may be satisfactory in practice. On the basis of sufficient conditions for optimality of a full-feedback control, an algorithm for constructing a suboptimal control in conditions of uncertainty is developed. Counterexamples of linear---quadratic problems of control of automaton-type systems in which the principle of separation fails are presented.

Optimal and suboptimal control over bunches of trajectories of automaton-type deterministic systems

A dynamical system simulating the operation of a switching device (switch) is considered. When functioning, the system changes its state a finite number of times. The change in the state (switching) is described by a recurrent activation when the switch is represented as a dynamic automaton with memory and instantaneous multiple switchings. The times at which switchings occur and their number are not prescribed. These values are determined from the optimization of a functional that calculates the number and cost of switchings. We have proved the sufficient conditions of the optimality of these systems and developed a method for synthesizing optimum switches. This method builds a family of auxiliary functions (conditional cost functions and conditional positional controls) that serve to form a cost function (a Hamilton--Jacobi--Bellman function) and an optimal scheme of the switch. The application of this method is illustrated by some examples.

Synthesis of optimal switched 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 optimality conditions for discrete automaton-type systems

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.

A. S. Bortakovskii

Papers

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