About conditions of controllability of ensamble trajectories of differential inclusion with delay

Introduction

In real situations, effective management of complex technical objects and processes is impossible without taking into account factors such as incomplete information on external parameters and inaccuracy of the initial a priori data on the initial state of the system. The control problems in conditions of limited information of various types lead to the so-called information models of control systems. Information models for dynamical systems described by ordinary differential equations can be represented in a general form in the form

is the parameter of unknown external influences. Known methods for studying a model of the form (1) are developed taking into account the degree of limited information regarding external parameters and the initial state of the system.

When the probability distribution functions of the parameter ) (t w w  and the initial data are known, system (1) is studied by methods of the theory of stochastic control. However, this assumption cannot always be considered admissible in the control process, and in addition, the use of probabilistic-statistical characteristics does not always guarantee the desired quality of system control. In applied problems, a more common situation is when a priori data on the initial state of the system and the parameters of external influences are minimal, i.e. there are no statistical descriptions of them, and information is limited to setting only the set of possible values of unknown parameters. In such cases, we can say that the model under consideration (1) represents a control system in conditions of uncertainty. It is well known from the theory of differential inclusions and multi-valued analysis [3,15] that under fairly general assumptions, control system (1) with indefinite parameters

The resulting mathematical model belongs to the class of so-called controlled differential inclusions [6,9,16]

For control systems under conditions of uncertainty, the properties of the ensemble of trajectories, methods for estimating the reachability set and forecasting the phase state of the system and others are of great interest [2,4,5,16] . Depending on the criterion for assessing the state of the system, various problems of optimal control of the ensemble of trajectories are studied: control by speed, minimax, and other criteria [4,5,7,8,13,16] . The control problem of ensemble of trajectories of differential inclusions can have various statements. The control problem f or such systems can be posed, for example, as the problem of the complete or partial immersion of the end points of all possible trajectories of a system on a given terminal set. Some statements of controllability problems for differential inclusions, understood as controllability of an ensemble of trajectories with respect to given initial and terminal states, were considered in [6,9,16] . For such systems, the controllability conditions and some properties of the set of controllability points with respect to the ter minal set are studied, in particular, its topological structure is studied. The delay factor has a significant impact on the dynamics of the system and, therefore, the final result of research on control and optimization problems. This is also confirmed by studies on controlled differential inclusions with delays. Some properties of controlled differential inclusions with delays were studied in [10,11] . In particular, the conditions of compactness, convexity of the set of absolutely continuous solutions, as well as their dependence on the control parameter and the initial state are revealed. The control problems for differential inclusions with delays are characterized by a feature related to the specifics of the problem of controlling the ensemble of trajectories and the delay factor. Separate results on the control of the ensemble of trajecto ries for differential inclusions with delay were obtained in [10] . In the case of linearity of such systems, the some controllability conditions of the ensemble of trajectories are studied. In [12,14,17] , for such systems, the minimax optimal control problem for the ensemble of trajectories, and in [17] the nonsmooth problem with terminal constraints was considered.

Statement of the problem

Consider a differential inclusion with delay of the form

Here, the parameter u plays the role of control actions.

By an admissible control for system (3) we mean a measurable bounded m -vector function

We consider the controllability problem for controlled differential inclusion with delay (3) in the sense of complete immersion of the ensemble of trajectories on a given terminal set. More precisely, assuming given a convex closed set of terminal states

Problem: it is required to find out the conditions under which there is an admissible control  

consisting of all points of the space of phase states

is called the ensemble of trajectories of the system (3) -( 4). Definition 1. We say that the ensemble of trajectories of system ( 3) is controllable from the initial state  

such that the corresponding ensemble of trajectories satisfies the boundary condition.

 

Definition 2. An ensemble of trajectories of system (3) is called completely controllable into a terminal set

-the controllability of the system means the solvability of the boundary value problem ( 3) -( 4) in the class of admissible controls

. Clarification of the conditions for the solvability of this problem is the main goal of the study provided for in this paper. In addition, we will study the conditions for the complete Y-controllability of the ensemble of trajectories of the system under consideration.

Research methods Auxiliary Results

It is clear from the above definitions that for the controllability of the ensemble of trajectories of system (3), the conditions for the compactness and convexity of the set 3), we impose some conditions that must ensure that these requirements are met.

From the results of [11] it easily follows Lemma 1. Let Assumption 1 hold Then:

is the space of square summable functions. Let the right-hand side of differential inclusion (3) have the form:

Those consider the following linear model:

Under the conditions of Assumption 2, the following representation of the ensemble of trajectories of system ( 7) is valid through its parameters:

Lemma 2. Let Assumption 2 be satisfied, and for any

(not necessarily convex). Then all the statements of Lemma 1 are true and, moreover, the support function of the form (6) satisfies the conditions of Assumption 1. Therefore, for the control system (7), all the statements of Lemma 1 remain val id. Using the representation of the ensemble of trajectories of the linear system (7), and taking into account the properties of the integral of multi-valued mappings, we verify that the convexity property of each set

without requiring the convexity of the values of the multi-valued mapping     u t b u t , ,  . Further, using formula (8) and the properties of the support functions, we have:

From this formula for the support function of the set  

The main results

According

Therefore, by virtue of Lemma 2 and the results of [6] , the following statement is true. 3) it is necessary, and for linear system (7) it is necessary and sufficient that there exists

Where f conc is the concave closure [2] of the function f

The above theorem gives an implicit criterion for controllability of the ensemble of trajectories in the form of relation (10 ).

Although it is complicated by the sequence of three operations:

, Theorem 1 will be the theoretical basis for the subsequent results we obtained on controllability conditions. Using formula (9) for the support function of the set

and using the properties of the operation of transition to concave closures, from Theorem 1 we obtain the following result.

Theorem 2. For   Y , 0  -controllability of system (7), it is necessary and sufficient that the relation

Consider another special case of a system with delay (3).

) (t Q -convex closed and bounded subsets n R ;

3) The multi-valued mapping

Under Assumption 3, all conditions of Assumption 2 are satisfied. Therefore, since

Then from Theorem 2 it follows that controllability criterion for the ensemble of trajectories of system (11) have the form:

We introduce the notation:

Therefore, condition (13) can be written as follows:

Thus, we have obtained the following controllability criterion for the ensemble of trajectories of system (11)

Then system (11) is completely Ycontrollable.

Proof. According to Theorem 3, it is enough for us to show that relations (14) holds for each initial function  

, and for some 0 1 t t  . We have: 16) we obtain (14). And this completes the proof of the theorem.

Discussion of results and conclusion

The problem of controllability is one of the important problems in the theory of optimal control. Here, for a controlled differential inclusion with delay, the controllability problem was considered as the problem of the complete immersion of an ensemble of trajectories on a given terminal set. The goal of the study was set: clarification of the controllability conditions expressed in terms of the parameters of the system under consideration. When studying the controllability ensemble trajectory control problem, the main conditions on the right-hand side of the considered differential inclusions are given in the form of assumptions 1-3. They cover a wide class of differential inclusions with delay (3), as well as their linear models of the form (7) and (11). Of the results obtained, the most general is Theorem 1. Although it is an implicit controllability criterion for the systems under consideration, the remaining results were obtained based on this theorem. In this case, the properties of the ensemble of trajectories of linear systems were taken into account. Among the important characteristics of the ensemble of trajectories of differential inclusions used in the subject under study, it is worth noting the properties given in the auxiliary lemmas. Theorem 2 gives necessary and sufficient conditions for   Y , 0  -controllability of the ensemble of trajectories linear in state of system (7). And in Theorem 3 they are refined for a linear system, both in state and in control. Here, formula (8) of the representation of the ensemble of trajectories and formula (9) of its support function are essentially used. We note that a feature of the linear structure of systems found their expression in the study of the controllability problem, as evidenced by the results of Theorem 4. The studied controllability property of the ensemble of trajectories of differential inclusions is of immediate interest for systems of the form. then the condition of non-degeneracy of the governing equations [1] of the system

is sufficient to satisfy condition (15). Therefore, this condition will be sufficient for controllability of the linear system (17)

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