To the problem of optimal allocation for physical control resources for separate dynamic systems

I have in this paper studied the problem of optimal allocation of the control resources between the separate control objects limited by the flow rate. As a resource needed to create control actions I consider power – in the form of instantaneous power – and material – in the form of the consumption rate of “fuel”– control resources. Examples are given of solving this problem on a fixed interval for a system with control objects – double integrators, local control objectives for which are given by boundary conditions of a general form. Based on the results of solving these problems, the main features of the optimal allocation of the control resource are revealed depending on its type and Ihave noted that the considered control problems can be attributed to the class of mixed problems by A.M. Letov.


Introduction
In [1] it is noted that the solution of the problems of creating effective control systems for various objects and control processes is possible with maximum use of physical models and variables, and also taking into account real physical limitations (energy, information, material and others) that play an important role in control tasks, when it becomes necessary to distribute any "control resources". Another such example is the so-called mixed optimal control problems [2] (see pp.193-194), in which a strict separation of control resources was assumed for solving independent variational problems for the same object (for example, in the problem of the landing of a spacecraft [2]), which has a strongwilled character and does not correspond to the spirit of two independent problems: the problem of programming optimal trajectories and the problem of synthesis of feedback laws. In this regard, the purpose of this article is to formulate the problem of distributing a physical control resource of a certain type between separate (independent) control objects that use this resource to create control actions while simultaneously solving their partial control tasks. As control resources, there we will considered power and material resources, the consumption rate of which is limited: in the first case it is the current capacity of the source of the power resource (spent to create control actions), and in the second casethe consumption rate of the material resource in the form of a certain "fuel", the consumption rate of which is proportional to the managemnet actions created on account of its costs to the control object.

Formulation of the problem
Let us consider a two-point boundary problem for a -th k control object ( 1, 2, ... , k N , 2 N t ) ( , ) k k k k d dt , with boundary conditions of general form as target conditions -th k of the partial control problem: where 0 k x , k f xinitial and final states -th k of the control object (1). The control parameters in (1) are constrained: where k Ucompact set. Constraints (3) in applied problems in model (1), as a rule, are determined by restrictions on the permissible values of control actions on the control object. In problem -th k (1) -(3) we are required to find any permissible control program ( ) which ensures the fulfillment of boundary conditions (2), that is, the corresponding control objective is achieved. If the solution of problem (1) -(3) is not unique, then it can be reformulated as an optimal control problem, in which it is required to minimize some functional ( , ) k k k k J J x u , which can also serve as a control objective, but more general in comparison with the local goal (2).The functional k J , as a rule, is chosen proceeding from the requirement of minimization of total expenditures of any physical resources, the costs of which are related to the need to create control actions in solving problem (1) -(3). If at any point in time the formation ( ) In applied control tasks, it is necessary to take into account not only constraints (3), but also restrictions on the consumable control resource: where 0 U fthe maximum permissible rates of its consumption. If ( ) . In addition, it should be specially noted that in (3) and in the case of k k u U the value of the control parameters can have different physical meaning, that is, their values will coincide only to within dimensionality.
Consider a set of separate control objects (1) that form a system with a single source of physical resource necessary for solving partial control problems (1) -(3), 1, 2, ... , k N . But then instead of (4) it is necessary to introduce the following joint constraint: Local control objectives for each of the partial tasks (1) -(3) can be supplemented with quality indicators k J , 1,2,..., k N , that is, for the resulting system, you can enter the general control objective in the form where 0 k D t , 1,2,..., k N ,some weight factors. If, with the help of k J the total control resource consumption for the kpartial problem solution, the coefficients k D in (6) can be considered as priorities in resource consumption by k control objects when solving the entire set of partial problems (1) -(3).
Thus, the optimal control problem (1) -(3), (5), (6) differs significantly from the problems (1) -(3) ( 1,2,..., k N ) by virtue of constraint (5), which, taking into account (6), necessitates an optimal resource allocation between separate objects (1) under the condition of coordinated and simultaneous solution of corresponding partial control tasks for them. Problems of this type belong to the class of 3 resource systems, which can also be attributed to the corresponding problems of both the physical control theory [1] and the theory of optimal control [2].
In the view of the sufficient generality of the above formulation of the problem (1) -(3), (5), (6) to obtain meaningful results of its solution, revealing the features of the optimal allocation of the control resource, depending on its type, it is advisable to specify its elements accordingly, namely: models of control objects (1); the partial problems solved for them, that is, taking into account (2) and taking into account the restrictions on the sets of admissible control actions. Therefore, further, the solution of problem (1) -(3), (5), (6) will be considered below only for control objects of the simplest kind.

The problem of optimal resource allocation in the system of double integrators
Let us consider a model version of the formulation of problem (1) -(3), (5), (6) for a given interval [0, ] T and 2 N by choosing double integrators as control objects for which local control objectives are determined by boundary conditions of a general form. Accordingly, the integral quality indicators for solving partial problems are selected depending on the type of consumable control resource. If it is required to minimize the total consumption of "control energy", then such an indicator has the form where the value 2 ( ) / 2 u t is proportional to the current power consumption required to create a control action ( ) u t . If it is required to minimize the consumption of any material resource in the form of "fuel", instead of (7), we should consider the functional where | ( )| u t instantaneous "fuel" consumption rate. Now, let us consider the optimal control problem for the following system: in which the control parameters of the corresponding subsystems are constrained: where 1 m and 2 mthe maximum possible levels of control actions, and the boundary conditions are given in general form: where 10 x , 20 x , 1 f x , 2 f x , 10 y , 20 y , 1 f y , 2 f yare the set parameters.
Taking into account (7), for the problem (9) -(11) it is required to minimize the functional where 1 0 D t , 2 0 D t . Limitations (10) specify the following sets of admissible control actions: Ou u with the center at its beginning.
In accordance with the statement of problem (1) -(3), (5), (6), we note that the control parameters 1 u and 2 u in both (9) and (10) correspond to control actions on control objects, and in (7) and (8) (10) or on the consumption rate of the control resource of the type (5) can be significant. Therefore, taking (5) and (7) into account, we introduce one more restriction for control parameters, namely: where 0 Epower of control resource source, а 1 1 K t , 2 1 K tcoefficients that take into account the non-productive costs of the resource in the formation of control actions, for example, due to losses in the executive bodies. The restriction (13) is set by: Taking into account (10) and (13) (10), (13) are substantial or effective. Thus, we can formulate the following problem of optimal allocation of the control energy resource: for a two-point boundary problem (9), (11), find optimal control programs 1 ( ) u t 1 ( ) u 1 ( and 2 ( ) u t 2 ( ) u 2 ( , that satisfy constraints (10), (13) and provide a minimum to the functional (12).
In the case when constraint (5) is taken into account together with exponent (8) in problem (9) -(11), it is necessary to minimize the functional where 1 0 D t , 2 0 D t , and for the control parameters in (9) we introduce the joint constraint in the form: where 0 Mmaximum rate of resource consumption, and 1 1 K t , 2 1 K t . According to (10), the set of admissible control actions 12 1 2 U U U 2 U corresponds to the control parameters in (9), and in (15) they characterize the resource consumption rates, that is, the restrictions (10) and (15) here have a different physical meaning. In the plane 1 2 Ou u the restriction (15) corresponds to the following set: Obviously, M U is a diamond-shaped region in the plane with the center at the beginning and with the semi-diagonals combined with the coordinate axes and equal to 0 1 / M K and 0 2 / M K . As in the case of constraints (10) and (13), there is also a mutual effectiveness (or inefficiency) of constraints (10) and (15), and in the most general case they will be effective if 12 ) / ( ) / ( 12 z ) / ( ) / ( 12 ) / ( 12 . So, we can formulate one more problem of optimal control resource allocation, which is represented by a material resource (of the "fuel" type), namely: for a two-point boundary problem (9), (11) find optimal control programs 1 ( ) u t 1 ( ) u 1 ( and 2 ( ) u t 2 ( ) u 2 ( that satisfy constraints (10), (15) and give a minimum to the functional (14).

To the solution of the problem of optimal allocation of the power resource control
Let us consider the problem of the optimal allocation of the power resource control (9) -(13). First we will assume that the constraints (10) and (13 are ineffective. When applying L.S. Pontryagin's maximum principle [3,4], we are able to record the Hamiltonian for this problem 2 2 1 1 2 2 1 2 2 1 3 2 4 2 1 2 where the conjugate variables k \ , 1, 2,3, 4 k , satisfying the system of equations From the maximum condition H (16) by 1 u and 2 u we will receive the programof optimal control where 2 ( ) t \ and 4 ( ) t \ are the equation solutions (17) with initial values 0 k \ , 1, 2,3, 4 k , obtained from the solution of the corresponding boundary-value problem [4], namely: ( 1 9 ) If the program (18) does not satisfy constraints (10) In the latter case, in order to find the optimal values 1 u 1 u and 2 u 2 u corresponding to the values of 2 \ and 4 \ , the maximum of (16) with respect to u should be sought with allowance for the restriction To do this, we use the method of Lagrange multipliers [5], introducing an auxiliary function 2 2 1 2 1 2 1 1 2 2 0 1 2 where O is the Lagrange multiplier. Finding the maximum of the function F with respect to 1 u and 2 u , we obtain such relations for calculating the optimal control parameters: It can be shown that, in the case of equality 2 2 1 1 2 2 0 2 u u E K K , there must be 0 O . The same equation must be satisfied for the values of (20), that is, from this, then we obtain the following equation relative to O : In the general case, it is necessary to solve equation ( 4 4 / . The latter means that a non-fully utilized resource is then allocated appropriately among these subsystems in (9). ; in the second case in accordance with (20) and taking into account the solution of equation (21). Ultimately, the resulting relationships for calculating 1 u 1 u and 2 u 2 u depending on the values 2 \ and 4 \ are used in the formation of the optimal control program 1 ( ) u t 1 ( ) u 1 ( and 2 ( ) u t 2 ( ) u 2 ( and in the process of correction of the initial values 0 k \ , 1, 2,3, 4 k in (19). Thus, a general algorithm for solving the optimal control problem (9) − (13) is obtained. An analysis of the results of solving this problem shows that the allocations of the energy control resource for separate control objects in the system (9) obtained for it are characterized by continuous and consistent allocations of such a control resource

To the solution of the problem of optimal allocation of the material resource control
We consider here the optimal control problem (9) − (11), (14), (15), in which the optimal allocation of the material resource between separate control objectsdouble integrators is realized. Beginning with its solution, we first write down its Hamiltonian [3,4]: where the conjugate variables k \ , 1, 2, 3, 4 k , also satisfy the system of equations (17) That is, in case of M U inefficiency, the program (23) realizes an independent optimal control for the corresponding subsystems in (9).
We now will consider the solution of the problem when 12 M U U , by eliminating (10) from consideration. In this case, the maximum H (22) with respect to 1 u and 2 u must be sought with allowance for the constraint (15). Let 0 1 [ d be the share of using the available resource; at 0 [ Starting from (15), we introduce the following relations: where 0 1 d U d is share of the resource used to create the control impact 1 u , and its remaining share, i.e. 1 U , is intended to create 2 u . Taking into account (24) from (22), we obtain that it requires a separate investigation because of the arbitrariness in the choice U , since a special optimal control is possible here [5]. Note that at 0 F in 2 4 O\ \ a corresponding partitioning of the area 24 Ψ in which there is a "competition" for the use of the control resource between the separate control objects in (9), is given.
In each quadrant