1. A multi-objective synthesis of optimal control system by the network operator method 1 A.I. Diveev Institution of Russian Academy of Sciences Dorodnicyn Computing Centre of RAS, Moscow, Russia (e-mail: aidiveev@mail.ru).

2. Classical problem of optimal control 2 The mathematical model of object of control is set Initial values Functional is criterion optimization A solution is optimal control Terminal condition Synthesizing function

3. 3 Synthesis of optimal control

4. Problem statement 4 Initial values are in closed bounded domain It is necessary to find the admissible control that satisfies the restrictions oris vector of parameterswhere Pareto set is considered to be the solution of problem

5. Theorem 1. Assume 5 is a finite denumerable set Supposeis the solution of the multi-objective synthesis problem. and Then is the solution of the same problem, but with the set of initial condition

6. The network operator 6 The set of variables The set of parameters The set of unary operations The set of binary operations Commutative Associative Unit element

7. Унарные операции 7

8. Binary operations 8

9. The network operator 9 Network operator is a directed graph with following properties: a)graph should be circuit free; b) there should be at least one edge from the source node to any non- source node; c) there should be at least one edge from any non-source node to sink node ; d) every source node corresponds to the item of set of variables or of parameters ; e) every non-source node corresponds to the item of binary operations set ; f) every edge corresponds to the item of unary operations set.

10. 10 Definition 1. Program notation of mathematical equation is a notation of equation with the help of elements of constructive sets X, Q,O 1,O 2. Definition 2. Graphic notation of mathematical equations is the notation of program notation that fulfills the following conditions: a) binary operation can have unary operations or unit element of this binary operation as its arguments; b) unary operation can have binary operation, parameter or variable as its argument; c) binary operation cannot have unary operations with equal constants or variables as its arguments. Theorem 2. Any program notation can be transformed in graphic notation.

11. 11

12. An example Program notation Graphic notation Network operator 12

13. Network operator matrix (NOM) 13 Definition 4 Network operator matrix (NOM) is an integer upper- triangular matrix that has as its diagonal elements numbers of binary operations and non-diagonal elements are zeros or numbers of unary operations, besides if we replace diagonal elements with zeros and nonzero non-diagonal elements with ones we shall get an vertex incident matrix of the graph that satisfies conditions a-c of network operator definition.. NOM vertex incidence matrix

14. Network operator matrix (NOM) 14 vector of numbers of nodes for input variables vector of numbers of nodes for parameters vector of numbers of nodes for output variables a node vector if

15. Theorem 3. If we have the NOM numbers of nodes for variables parameters and outputs the mathematical expression is described by NOM and vectors of then it is sufficient to calculate 15

16. 16 Small variations of network operator: 0 - replacement of unary operation by the edge of the graph; 1 - replacement of binary operation in the node; 2 - addition of the edge with unary operation; 3 - removal of unary operation with the edge of the graph. Vector of variations specifies the number of variation, is the number of node that the edge comes out, is the number of node that the edge comes in, is number of unary or binary operation.

17. The principle of basic solution 17 When solving optimization problems, initially we set the basic solution that is one of admissible solutions, then define small variations of basic solution and create search algorithm that searches for the optimal solution on the set of small variations

18. The genetic algorithm 18 The basic solution is a basic NOM A structural part of chromosome NOM for chromosome i A parametrical part of chromosome is number of bit for integer partis number of bit for fractional part Vector of parameters Grey code

19. An example 19

20. 20

21. Pareto set 21

22. the solution 22

23. 23

24. 24