Udc 17. 977+644. Bashkir State University, Republic of Bashkortostan

Algorithm for solving the optimal control problem with terminal constraints. Let us formulate a combined algorithm
for solving the optimal control problem with terminal constraints based on the penalty method and genetic algorithms. The main
idea of the penalty method is to replace the problem with constraints by the problem
,
where
H i
At each k-th iteration, age k is determined using genetic algorithms . Nay
Hi(u) =
.
is the penalty parameter determined at the kth iteration (the larger z,
the greater the penalty for non-compliance with the constraints). The penalty function S(u, zk ) is defined to be equal to zero if
the constraints are satisfied and greater than zero if
The work of the genetic algorithm is to sequentially change the set of vectors, called the population. Each vector is an
analogue of an individual (chromosome) in a population and consists of a set of numerical values, which are called genes or
traits [15, 16]. According to a certain rule, two individuals are selected from the population (by the selection operator), which
are crossed with each other, i.e., a new vector-individual is generated, which is subsequently subjected to the action of the
mutation operator, as a result of which one or several genes (the coordinate values of the vector ). The fittest individuals from
the current population pass to the next generation. The fitness of an individual is determined by calculating the fitness function,
which is the criterion of optimality. This procedure for changing the generations of the population continues until the criterion for
terminating the search is reached [17, 18].
The main idea of genetic algorithms is to imitate the evolutionary processes of living organisms: the fittest individuals
pass to the next generation, while retaining the best properties of their parents and acquiring new useful properties under the
influence of changing environmental conditions.
) = max g0(x(t1)) = ÿ min [ÿg0(x(t1))].
the
torus u ÿ (t) is used as the initial torus at the next iteration for the withdrawn vector u of the personal value of the penalty
parameter z
ÿ (t), which provides the minimum of the optimality criterion (5) for a given z
2
uÿU
m
k
k
uÿU
ÿ
i=1
134
AUTOMETRY. 2020. V. 56, No. 6
k
Machine Translated by Google

---

Rp
.
ÿ
.
ci = ÿai + (1 ÿ ÿ)bi ,
where ÿ is a random number from the range (0, 1), i = 1, r.
l and r are the numbers of individuals with the highest selection probabilities.
ÿ
i = 1, Rp.
u(t) =
.
(With
The selection operator "tournament selection" performs two tournaments. The first
tournament is a random selection of two individuals from the population. At the second
tournament, an individual is selected at random from two individuals selected at the first tournament.
Q(u j
Step 3. Crossing - generation of offspring individuals c = (c1, c2, . . . , cr) and d =
.
ÿ
ÿÿÿÿ
.
as a population of individuals, a set of vectors uj (t) = (uj1(t), uj2(t), . . . , ujr(t)), where j = 1, Rp, Rp is the size of the population. The coordinates of the
vector uj (t) are called genes. We will determine the fitness of individuals by calculating the optimality criterion (5). Then the algorithm for solving the optimal
control problem with terminal constraints will consist of the following steps. Step 1. Set the initial parameters of the algorithm: N is the number of partition
points of the time interval [0, t1], Rp is the population size, Max P is the maximum number of populations, the number of the current iteration of the
genetic algorithm q = 0, initial value = 0, 01; 0.1; 1) [13], a constant G ÿ [4, 10] to increase pa
Step 2. Selection - selection of a pair of individuals from the current population for the next
) = Q(u
d = (b1, . . . , bs, as+1, . . . , ar) ,
ÿÿÿÿ
ur(t)
.
(t), u0
di = ÿbi + (1 ÿ ÿ)ai ,
u1(t)
u2(t)
,
= (d1, d2, . . . , dr) from the parent pair.
Arithmetic crossover generates children according to the rule
. . .
),
(t) = (u
j1 on the set of admissible control values U, determine fitness for each individual. To
do this, it is necessary to find a numerical solution to the system of differential
equations (1) with initial conditions (2).
ÿ
ÿÿÿÿ
.
crossing.
The panmixia selection operator performs a random equiprobable selection. The
selection operator "roulette" selects two individuals with the best value of the fitness
function. For the minimum problem, this best value is the minimum value of the
optimality criterion (5). For each individual, the probability of selection is calculated by
the formula
)
q and b = u
r, where
ÿÿÿÿ =
As individuals of living organisms, we will consider the vector of control functions
.
.
p(u
Two individuals are selected with the highest probabilities a = u
The simplest crossover creates children according to the rule
c = (a1, . . . , as, bs+1, . . . , br) ,
where s is a random number from the range [1, r ÿ 1].
0
0 of the penalty parameter z of the
penalty parameter, the number of the current iteration of the penalty method algorithm k = 0, the parameter ÿ > 0 to terminate the algorithm.
Randomly set the initial population u j
q
u0 jr(t))
(t), . . . , j2
u21 . . . u2N
ur1 . . . urN
u11 . . . u1N
q
j=1
0
q
i
i
0
q
l
E. V. Antipina, S. I. Mustafina, A. F. Antipin, S. A. Mustafina
135
Machine Translated by Google

---

k+1
X1 ÿ X2, X2 ÿ X4, X1 ÿ X3, X1 ÿ X4, X2 ÿ X3, X3 ÿ X5,
0 u
y
kj (T) = k0je ÿEj/(RT)
136
AUTOMETRY. 2020. V. 56, No. 6
Random mutation transforms a randomly selected gene of each of the descendants c, d
anhydride.
dx4
= k2x2 + k4x1;
,
population to a local extremum.
dx2
= k1x1 ÿ k2x2 ÿ k5x2; dt
dx3
= k3x1 + k5x2 ÿ k6x3; dt