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ДИНАМИЧЕСКИЕ СИСТЕМЫ И ОПТИМАЛЬНОЕ
УПРАВЛЕНИЕ
DYNAMIC SYSTEMS AND OPTIMAL CONTROL
Серия «Математика» 2021. Т. 35. С. 3—17
Онлайн-доступ к журналу: http://mathizv.isu.ru
ИЗВЕСТИЯ
Иркутского государственного университета
УДК 517.977 MSC 49J20, 49M05
DOI https://doi.org/10.26516/1997-7670.2021.35.3
An Optimal Control Problem by a Hyperbolic System with Boundary Delay *
A. V. Arguchintsev1, V. P. Poplevko1
1 Irkutsk State University, Irkutsk, Russian Federation
Abstract. The paper deals with an optimal control problem by a system of semilinear hyperbolic equations with boundary differential conditions with delay. This problem is considered for smooth controls. Because this requirement it is impossible to prove optimality conditions of Pontryagin maximum principle type and classic optimality conditions of gradient type. Problems of this kind arise when modeling the dynamics of non-interacting age-structured populations. Independent variables in this case are the age of the individuals and the time during which the process is considered. The functions of the process state describe the age-related population density. The goal of the control problem may be to achieve the specified population densities at the end of the process. The problem of identifying the functional parameters of models can also be considered as the optimal control problem with a quadratic cost functional. For the problem we obtain a non-classic necessary optimality condition which is based on using a special control variation that provides smoothness of controls. An iterative method for improving admissible controls is developed. An illustrative example demonstrates the effectiveness of the proposed approach.
Keywords: hyperbolic system, boundary differential conditions with delay, necessary optimality condition, optimal control.
* The reported study was funded by RFBR and the Government of the Irkutsk Region, project number 20-41-385002, and by RFBR, project number 20-07-00407
1. Introduction
Optimal control problems with time delays play an important role in various fields of applications. The study of such problems began in the 60s of the XX century (see, for example, [6]) from ordinary differential equations.
Optimal control problems for specific classes of distributed parameter systems with delay were investigated in the following areas:
1) existence of optimal control [10; 11];
2) necessary optimality conditions [9];
3) computational methods based on ideas of dynamic programming, reductions to simpler problems, etc. [3;5].
The study of optimal control problems for distributed parameter systems is closely related to the study of partial differential equations and/or integral equations, including generalized solutions of such systems. At the same time, it makes sense to consider a number of optimal control problems in a smooth control class. In this case solutions of the corresponding boundary value problems for differential equations could be considered in the classical or "almost" classical sense. In particular, the solution of inverse problems for partial differential equations can often be reduced to the corresponding problems of optimal control in the class of smooth control functions. In this paper we consider an optimal control problem by boundary differential conditions with delay for semi-linear hyperbolic equations. Such problems arise in modeling of age-structured biological populations. Boundary delay is connected with environmental factors, reproduction peculiarities, etc.
A principle difference from classical statements of optimal control problems is the investigation of problems in the class of smooth admissible controls. In this paper, we study the problem in the class of smooth controls that satisfy the pointwise constraints in each point. It is impossible to use the optimal control methods based on the Pontryagin maximum principle or the classic gradient optimality conditions. These methods are focused on the classes of discontinuous controls. We apply the idea of the general approach [2] based on using a special variation that provides smoothness of controls and satisfaction the constraint. This approach is based on the "inner" variation idea, originally applied by L. Zabello for optimal control problems in ordinary differential equations [12]. However L. Zabello used a combination of inner and needle variations. His approach, generally speaking, is effective for researching problems with lag, but in the class of discontinuous controls.
Another way of deriving constructive necessary optimality conditions was suggested in [4; 7; 8]. The new class of optimal control problems governed by the dissipative and discontinuous differential inclusion of the sweeping/Moreau process with controlled moving sets was investigated. Controls actions of moving sets must be continuous. Authors used the
method of discrete approximations and combined it with advanced tools of first-order and second-order variational analysis and generalized differentiation. These techniques can be useful for the class of problems considered below.
2. Problem statement
Consider the system
xt + A(s,t)xa = f (x,s,t), (2.1)
(s,t) e n, n = S x T, S = [so,si],T = [to,h],
Here x = x(s,t) is a n-dimensional vector-function of state variables, A = A(s,t) is a (n x n) - matrix. The system (2.1) is written in an invariant form, i.e., A is a diagonal matrix. In addition, we assume that the diagonal elements ai(s,t) of the matrix of coefficients possess constant signs in the rectangle n:
ai(s, t) > 0, i = 1, 2,... ,m1; ai(s, t) = 0, i = m1 + 1, m1 +2,..., m2; ai(s, t) < 0, i = m2 + 1,m2 + 2,... ,n. Respectively, the state vector x = x(s, t) contains two subvectors
x+ = (xi , X2, . . . , Xmi — (Xm2+1, %m2+2, . . . , %n),
which correspond to positive and negative diagonal elements of the matrix of coefficients.
Let the controlled initial-boundary conditions for the system (2.1) be given in the following form:
dX+^t0,t^ = g(x+(so,t),x+(so,t - h),u(t),t), t e T, (2.2)
x(s,t0) = x0(s), s e S, x~(s1,t) = v(t),t e T,
x+(so, t) = q(t), t e [-h; to]; h> 0,
where h is a constant delay.
Control u = u(t) is a smooth r-dimensional vector-function on segment T. It satisfies the constraint
u(t) e U, t e T, (2.3)
where U is a compact set.
The problem is to minimize the functional
J(U) = J <fi(x(s,h),s) ds, (2.4)
defined on the solutions of the problem (2.1), (2.2) for admissible control functions (2.3). Denote y(t) = x+(s0,t — h).
The optimal control problem (2.1)-(2.4) is considered under the following suppositions:
1) the diagonal elements ai(s,t) of the matrix A are continuous and continuously differentiable in n;
2) functions x°(s), v(t), q(t) are continuous with respect to their arguments on the sets S and T respectively and satisfy the conditions
^ (to) = (x°(Sl))-, q(to) = (x°(so))+;
3) functions f (x, s, t) and <p(x, s) are continuous with respect to their arguments, and they have continuous and bounded partial derivatives with respect to the state function x;
4) function g(x+, y, u, t) is continuous and continuously differentiable, and has bounded partial derivatives with respect to x+, y and u.
Despite the smoothness of the controls, the solution of the initial-boundary problem for the hyperbolic system is understood in a generalized sense. It is suitable to use the definition of a generalized solution in terms of characteristics of the system. Let us consider characteristic curves determined by the ordinary differential equations
^ = ai(s,t), i = 1, 2,...,n. (2.5)
Let Si = Si(£,r;t) be a solution of (2.5), which passes through the point (£, t) £ n. If there exists a classical solution of the system under consideration, then the given system is equivalent to the following one:
t
xi(s,t)= Xifa^^ + J fi(x(£,T),£,T)\c=s.{s,t.r) dт, (2.6)
Ti
(s,t) £ n, i = 1,2, ...,n,
where ((i, Ti) is the initial point of i-th characteristic curve passing through (s,t).
By means of integral system (2.6) it is possible to prove the existence and uniqueness of a continuous in n a generalized solution. Each component of this solution is continuously differentiable along the corresponding characteristic family [1]. So, instead of the left side of system (2.1) we consider the differential operator
idx\ //dx\\ idx2\ fdxn\ \
W a = U ~dt) A A J ,
where (dxi/dt)A is the derivative of i-th component of the state vector along the corresponding family of characteristic curves.
3. Increment formula
Consider two admissible processes, namely, the initial admissible process {u,x} and the perturbated one {« = u + Au,x = x + Ax}. Write the problem in the following form
(dAx\ . .. .
{-W) A = A/ <*•»•«>•
Ax+(so,t) = 0, t e [-h; ioj; Ax(s,to) = 0, s e S; Ac" (s1,t)=0, t e T;
Ax+t (so,t) = Ag(x+(so,t), y(t), u(t), t), (3.1)
where
Ag(x+(so,t),y(t),u(t),t) = g(x+,y,u, t) - g(x+,y,u,t) = = Azg(x+ ,y, u, t) + A5+ g(x+, y, u, t) + A$g(x+, y, u, t),
AJ(u) = J Aip(x(s, t1), s) ds. s
Add the following terms to the increment formula for the cost functional
//w^), - Af (x,s,t)) dsdt,
n A
J (p(t), Ax+t(so,t) - Ag(x+(so,t),y(t),u(t),t)) dt,
T
where ^(s, t) and p(t) are still undefined n-dimensional and mi-dimensional vector-functions having the same smoothness properties as x(s, t) and x+(t) respectively. Here (.,.) is a designation of a scalar product in Euclidean space of a corresponding dimension.
Applying integration by parts, we have
AJ(u) = J Aip(x(s,ti),s) ds + J[(^(s,ti), Ax(s,ti))-
s s
-(^(s, to), Ax(s, to))] ds - jj( ^^ + As^, Ax(s, t)) ds dt+
n A
+ J№(si,t),A(si,t)Ax(sht)) - (^(so,t),A(so,t)Ax(so,t))] dt+
T
+(p(t 1), Ax+(So, ti))-(p(io), Ax+ ( s0, to))- j(pt, Ax+(so, t)) dt-
T
U «М),А , (x,s, t))^ - / (P (t), Ag(x+,y,u,s, t))dt.
n T
Introduce the following auxiliary functions
H(ip(s, t),x(s, t),s, t) = (ip(s, t), f(x, s, t)),
h(p(t),x+(so, t), y(t),u(t), t) = (p(t), g(x+(so, t),y(t),u(t), t)).
Then
Ah(p, x+, y, u, t) = A^h(p, x+, y, u, t) + h(p, x+ ,y, u, t) + A^h(p, x+,y, U, t),
where
Ayh(p, x+,y, u, t) = h(p, x+ ,y, U, t) — h(p, x+, y, u, t), h(p, x+, y, u, t) = h(p, x+,y, u, t) — h(p, x+,y, u, t), Ayh(p, x+,y, u, t) = h(p, x+,y, u, t) — h(p, x+, y, u, t). Use the following expansions
Ap(x(s, ii), s ) = (S), Ax(s, ii)) + ov(IAx(s, ii)|),
A H(ip,x,s, t) = (dH^ , Ax(s, t)) + oh(|Ax(s, i)|),
Aj+ h(p, x+( So, t), y(t),u, t) = = ( dh(p,x+(s d-OMt^, t) Ax+( s o, t)) + oh(IAx+(s o, i)|),
,dh(p, ж+(s0, t),y(t),u, t) dy
Transform the term
Аф(р ,5?+(e o, t), y(t),ù, t)=(Uh(P ,x (b 0^),У(b),u, b), Ay(t))+ oh(\Ay(t)\).
dh(p,x+(so, t), y(t),u, t) _ dh(p, x+(so, t),y(t),u, t) +
о — ^'U о +
ду ду
+ dh(p,X+(So, t), y(t),u, t) ду ,
where
dh(p,ж+(so, t), y(t),u, t) _ ^ dh(p,x+ (so, t), y(t),u, t) + ду x ду
+ dh(p,x+(So, t), y(t),u, t) dy '
Then, we obtain
j(dh(p, x+(so, t),y(t),u, t) dt = t ^
= i(9h(P(t),x+(*0, t),x+(s^ — h),u(t),f), Ax+(so,t — h))dt =
J 9y
t
tf.dh(p(d+h),x+(so,e + h),x+(so, d),u(d + h),9 + h) . + =J (--q^-, Ax (so, v))dv+
t0-h
t-i-h
+ - (dh(p(P + h), x+(so, 0 + h),x+(so, 0),u(Q + h),d + h), Ax+ ^
J -y , 0, '
to
Here we have used the following designation: d = t — h, de [io — h, t\ — h]. Then, reverting to a variable t, we get
ti-h
i ,dh(p(t + h),x+(so,t + h),x+(so, t),u(t + h),t + h) ax+(s t)) dt
J dy , x So, .
to
Let functions ^(s, t), p(t) be the solutions of the following adjoint problem
= -H*(^,x,s, *), , fi) = -VrMs, ti), s), (3.2)
^+(s i, t) = 0; so, t) = 0, teT;
f -hx[t] - hy[i + h] - A+(so, t)ip+(so, t), t e [io; h - h], Pt = \ -hx[t] -A+(so, t)ip+(so, t), t e [ti - h; ti].
p(ti) = 0; p(t) = 0, t > ti. (3.3)
Here
hx [i] = hx (p(t), x+ (so, t), y(t), u(t), t), hy [i + h] = hy (p(t + h),x+( so,t + h),x+ (s o, t),u(t + h),t + h). Then, the increment formula for the functional takes the following form:
A J(u) = - j A^h(p(t),x+(so, t), y(t),u(t), t)dt + rj, (3.4)
T
where
r]_ J ov(|Ax(s, h)l)ds + JJ(он(\Ax(s, t)\) dsdt+
S П
+ J [ oh(\Az+(so, t)\) + (Aahx+ (p(t),x+(so, t), y(t), u(t), t), Ax+(so, i))] dt+
T
+ J[ Oh(\Ay(t)\) + (Ashy(p(t), S+(so, t), y(t),u(t), t), Ay(t))] dt+
T
+ j(A?+ hy(p(t),x+(so, t), y(t),u(t), t), Ay(t))] dt.
T
Lemma 1. Under the condition (3.1) the estimation of a state increment (analogously to [1]) takes the form
t
7(t) = max ,r)\ <L3J\Au(r)\dr, (3.5)
to
n(i) = {(£, t) £ n : t <t}.
Proof. Consider the following system:
t
Axt(s, t) = Axt(£t, n) + J Afi(x(£, t),£, T)dr, (3.6)
Ti
where ((i, Ti) is the initial point of i-th characteristic curve passing through ( , ). Denote
7+(i) = max \Az+(so,r)\. (3.7)
to <t <i
Under assumption of the problem
\Af(x,s, i)\ = \ f(x,s, t) — f(x,s, t)\ < L|z — x\,
\Ag(x,y,u, i)\ = \g(x,y,u, t) — g(x,y,u, i)\ < Lx(\x — x\ + \y — y\ + \u — u\),
where L is a Lipschitz constant for the function f, L1 is a Lipschitz constant for the function g. Taking into account (3.6), (3.7) and conditions A;c-(s1, t) = 0, Ax(s, t0) = 0, we get
i
I Axi(s, t ) | <f+(t) + L J f(T)dT, i_1,2,...,mx;
t0
t
lAxi ( s, t)\ < L J j(t) dr, i = m2 + 1,m2 + 2,... ,n.
to
The right sides of the inequalities do not depend on i and s, then the following inequalities is valid
t
l(t) < Vnj +(t) + LV^J 7(r) dT. (3.8)
to
Further, from (3.1),we get
t t A+(s0, t)l< j lAg(x+,y,u, T)ldr<Lij (\Z+(s0, t) -x+(s0, r)| +
to io
+ W) -y(r)l + |u(r) -u(r)l)dr.
Consider
t t j lAy(r)ldr = j \y(r) - y(r)ldr =
to to
t
= J \x+ ( so,t - h) -x+(sq,t - h)\dr =
to
to t
= f re,*)-,+(,,0)Uw+Jwn«.» -^ =
to—h to
t
= J\i+(s,, r) -x+(. C T)\dT.
to
Here we have used that
T- h = 6, 6 e [io - h, ti - h].
Then
t t A+(s0, i)| < 2Li J A+(s0, r)\dr + Li J |Au(r)|dr.
to to
From the inequality, we get
t t 7 +(t) < 2Li J 7 +(r) dr + Li J |Au(r)| dr.
to to
Using Gronwall-Bellman inequality, we obtain
i
7+(i) <L2J |Au(t)| dr, L2 _ Li ■ e2Ll(tl-t0).
to
Take the inequality in (3.8) and reuse Gronwall-Bellman inequality. Then, we get estimation (3.5), where L3 = ^JnL2 ■ e^L(tl-o). □
4. Optimality condition
Apply the idea of the general approach [1] based on using a special variation that provides smoothness of control and satisfaction the constraint. The varied control takes the form
U£,s(t) =u(t + e5(t)), t£T,
where e £ [0,1] is a parameter of variation, S(t) is a continuously differen-tiable function and satisfies the following conditions t0 < t + S(t) < t1, t £ T. Since admissible controls belong to the class of smooth functions, the increment formula for the control function takes the form
Au = u(t)e5(t) +o(e).
Using (3.5), we get
A J(u) = —e J(hu, u)S(t) dt + o(e). (4.1)
T
Formulate a necessary optimality condition (analogously to results obtained in [1]).
Theorem 1. If {u,x} is the optimal process in the problem, there is valid the following condition
u(t) = (hu(p(t),x+(so, t), y(t),u(t), t),ii(t)) =0, t£T,
where p(t) is a .solution to the adjoint problem (3.2), (3.3).
The convergence result is given in [1].
5. Iterative method
Describe the general scheme of the method (analogously to [1]).
1) Let uk(t) be an admissible control calculated on the k-th iteration.
2) For the control uk(t) solve problems (2.1), (3.2) and (3.3) to get functions xk(s, t) and (s, t), pk(t) respectively.
3) Calculate the value of the functional Jk = J(uk) and construct the function
(t) = {hu(pk(t),xk(so, t),uk(t), t),uk(t)).
The function Wk(t) consideres as a discrepancy of the fulfillment of the optimality condition. If (t) = 0, then the control function uk satisfies the optimality condition and the iteration process finishes.
4) If Wk(t) = 0, then consider a smooth variation of uk(t)
ukk (t) = uk (t + ek5k (t)),
= (t -tQ)(ti - t)uk(t),
k (11 — to) max luk(í)|, t£T
where ek is a solution of the minimization problem £k : J(uk) ^ min, e e [0,1].
5) The next approximation is given by the formula
uk+i(t)=uk£k (t).
One of the following conditions can be considered as the stop criterion on some k-th iteration of the method.
a) The function uk(t) satisfies (with a given accuracy) the necessary optimality condition. For example, by the condition max l^k(í)| < 10-5.
b) The value of the functional calculated on the previous iteration (that with the number k — 1) is not improved, for example, Jk — Jk-i > 10-6.
c) £k is closed to zero. This case is considered as no improvement of the functional on the method step.
6. Illustrative example
Consider the application of the described method to one test example. The algorithm was coded in Matlab 7.0. In the square [0, 4] x [0,4] we consider the optimal control problem
Xlt + Xis =Xi +X2 + fi(s, t),
X2t - X2s =X2 - ¡2(s, t),
Xit(0, t)=u ■Xi(0,t — 0.5), u(t) £ [0, 3]; Xi(0, t) = 0.3 ■ t, t£ [—0.5; 0];
X2(4, t) = 0.1 ■ t, xi(s, 0) = 0, X2(s, 0) = s — 4. The cost functional takes the form
J(u) = 1 J(xi(s, 4) — xi(s))2 + (x2(s, 4) — x2(s))2 ds ^ min, u £U, s
where ^i(s) = xi(s, 4), x2(s) = x2(s, 4) are evaluated for the control
t t u(t) = 2 cos — — si^ —.
16 2
The auxiliary functions and the adjoint problem:
H(0, X, S, t) = ^i ■ (xi + ^2 + fi(s, t)) + ^2 ■ (X2 — f2(s, t)), 0it +0is = —0i, 02t —02s = —0i —02,
0i(s, 4)= Xi(s) —Xi(s, 4), i = 1, 2; 0i(4, t) = 0; 02(0, t) = 0; h(p, x,y,u, t) = p ■ u ■y, where Xi(0, t — 0.5) = y(t); p(4) = 0,
—p^u — 0i(0, t), t £ [0; 3.5),
Vt
Pt \—0i(0, t), t £ [3.5; 4].
We solve the problem under following functions: /i( s, t) = es cost and 2( , ) = sin . The initial control is
u0(t) = 1 + cos1.2i.
The value of the functional is J(u0) = 7.61749. We solved the problem by the described method. Results of calculation are presented at the Table 1. We have obtained: J(vk) = 0.03854, max l^k(i)| = 0.02791, the total
number of iteration equals 48. The stop criterion is £k < 10-5 (there is no improvement of the functional on the method step).
7. Conclusion
In this paper we considered the optimal control problem by hyperbolic system with a special type of boundary delay. The optimality condition in a class of smooth control is proved. We applied approach [1] which is based on using a special variation that provides smoothness of control and
Table 1
t w(i) u°(t) uk(t)
0 2 2 2
0.3 1.850 1.935 1.631
1.075 1.483 1.277 1.526
1.65 1.254 0.602 1.372
2.1 1.115 0.187 0.989
2.975 0.968 0.090 0.843
3.3 0.960 0.316 0.875
4 1.028 1.087 1.032
satisfaction the constraint in each point. We consider the application of the iterative method to solving the optimal control problem with boundary delay. Numerical experiment showed the efficiency and applicability of the method.
References
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Alexander Arguchintsev, Doctor of Sciences (Physics and Mathematics), Professor, Irkutsk State University, 1, K. Marx st., Irkutsk, 664003, Russian Federation, tel.: +7(3952)521-282, email: arguch@math.isu.ru, ORCID iD https://orcid.org/0000-0002-9314-485X
Vasilisa Poplevko, Candidate of Sciences (Physics and Mathematics), Associate Professor, Irkutsk State University, 1, K. Marx st., Irkutsk, 664003, Russian Federation, tel.: +7(3952)521-282, email: vasilisa@math.isu.ru
Received 17.08.2020
Задача оптимального управления гиперболической системой с запаздыванием на границе
А. В. Аргучинцев1, В. П. Поплевко1
1 Иркутский государственный университет, Иркутск, Российская Федерация
Аннотация. В статье рассматривается задача оптимального управления системой полулинейных гиперболических уравнений, в которой граничные условия определяются из системы обыкновенных дифференциальных уравнений с запаздыванием. Задача рассматривается в классе гладких управляющих воздействий. В силу данного условия невозможно доказать условие оптимальности типа принципа максимума Л. С. Понтрягина и классические условия оптимальности градиентного типа. Задачи такого рода возникают при моделировании динамики невзаимодействующих между собой популяций с учетом возрастного распределения особей. Независимыми переменными в этом случае являются возраст особей и время, в течение которого рассматривается процесс. Функции состояния процесса описывают возрастные плотности популяций. Целью задачи управления может быть достижение заданных плотностей популяций в конечный момент времени. Проблема идентификация функциональных параметров моделей может также рассматриваться как задача оптимального управления с квадратичным целевым функционалом. Для указанной задачи получено неклассическое необходимое условие оптимальности, которое основано на применении специальной вариации управления, обеспечивающей гладкость управляющих функций. Предложен метод улучшения допустимых управлений. Эффективность предлагаемого подхода проиллюстрирована примером.
Ключевые слова: гиперболическая система, граничные дифференциальные условия с запаздыванием, необходимое условие оптимальности, оптимальное управление.
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Александр Валерьевич Аргучинцев, доктор физико-математических наук, профессор, Иркутский государственный университет, Российская Федерация, г. Иркутск, 664003, ул. К. Маркса, 1, тел.: +7(3952)521-282, email: arguch@math.isu.ru, ORCID iD https://orcid.org/0000-0002-9314-485X
Василиса Павловна Поплевко, кандидат физико-математических наук, доцент, Иркутский государственный университет, Российская Федерация, г. Иркутск, 664003, ул. К. Маркса, 1, тел.: +7(3952)521-282, email: vasilisa@math.isu.ru
Поступила в 'редакцию 17.08.2020
