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0Russian Journal of Nonlinear Dynamics, 2023, vol. 19, no. 1, pp. 35-48. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd220801
NONLINEAR PHYSICS AND MECHANICS
MSC 2010: 34H05
Methods of Simplifying Optimal Control Problems, Heat Exchange and Parametric Control of Oscillators
A. M. Tsirlin
Methods of simplifying optimal control problems by decreasing the dimension of the space of states are considered. For this purpose, transition to new phase coordinates or conversion of the phase coordinates to the class of controls is used. The problems of heat exchange and parametric control of oscillators are given as examples: braking/swinging of a pendulum by changing the length of suspension and variation of the energy of molecules' oscillations in the crystal lattice by changing the state of the medium (exposure to laser radiation). The last problem corresponds to changes in the temperature of the crystal.
Keywords: change of state variables, problems linear in control, heat exchange with minimal dissipation, parametric control, oscillation of a pendulum, ensemble of oscillators
Dedicated to the memory of V. F. Krotov and V. I. Gurman
1. Introduction
The solution of the optimal control problem
T
I = J f0(x, u, t) dt ^ max (1.1) 0
under the conditions
Xi = fi(x,u,t), x e Rn, u e Vu C Rm, i = 1,...,n, (1.2)
Received January 31, 2022 Accepted July 08, 2022
This work was supported by the Russian Science Foundation, project No. 20-61-46013.
Anatoly M. Tsirlin tsirlin@sarc.botik.ru
Program Systems Institute of RAS,
ul. Petra Pervogo 4a, Pereslavl-Zalessky, Yaroslavskaya obl., 152020 Russia
in which fi are continuous in u and continuously differentiable in x, t and Vu is a compact, can be considerably facilitated by transforming to new state variables y(x). Below a general scheme of this approach is presented and the possibilities it opens are demonstrated using as examples the solution of the problem of heat exchange and two oscillator control problems.
Let us call the problem (1.1)—(1.2) the initial problem, and the problem with the state variables y the transformed problem, and let us list the aims one pursues by such a transformation:
1. Make some of the variables y independent of part of the control actions and convert part of these variables to the class of controls with a decrease in the dimension of the problem.
2. For problems in which the right-hand sides of differential equations do not explicitly depend on the independent variable, in particular, on time, the dimension of the problem can be decreased if the rate of change of one of the variables yv retains its sign on the set of admissible solutions. In this case, dt can be replaced with dyv.
3. Reveal the functions of the phase variables that are constant along the trajectories of the system in the class of controls satisfying the optimality conditions.
4. If the right-hand sides of the differential equations for y-, j = 0, ..., n do not depend on the state variables yv, then the vth equation can be discarded or replaced with an integral restriction.
Some of the aims listed above can be achieved in the evaluation problem with a wider set of admissible solutions rather than in the initial problem (1.1), (1.2). This is how the change of state variables is used in the work of V. I. Gurman [1].
By virtue of equations (1.2) the rate of change of the phase variable y- (x) is
A dy.
= (1.3)
i=1 i
To attain these aims, some conditions, for example, conditions for sign constancy or conditions for independence of the right-hand sides from specific variables, must be imposed on the right-hand sides of these equations. These requirements lead to partial differential equations whose solution determines the desired replacement.
The decrease in the dimension simplifies the problem considerably, and such simplifying transformations are appropriate if the complexity of the solution of the partial differential equations for the choice of a transformation in combination with the solution of the transformed problem is not greater than the complexity of the solution of the initial problem. Therefore, it is important to single out a class of problems in which the type of transformations has been defined. These include problems in which one or several phase coordinates monotonically depend on t and problems linear in control. How the initial problem can be reduced to this class is nonobvious and requires a certain skill. The examples given below are meant to contribute to the development of this skill:
• the problem of optimal heat exchange;
• the problem of braking (swinging) of the pendulum by changing the distance of its center of gravity from the suspension point;
• the problem of extraction of work from the system of quantum oscillators (their cooling) by changing the parameter of the Hamiltonian.
The problem of optimal heat exchange in this setting has not been addressed. The last two problems can be interpreted as problems of extraction of a given energy in minimal time or, in terms of "optimization thermodynamics" (see [2]), as problems of maximal power. The parametric braking of the pendulum is considered in [3] and [4]. The solution obtained in [3] is erroneous, and the solution found in [4] was obtained without using a simplifying transformation. The problem of braking the ensemble of oscillators was solved for a particular type of boundary conditions numerically in [5] using Maple.
Methods of simplifying optimal control problems which precede their solution are considered in [6]. There the transition from the system of linear differential equations to the integral equation of convolution is considered along with the transition to new phase coordinates and the change of an independent variable. Such a transition is particularly efficient for equations with a retarded argument.
1.1. Conversion of state variables to the class of controls for problems linear in control
1.1.1. The scalar problem
Consider a problem of the form
T
I0 = j [M0(x, t) + N0(x, t)v] dt ^ max (1.4)
0
under the conditions defining the set of admissible solutions D:
x = v, ,t(0) = x0, x(T) =x, Vx, v e R1, (1.5)
where x(t) and v(t) are scalar functions, M0 and N0 are continuous in x and continuously differentiable in t, and Vx is a compact.
Let us introduce a scalar function y(x, t) that is continuous in x and smooth in t, and let us form an auxiliary problem with the optimality criterion
T
J = h + / ^jf^dt ~ T) " 3/(z(°)> (L6)
0
It is obvious that for any such function I0 = J and it can be specified according to some additional requirement. We write the criterion J in more detail:
T
I M0(x,t) + N0(x,t)v + ^v + C^
dt - (y(T) - y(0)). (1.7)
0
Let us specify y(x, t) from the condition for independence of the integrand in J from v. Thus,
9y = -N0(x, t) ->■ y(x, t) = - j N0(x, t) dx, (1.8)
0
x
and
x(T) x(0)
y(T) = -! N0>dx y(0) = -/ Ndx d-9)
00
Using Eqs. (1.8) and (1-9), the problem (1.4)—(1.5) can be transformed as follows:
T
J = J
0
x
d i
M0(x, t)--gt N0(x, t) dx 0
dt - (y(T) - y(0) ^ max. (1.10)
xeVx
The coincidence condition has been fulfilled: On the set D defined by conditions (1.5), the junctionals I0 and J coincide.
The phase variable x of the initial problem in the auxiliary problem has become a control, and there is no differential equation in the auxiliary problem.
If the boundary conditions x0 and x in the problem (1.4)—(1.5) are not fixed, but are subject to a choice, then they are defined by the requirements
x*(0) = argmink(0, x(0)), x*(T) = argmaxk(T, x(T)), (1.11)
where k(0, x(0)) and k(T, x(T)) are additional terms to the functional I0.
Conditions (1.5) are absent in the problem (1.10). The control corresponding to its optimal solution and found using Eqs. (1.5) is optimal.
The situation changes when the set of admissible controls in the initial problem is restricted by v G Vv C R1. In this case, the following inequality (estimate) holds:
J * = J (x* (t)) > I* = Io(v*(t)). (1.12)
If the function x*(t) turns out to be such that, v = e Vv, then inequality (1.12) becomes an equality and
dx*
In this case, the auxiliary problem is called an equivalent extension of the initial problem.
The transformation (1.7), (1.9) for the Euler problem (1.10) was proposed by V.F.Krotov (see [7, 8]), so it is natural to call it Krotov's transformation.
If on x(t) there are no other restrictions than the boundary conditions, and the set of admissible velocities Vv is bounded, then the equivalence of the extension is unlikely. However, one can preliminarily single out, for each t, those values of x(t) beyond which the function x(t) will not go on the solution of the initial problem. We will call the set of such values Wx(t) an external, admissible set or a set of attainability. Then in the problem (1.10) the maximum in x of the integrand needs to be searched for on the intersection of the set of attainability and the set VX. In this case, the estimate (1.12) will be more accurate. The set of attainability Wx(t) is bounded by the trajectories for which, by virtue of conditions (1.5), v reaches the maximum and minimum at each x, t with the boundary conditions taken into account.
In the case where the function N0(x, t) in the problem (1.4)—(1.5) is zero, the extended problem (1.10) reduces simply to the requirement of the maximum in x G Wx(t) of the function M0 for each t.
When N0(x, t) does not depend on x and is equal to N0(t), the extended problem (1.10) takes the form
T
J = [M0(x, t) - N0(t)x] dt + N0(T)x(T) - N0(0)x(0) — max . (1.13)
J xewx(t) 0
And in the case where N0(x, t) = N0(x) and does not depend on t
T x(T)
J = M0(x,t) dt + N0(x) dx — max . (1.14)
J J xeWx(t)
0
x(0)
Example.
(1.15)
M0(x, t) = -x2(t), N0(x, t) = 0,
x0 = x = 0.5, T = 2, \v\ < 1.
The set of attainability Wx(t) is bounded by the straight lines emanating from the points x(0) and x(T) with the inclination +1 and -1; this is a rhombus on the plane x, t with vertices at points (x(0); 0), (x(2); 2), i.e., (0.5; 0), (0.5; 2). The maximum M0 on the set Wx(t) is achieved on the solution
( 0.5 - t for 0 < t < 0.5,
x*(t) = < 0 for 0.5 <t< 1.5,
[ t - 1.5 for 1.5 < t < 2.
Since this solution satisfies the restrictions of the initial problem, it is optimal, and the extension taking into account the set of attainability Wx(t) is equivalent.
In real-world problems, the right-hand sides of equations for phase coordinates depend on state variables and control actions. We show that the problems in which the controls are involved linearly, or those brought to such form in making a change of state variables, can be transformed similarly to the simplest problem of calculus of variation in such a way that in the auxiliary problem the phase variables become controls and the dimension of the phase space decreases.
1.1.2. Generalization of Krotov's transformation to vector problems linear in control
J ^o(x, t) + è r0i(x, t)u^ dt, (1.16)
In the case where x is a vector, the problem linear in control can be written as
I = /u,xt>+> ,,.,), 0
and the change in the state vector is defined by the equations
obi = si(x, t) + ri(x, t)uj,, i = 1, ..., n, u G Vu C Rn, x G Vx C Rn. (1.17)
The following theorem holds.
Theorem. In order for the function x*(t) to be the optimal solution to the problem (1.16) — (1.17), it suffices that it furnish, at each t, the maximum on the intersection of the set Vx and the set of attainability Wx to the integrand of the functional
J = [ (s0{x, + t)Si{x, t) + M^lll j dt-y{x, T) + y(x, 0) ->■ max (1.18)
0 V i=i /
0
for a function y(x, t) continuously differentiable with respect to a collection of arguments and satisfying the conditions
dy(x,t)_ rm(x,t)=<f)i{Xit)i . ,.....(L19)
dXi r¿x, t)
and be realizable using the controls u £ Vu.
Proof. Similarly to the Euler problem, we add to the optimality criterion the total time derivative of the function y(x, t) and subtract the difference of the values of this function at t = T and t = 0.
- ± »>+*«* t)«,) + «S|í>. (1.20)
i=1 i
On the set of admissible solutions this additional term is zero. The function satisfying conditions (1.19) is
*) = E / f - E ^É-^W (L21) i=i{ \ j=i,j=i dxj )
which can be verified by differentiation.
The optimality criterion (1.18) of the transformed problem in this case contains no control actions and is equal to the optimality criterion of the initial problem on the set of its admissible solutions. Thus, if x* (t) is realizable in the initial problem, it is optimal.
This proves the theorem. □
If the found solution is not realizable using admissible controls, then the value of the functional J on it gives an upper estimate for the values of the initial problem.
In the case where only part of the differential equations linearly depends on the control variables and these controls are involved linearly in the optimality criterion and are not involved in other differential equations, Krotov's transformation and the formulae presented above for y(x, t) can be used to decrease the dimension of the problem. As a result, the controls involved linearly are excluded and the corresponding state variables are converted to the class of controls.
2. Examples of the use of simplifying transformations
2.1. The problem of optimal heat exchange
Consider the problem of heat exchange of two bodies: a heat body having the initial temperature T+0 and a cold body with the initial temperature T-0 < T+0. The heat flow q(T+, T-) does not change its sign and is directed from the hot body to the cold body (the second law of thermodynamics [11]). It is desirable to carry out the process in such a way that the increment of the entropy of system S for a given duration of the process t and for a given amount of the transferred heat Q is minimal. Such a setting corresponds to the minimum of losses of the efficient energy (exergy).
The equations of motion are
dT+ q(T+, T_) dT q(T+, T ) , x
+ - T+(0)=T+o, = T+; T_(0)=T_o. (2.1)
dt W+ ' +w +u' dt W-
The amount of heat Q transmitted by time t satisfies the equation
dQ
dt
= q(T+,T_), Q( 0) = 0, Q(t)=Q.
(2.2)
The increment of entropy is
T
S = j q(T+, T_) -l^jdt^ min.
(2.3)
W+, W- are the heat capacities of the bodies.
Since the right-hand sides of Eqs. (2.1) and (2.2) do not change sign and time is not involved explicitly in these equations nor in the integrand for S, it follows that any of the three phase coordinates can be used as an independent variable. For this, it is convenient to choose the variable Q.
Let us make the transformation dt = ^. After this transformation the problem takes
the form
dT+ dQ
tq=~W;< T+<°> = r+»-
whence
T+(Q) = T+o -
Q_
dT_ = q(T+, T_) dQ W_ '
T_ (Q) = T_ o +
T_ (0)= ^ o,
+
Q_
w_
Hence, the problem (2.1), (2.2), (2.3) can be written as
(2.4)
Q
S= [Q(T+O-^-,T_0 + 3-
W+
+
W
rp , Q
—0 ' w_
1
T__21 +o w+
dQ ^ min
W+,W_
(2.5)
under the condition
Q
dQ
<7(^+0 o +
W
= t.
(2.6)
Thus, after the simplifying transformation the problem reduces to the problem with one integral restriction. In this case, the type of the simplified problem does not depend on the kinetics of heat exchange q, which for the radiant heat exchange has the form q = k(T+ - T4), for the Newtonian heat exchange the form q = k(T+ -T_), for the Fourier heat exchange the form q = k(1/T_ -T+), etc.
1
0
0
2.2. Parametric braking of the oscillator
The problem of braking/swinging of a pendulum has been studied many times for the case where control corresponds to the force applied to the pendulum. In this case, the control is involved on the right-hand side of the equation of motion additively. In the case of parametric control it is involved in the equation of motion multiplicatively.
Consider the problem of optimal operation speed for a controlled system of the form
q = p, p = -uq, 0 <u1 ^ u ^ u2, (2.7)
where q is the deflection of the pendulum, p is its velocity, and the coefficient u corresponds to the square of the frequency of self-oscillations of the pendulum, it depends on the mass and the distance from the suspension point to the center of gravity. The latter can be controlled as each person does when rocking a swing.
Let the initial state of the system p0, q0 and the value of the total energy be given at the final time t:
p2(T)+u0q2(r)=E. (2.8)
The value of u0 is fixed and it can be taken to be equal to unity without loss of generality, and the restrictions are u1 ^ 1, u2 ^ 1. It is required to find a law of control u*(t) for which
t ^ min . (2.9)
u
The restriction (2.8) can be rewritten in the integral form
T
j V2 + UoQ2) dt = j pq(u0 - u) dt = (2.10)
0 0
Since the initial state is given and hence the initial value of energy is E0 = p2 + u0q°, it
_
follows that the problem corresponds to extraction of the maximal power N* = On the
phase plane, with u constant, the trajectories form a phase portrait of center type (a system of ellipses), the change of the phase state in time corresponds to clockwise motion where e(t) = = p2(t) + uq2(t) = const. Time is not explicitly involved on the right-hand sides of Eqs. (2.7), but the sign of the right-hand sides changes, and so it is impossible to decrease the dimension of the problem by rescaling the time of one of the state variables.
The solution of the problem (2.7), (2.9), (2.10) using the maximum principle [9] requires solving the boundary-value problem for a system of four equations (two of them corresponding to conjugate variables). To these equations one needs to add transversality conditions, find a singular solution or prove its absence.
Transformation to new state variables allows the solution to be greatly simplified. Let us introduce new variables z(p, q) and e(p, q) so that, by virtue of Eqs. (2.7), the rate of change of one of them does not change sign. This will allow the dimension of the problem to be decreased. It is also desirable that one of the phase variables be not involved on the right-hand sides of Eqs. (2.7); this will make it possible to replace the condition in the form of differential equations by an integral equality.
Choose new variables as follows:
z = ~> e = ln{p2 + q2). (2.11)
By virtue of Eqs. (2.7), the rate of change of these variables is
22
. qp - pq p2 + uq2 o p0 . .
z =-5— =-5-= z + u, z0 = —(2-12)
q2 q2 0 q0
. 2(pp + qq) 0 -pqu + qp z{u - 1)
6 p* + q2 q\l + z2) l + z2 ' 1 j
e0 = ln(.E'o), e = In E.
The right-hand side of Eq. (2.12) is positive for all admissible z and u and hence the variable z can be used as an argument instead of t by decreasing the dimension of the problem by unity. The variable e is not involved on the right-hand sides of Eqs. (2.12) and (2.13), and the values of this variable are given at the initial and final time, which allows Eq. (2.13) to be replaced by an integral restriction.
It follows from conditions (2.12) and (2.13) that
dz 2z(u — 1)dz
at = -, de =
z2 + u' (z2 + u)(1 + z2):
so that the problem takes the form
dz
z2 + u u,z
T = I \--> min (2.14)
under the condition
f z(u — 1) dz 1 . 1 , E ,
¿0
and under restrictions on the control.
Thus, the problem has turned out to be transformed to a much simpler one, containing, instead of two differential equations, one integral restriction. The conditions for its optimality [10] are expressed in terms of the Lagrange functional
5 = A0t + AJ (2.16)
and its integrand
1 » z(u — 1) ,
L = A0^—+ A 1 1 2.17
z2 + u (z2 + u)(1 + z2)
For the problem with an integral functional and an integral restriction, the following statement follows immediately from Pontryagin's maximum principle: Let u*(z) be the optimal solution of the problem (2.14)-(2.15). Then there are values of A and A0, not equal to zero simultaneously, such that, for each z, u*(z) furnishes the minimum to the function L.
In the general case, in such problems there may exist singular solutions. To them there correspond solutions on which the value of the functional J is extreme. For the singular solution in the Lagrange functional one takes A0 = 0. In this case there is no singular solution, since the derivative of the second term with respect to u is equal to A^r^p-, it does not change sign, so that J has no extremum. Therefore, the multiplier A0 can be taken to be equal to unity. From the condition of the minimum L it follows that:
a) the optimal control exhibits switches ; the switching lines on the plane p, q are the ordinate axis and the line with inclination zr = — ^ which passes through the second and fourth quadrants of this plane ;
b) the Lagrange multiplier A in the expressions (2.16) and (2.17) is positive.
c) since the optimal control is unique, the necessary optimality conditions are sufficient.
z
0
The zones in which the optimal control is constant are shown in Fig. 1. In [3], the authors assumed without any grounds that zr =0 and the switching lines coincide with the coordinate
In order to find J and A (or zr, which is the same), we have the condition for stationarity of the functional S and condition (2.15).
2.3. Extraction of work from the system of quantum oscillators
This problem of optimization quantum thermodynamics concerning the laser cooling of bodies with crystal structure is formulated and solved numerically in the paper [5] using the package of analytical transformations, Maple, for a particular case of specification of initial conditions. In the same paper, the physical meaning of the main variables and the conditions imposed on them is examined in detail. We show that the transformation of the state space simplifies the problem and makes it possible to obtain the structure of the optimal solution, and the control is obtained as a function of the phase coordinate z.
Initial problem statement: Drive, in minimal time, the system characterized by the equations
' E = u(E - L), L = -u(E - L) - 2wC, < C = uC + 2wL, (2.18)
(j = uw, w1 ^ w ^ u2, ,E{t)=E
from a given initial state to a given final state. Here E, L, C, w are state variables having the meaning of the Hamiltonian, Lagrangian, moment-deviation correlation and the oscillation frequency averaged over the ensemble of oscillators, with E = E{t) < E0. The system (2.18) has a first integral X that does not change in time along its trajectories:
(2.19)
w2 w2
Physically the quantity X defines the von Neumann entropy SN, which monotonically depends on it:
l' / — x
SN = In x ~ ^ J + v X arg sinh ^y^j J '
The constancy of X suggests that the process of extraction of energy from the system by changing the frequency of oscillations is adiabatically reversible. The existence of the minimal time corresponding to an adiabatic transition from one energetic level to another is equivalent to the statement that, in a time smaller than this minimum, this energy can be extracted only in an irreversible process accompanied by the so-called "quantum friction".
The initial values of all state variables, and hence the quantity X, are given. The initial problem contains an unbounded, linearly involved control u, and its state variables are related by condition (2.19). In problems with unbounded, linearly involved controls the solution can contain delta functions. This circumstance, as well as the relation between the state variables, makes impossible the use of the maximum principle in its traditional form for the solution.
The variables E, L, C depend on the oscillation frequency w, deviations from the equilibrium state qi and the velocities pi of the oscillators as
j2
(P2 = £ p2 ; Q2 = £
<
E = P2 + 0.5w2 Q2, (2.20)
L = P2 - 0.5w2Q2, k C = 0.5w(QP + PQ).
In order to exclude the dependence of the velocities of the phase variables on the unbounded control u, we transform the space of states by transforming to new variables
E - L C
z1=E + L, z2 =-—, % = (2.21)
w2 w
The choice of these variables is made in such a way that in expressions of the form (1.3) the sum of terms depending on u, by virtue of Eqs. (2.18), vanishes.
The initial variables and the value of X are related to the state variables in the transformed problem by
C = wz3,
E = 0.5(zi + w2 z2),
2 (2.22) L = 0.5(zi — w z2),
- X = zi zi — z3.
Since the value of X is given, only two variables are independent.
Let S denote the initial state of the system on the plane z1, z2. This initial state lies above the hyperbola z1 z2 = X. Eliminating the variable z3 via X, z1, z2 and rewriting the equations for z1 and z2 by virtue of the system (2.18) taking (2.21) and (2.22) into account, we obtain
(z1 = -2u z3 = Vzizi " (2 23)
\z2 = 2 z3 = ±2x/z1z1 -X.
Since u is not involved in these equations, we will consider w2 > 0 as control and decrease the number of state variables to two, and the number of controls to one.
Time is not explicitly involved on the right-hand sides of Eqs. (2.23), which makes it possible to simplify the system further by taking the variable z2 as an argument. Also, along the trajectories
^ = -w2. (2.24)
dz2
For a constant value of w the trajectories on the plane
Z\, Z2 are straight lines with a negative
inclination.
Changes in the frequency w can occur instantly, therefore, the minimal duration of the process does not depend on the value of the initial frequency. We will assume that the initial frequency is w0, but it can be instantly changed to w1 with a decrease in energy. The same is true of the final value of frequency, here the minimum of energy always corresponds to the frequency equal to w1.
The initial conditions for the variables z1 and z2 are given and are equal to
zw = E0 + L0, z20 = ° 2 (2.25)
wo
The final values satisfy the equation
zi + ufzZ = 2E. (2.26)
The value of E is bounded from below by virtue of the inequality z1z2 ^ X. The lower limit is achieved at the point of contact of the hyperbola z1 z2 = X and the straight line defined by Eq. (2.26). For any fixed value z3, and hence the product z1z2, the points corresponding to the minimum of energy lie on the straight line z1 = w2z2. The lower boundary of the attainable energy is reached at z3 = 0 and is equal to
^min = UiVX.
Values of energy smaller than Emin are unattainable from. any initial state. That is to say, there exists a minimal temperature that can be achieved in the adiabatic process. The duration of the transition from the initial state to the final state
dz2
z1z2 — X
— min (2.27)
depends on the trajectory z1(z2).
The conditions of the maximum principle for the transformed problem (2.27), (2.24), (2.26) under the assumption of nondegeneracy of the solution (é0 = —1) are as follows: The Hamiltonian function is
H = ~o / ? v - (2-28)
2v z1 Z2 — A
The optimality conditions are:
d^ dH
dz2 dz1 4(z1 z2 — X )3/2
< 0, (2.29)
for é < 0,
w2*(z2) = arg max H =J ^ (2.30)
I w2 for é > 0.
2
2
20
z
2
Since the function ) decreases monotonically and, by virtue of the condition imposed on the final value z1, does not vanish at the end of the process, it follows that in the course of the optimal process the frequency can change once from w1 to w2 at any final energy smaller than E0. Let R denote the switching point.
Fig. 2. The region of attainable values of the state variables and the optimal solution
Thus, on the optimal solution the dependence of z1 on z2 is linear up to the switching point R with minimal inclination, and after the switching point, with maximal inclination. For each of the segments, the integral (2.27) can be calculated.
In Fig. 2, S and F denote the initial and the final state, respectively. The sets that are attainable from the initial state and ensure that the final state is reached are bounded by the straight lines emanating from the initial and final states with inclinations corresponding to the extreme frequency values. The optimal process corresponds to the trajectory passing along the boundaries of the sets of attainability or through the switching point R shown in the figure, or through the switching point lying at the opposite vertex of the parallelogram. At point F the frequency changes instantly to w1. The tangent line drawn to the hyperbola z1z2 = X and having an inclination corresponding to u1 cuts off on the ordinate axis a segment equal to the doubled minimum of the attainable energy.
Thus, the coordinates of the switching point R are defined by the equations
Z1R = Z10 — ~~ z2o)) Z1R = Z1 + wi(z2 ~~ Z2R)i (2-31)
„ ï _ ^22z2-uj'21z20-z10 + z1 _ ^ 2{E0 - Ë)
W2 — W1 W2 — W1
After substitution of the optimal solution into (2.27), the value of the smallest possible time of cooling in the reversible process t* is equal to
t* = 0.5
dz2 f dz2
j y/{2E0 - ujjz2)z2 - X J J(2E - uj22z2)z2 - X
20 z2R\ 222
It depends on ~z2 and given values of energy E at the beginning and at the end of the process.
(2.33)
3. Conclusion
This paper lists the aims and gives examples of transformations of the phase space of optimal control problems which allow its dimension to be decreased. It presents a generalization of Krotov's transformation proposed by him for the Euler problem, linear in the rate of change of the desired variable, to include vector problems of optimal control in which the rates of all or part of the state variables are linear in control. The paper also gives examples of solving three applied problems demonstrating the efficiency of simplifying transformations and confirming that the solution of the optimal control problem must be preceded by an attempt to simplify it.
Acknowledgments
The author would like to thank Professor P. Salamon (University of San Diego) for formulating the oscillator control problem and discussing the results of its solution.
References
[1] Gurman, V.I., The Expansion Principle in Control Tasks, 2nd ed., Moskow: Fizmatlit, 1997 (Russian).
[2] Tsirlin, A. M., Mathematical Models and Optimal Processes in Macrosystems, Moscow: Nauka, 2003 (Russian).
[3] Piccoli, B. and Kulkarni, J., Pumping a Swing by Standing and Squatting: Do Children Pump Time Optimally?, IEEE Contr. Syst. Mag., 2005, vol. 25, no. 4, pp. 48-56.
[4] Andresen, B., Salamon, P., Hoffmann, K.H., and Tsirlin, A.M., Optimal Processes for Controllable Oscillators, Autom. Remote Control, 2018, vol. 79, no. 12, pp. 2103-2113.
[5] Salamon, P., Hoffman, K.H., Rezek, Y., and Kosloff, R., Maximum Work in Minimum Time from a Conservative Quantum System, Phys. Chem. Chem. Phys., 2009, vol. 11, no. 7, pp. 10271032.
[6] Tsirlin, A. M., Optimization Methods in Irreversible Thermodynamics, Moscow: Urait, 2019 (Russian).
[7] Krotov, V.F., Discontinuous Solutions of Variational Problems: 1, Izv. Vyssh. Uchebn. Zaved. Matem., 1960, no. 5, pp. 86-98 (Russian).
Krotov, V.F., Discontinuous Solutions of Variational Problems: 2, Izv. Vyssh. Uchebn. Zaved. Matem., 1961, no. 2, pp. 75-89 (Russian).
[8] Krotov, V.F., Bukreev, V. Z., and Gurman, V. I., New Variational Methods in Flight Dynamics, Moscow: Mashinostroenie, 1969 (Russian).
[9] Pontryagin, L. S., Boltyanskii, V. G., Gamkrelidze, R. V., and Mishchenko, E.F., The Mathematical Theory of Optimal Processes, L. W. Neustadt (Ed.), New York: Wiley, 1962.
[10] Alekseev, V.M., Tikhomirov, V.M., and Fomin, S.V., Optimal Control, Moscow: Nauka, 1979 (Russian).
[11] Kondepudi, D. and Prigogine, I., Modern Thermodynamics: From Heat Engines to Dissipative Structures, 2nd ed., New York: Wiley, 2014.
