ISSN 2079-3316 PROGRAM SYSTEMS: THEORY AND APPLICATIONS vol. 11, No 2(45), pp. 61–73
CSCSTI 27.37.17,28.15.19,89.23.41
UDC 517.977
Irina V. Rasina, Oles V. Fesko
Sufficient relative minimum conditions
for discrete-continuous control systems
Abstract. In this paper, we derive sufficient relative minimum conditions for
discrete-continuous control systems on the base of Krotov’s sufficient optimality
conditions counterpart. These conditions can be used as verification conditions for
suggested control mode and enable one to construct new numerical methods.
Key words and phrases: heterogeneous systems, optimal control problem, local extremum.
2010 Mathematics Subject Classification: 49M30; 49N10
Introduction
Hybrid systems, which include systems with variable structure [1],
discrete-continuous [2], logic-dynamic [3,4], impulsive [5], and some other
systems, firmly took their place in the theory of optimal control. They are
also sometimes referred to as “heterogeneous”. A considerable part of
scientific conferences and journals are devoted to that field. They propose
corresponding mathematical models and methods of investigation for each
type of such systems.
We consider a class of optimal control problems, which is characterized
by the change of descriptions with time in terms of controlled differential
systems. For this class, a two-level model of a discrete-continuous system
(DCS) is proposed in [2,6–8]. Its lower level describes the continuous
uniform controllable processes at the individual stages. The upper (discrete)
level integrates these descriptions into a unique process and controls the
functioning of the entire system as a whole to ensure the minimum of the
functional.
⃝c I. V. Rasina, O. V. Fesko, 2020
⃝c Program Systems Institute of the Russian Academy of Sciences, 2020
⃝c Program Systems: Theory and Applications (design), 2020 CC-BY-4.0
62 Irina V. Rasina, Oles V. Fesko
The sufficient optimality conditions for the model were established and
control improvement methods were developed [6–8]. These conditions
allow one to find the global minimum of the functional of optimal control
problem. However, in practice, the admissible control set may have a
complex structure with specific state constraints. Therefore, most of the
computational methods suggested in the literature yield to a relative
minimum of the functional. However, it is impossible to verify that the
suggested solution provides a relative minimum.
The purpose of this work is to fill the gap. Namely, we derive sufficient
relative minimum conditions that can be used for the evaluation of the
proposed solution and enable one to construct new numerical methods.
1. Model of the discrete-continuous system
We consider the abstract discrete controllable system [9]:
x(k + 1) = f
(︁
k, x(k), u(k)
)︁
, k (1) ∈ K = {kI , kI + 1, . . . , kF },
where k is the step (stage) number, time is not necessarily physical, x and
u are variables of state and control, respectively, f is an operator. All
these objects have an arbitrary (possibly, different) nature for different k,
U(k, x) is a set given for each k and x, kI , kF are the initial and final steps,
respectively. On some subset K′ ⊂ K, kF /∈ K′, there is a continuous
system of the lower level
x˙ c =
dxc
dt
(2) = fc(z, t, xc, uc), t ∈ T(z) = [tI (z), tF (z)],
xc ∈ Xc(z, t) ⊂ Rn(k), uc ∈ Uc(z, t, xc) ⊂ Rp(k), z = (k, x, ud).
Here Uc(z, t, xc) is the given set.
The right-hand side operator (1) is given by
f(k, x, u) = θ(z, γc),
where
γc = (tI , xcI
, tF , xc
F ) ∈ Γc(z),
Γc(z) =
{︁
γc : tI = τ (z), xcI
= ξ(z), (tF , xc
F ) ∈ Γc
F (z)
}︁
.
Relative minimum for DCS 63
Here z = (k, x, ud) is a set of the upper-level variables playing the role
of lower-level parameters, ud is a control variable of arbitrary nature,
tI = τ (z), xcI
= ξ(z) are given functions of z.
The solution of this two-level system is the set m =
(︁
x(k), u(k)
)︁
called
the discrete-continuous process. For each k ∈ K′, u(k) =
(︁
ud(k),mc(k)
)︁
,
where mc(k) ∈ Dc
(︁
z(k)
)︁
is a continuous process
(︁
xc(k, t), uc(k, t)
)︁
, t ∈
T
(︁
z(k)
)︁
. Here Dc(z) is the set of admissible processes mc, satisfying with
the specified differential system (2) with additional restrictions. It is
supposed that uc(k, t) is piecewise continuous and xc(k, t) is piecewise
smooth (at each discrete step k). Let us denote the set of elements m
satisfying all the above conditions by D and call it a set of admissible
discrete-continuous processes.
For the model (1), (2), we consider the problem of finding the minimum
on D of the functional I = F(x(kF )) under fixed kI = 0, kF = K, x(kI )
and additional constraints
x((3) k) ∈ X(k), xc ∈ Xc(z, t),
X(k), Xc(z, t) are the given sets.
The model (1), (2) is suitable to describe various nonuniform controllable
processes. Its lower level describes the continuous controllable
processes at the individual stages. The upper one integrates these descriptions
into a unique process. In the problems of optimization, both levels
are interacting. Interaction with each subsystem of the lower level is
through the boundary of this subsystem and the corresponding continuous
process γc.
2. Control improvement and optimality sufficient conditions
The sufficient optimality conditions for this model we derive by
analogy with Krotov’s sufficient conditions for discrete and continuous
systems by eliminating the discrete chain and the differential
system from the constraints of the sets D and Dc, scalar functions
(functionals) φ(k, x), φc(z, t, xc) are introduced. The latter is the parametric
family (with the parameter z) of the smooth functions, where
z =
(︂
k, x(k), ud(k)
)︂
. The following generalized Lagrangian is constructed:
64 Irina V. Rasina, Oles V. Fesko
L = G
(︁
x(kF )
)︁
−
Σ︂
K\K′\kF
R
(︁
k, x(k), u(k)
)︁
+
+
Σ︂
K′
(︃
Gc
(︂
z(k), γc (︁
z(k)
)︁)︂
−
−
∫︂
T(z(k))
Rc (︁
z(k), t, xc(k, t), uc(k, t)
)︁
dt
)︃
,
G(x) = F(x) + φ(kF , x) − φ
(︁
kI , x(kI )
)︁
,
R(k, x, u) = φ
(︁
k + 1, f(k, x, u)
)︁
− φ(k, x),
Gc(z, γc) = −φ
(︁
k + 1, θ(z, γc)
)︁
+ φ(k, x) + φc(z, tF , xc
F )−
− φc(z, tI , xcI
),
Rc(z, t, xc, uc) = φcT
xc fc(z, t, xc, uc) + φct
(z, t, xc),
μc(z, t) = sup
{︁
Rc(z, t, xc, uc) : xc ∈ Xc(z, t), uc ∈ Uc(z, t, xc)
}︁
,
lc(z) = inf
{︁
Gc(z, γc) : γc ∈ Γ(z), xc ∈ Xc(z, tF )
}︁
,
μ(k) =
sup
{︁
R(k, x, u) : x ∈ X(k), u ∈ U(k, x)
}︁
, t ∈ K\K′,
−inf
{︂
lc(z) : x ∈ X(k), ud ∈ Ud(k, x)
}︂
, k ∈ K′,
l = inf{G(x) : x ∈ Γ ∩ X(K)}.
Here φc
xc is the gradient of φc in the space (xc), T denotes transposition.
Theorem 1. Let there be a sequence of discrete-continuous processes
{ms} ⊂ D and functionals φ, φc such that
1) μc(z, t) is piecewise continuous for each z;
2) R
(︁
k, xs(k), us(k)
)︁
→ μ(k), k ∈ K;
3)
∫︁
T(zs)
(︂
Rc
(︁
zs, t, xc
s(t), uc
s(tc)
)︁
− μc(zs, t)
)︂
dt → 0, k ∈ K′ and t ∈
T(zs);
4) Gc(zs, γc
s) − lc(zs) → 0, k ∈ K′;
5) G
(︁
xs(tF )
)︁
→ l.
Then the sequence {ms} is a minimizing sequence for I on D.
The proof is given in [6,8].
Relative minimum for DCS 65
3. Relative minimum
Suppose that xc(k, tI ) = ξ
(︁
k, x(k)
)︁
, kI , kF , x(kI ), tI (k), tF (k) are
fixed, there are no constraints for state variables of both levels and
upper-level control variables, lower-level subsystems do not depend
on ud, X(k) = Rd(k), Xc(k, t) = Rp(k),U(k, x) = Rr(k), and the used
constructions of sufficient optimality conditions are such that all the
following operations are valid. Let ¯x(k) and ¯xc(k, t) be elements of D,
and there exists, at least one value u(k) and uc(k, t) with corresponding
¯x(k + 1), ¯xc(k, t), i.e., ¯u(k), ¯uc(k, t) are inner points of U,Uc.
Let us denote by Dϵ a subset of elements of D that satisfy additional
conditions
|x(k) − ¯x(k)| < ϵ, |xc(k, t) − ¯xc(k, t)| < ϵ, ϵ > 0.
We will say that on a discrete-continuous process ¯m a relative minimum
of I is attained on D if I( ¯m) = inf
Dϵ
I.
On the basis of the above-mentioned theorem, the sufficient conditions
for the relative minimum of the functions R, Rc are as follows
(4) dR = 0, dPc = 0,
(5) d2R < 0, d2Pc < 0,
where Pc(k, t, xc) = sup
uc∈Uc
Rc(k, t, xc, uc).
We can make these conditions more detailed for the problem with a
free terminal state, i.e., x(kI ) is fixed and x(kF ) is free, x(kF ) ∈ Rd(k). In
addition, the set Γc(z) is given by Γc(z) = {γc : xcI
= ξ(z), xc
F ∈ Rp(k)}.
It follows from (4) that
Rx = 0, Ru = 0, Pc
xc = 0, Pc
(6) x = 0,
where dR = RT
xΔx + RT
uΔu. The derivatives are calculated on the element
¯m.
From (6) one has
ψ(k) = Hx, Hu = 0,
H(k, ψ, x, u) = ψT (k + 1)f(k, x, u), k ∈ K\K′,
(7)
66 Irina V. Rasina, Oles V. Fesko
˙ψ
c(k, t) = −Hcx
c , ˙λ(k, t) = Hcx
,
H(k, ψ, xcI
, xc
F ) = ψT (k + 1)θ(k, xcI
, xc
F ), k ∈ K′ \ kF ,
(8)
Hc(k, t, x, xc, uc) = ψcT(k, t)fc(k, t, xc, uc), Hc = sup
uc
Hc.
The condition d2R < 0 or
d2R = ΔxTRxxΔx + 2ΔxTRxuΔu + ΔuTRuuΔu < 0
for k ∈ K\K′, as in [10], can be replaced by the equivalent conditions
max
Δu
(9) d2R < 0,
(10) ΔuTRuuΔu < 0.
Assuming that the maximum is reached in a stationary point, we
have RT
xuΔx + RuuΔu = 0, from which we get Δu = −R−1
uuRxuΔx. Then
max
Δu
d2R = ΔxT (Rxx − RxuR−1
uuRT
xu)Δx.
Let us take into consideration negative definite matrix Θ(k); with
regard to Rxx, Rxu, Ruu the condition (6) can be rewritten in the form:
σ(k) = fT
x σ(k + 1)fx + Hxx − HxuH−1
(11) uu Hux − Θ(k),
where σ(k) = φxx(k, x). Let us now consider the following condition
d2Pc = ΔxcTPc
xcxcΔxc + 2ΔxcTPxcxΔx + ΔxTPxxΔx < 0.
The matrix of second derivatives is:
M =
Pc
xcxc Pc
xcx
Pc
xxc Pc
xx
.
We introduce the matrix
Θ∗ =
(︄
Θ2 0
0 Θ3
)︄
,
where matrices Θ2, Θ3 are negative definite. Then we can represent the
Relative minimum for DCS 67
condition d2Pc < 0 in the form M = Θ∗. Hence it follows that
˙ σc = −Hcx
cxc − Hcx
cψcσc − σcHcT
xcψc − σcHcψ
(12) cψcσc + Θ2(k, t),
˙β
= −Hc
xx (13) − ωHψcx − Hxψcω − ωHψcψcω + Θ3(k, t),
ω˙ = −Hc
xxc − Hcx
cψcω − σcHcψ
cx − σcHcψ
(14) cψω.
It is easy to see that on the set K′
σ(k) = θT
x σ(k + 1)θx + Hxx + ξT
x θxcI
σ(k + 1)θx+
+ θT
x σ(k + 1)θxcI
ξx + ξT
x θT
xcI
σ(k + 1)θxcI
(15) ξx+
+ ξT
x σc(k, tI )ξx + ξxωT (k, tI ) + ω(k, tI ).
Here σc(k, t) = φc
xcxc(k, t, x, xc), β(k, t) = φc
xx(k, t, x, xc), ω(k, t) =
= φc
xxc (k, t, x, xc).
Consider now first and second-order minimum conditions for G,
Gc. We denote by Γ the subset of elements of the set D such that
|x(kF ) − ¯x(kF )| < ϵ, ϵ > 0 and by Γc the subset of elements of the set
D such that |x(kF ) − ¯x(kF )| < ϵ, |xc(k, tF ) − ¯xc(k, tF )| < ϵ, ϵ > 0. By
virtue of Theorem 1, the sufficient conditions for relative minimum of the
functions G, Gc on the sets Γ, Γc are as follows:
(16) dG = 0, dGc = 0,
d2G > 0, d2Gc > 0.
It follows from condition (16) that
GxF = 0, Gc
x = 0, Gc
xc
F
= 0.
Then
ψ(kF ) = −FxF , ψc(k, tF ) = −Hxc
F
(17) , λ(k, tF ) = 0.
From condition d2G > 0 for k ∈ K\K′ we have σ(kF ) = −FxF xF +Θ1,
where the matrix Θ1 is positive definite.
On the other hand, in analogy with the peceding condition d2Pc < 0,
the condition d2Gc > 0 can be represented in the form
σc(k, tF ) = θT
xc
F
σ(k + 1)θxc
F
+ Hxc
F xc
F
,
ω(k, tF ) = 0, β(k, tF ) = 0.
68 Irina V. Rasina, Oles V. Fesko
As before, all the derivatives are evaluated on element ¯m.
Theorem 2. In order for element ¯m to be a relative minimum of I on
D, it is sufficient that there exist vector functions ψ, ψc, λ, matrices σ,
σc, β, ω, and negative definite matrices Θ(k), −Θ1(k), Θ2(k, t), Θ3(k, t),
which satisfy conditions (7), (8), (11)–(15).
Proof. We define the functions φ, φc in the form
φ(k, x) = ψT (k)x +
1
2
(x − ¯x)T σ(k)(x − ¯x),
φc(z, t, xc) = λT (k, t)x + ψcT(k, t)xc+
+
1
2
(xc − ¯xc)T σc(k, t)(xc − ¯xc)+
+
1
2
(x − ¯x)T β(k, t)(x − ¯x) + (x − ¯x)T ω(k, t)(xc − ¯xc),
where ψ, ψc, λ, σ, σc, β, ω satisfy the theorem conditions. Fulfillment of
these conditions implies that the functions R, Rc, G, Gc attain a relative
extremum on element ¯m and at points ¯x(kF ), ¯xc(k, tF ). This means that
there exists a number ϵ > 0 such that the functions R, Rc attain extremum
on the set |x(k)−¯x(k)| < ϵ, |xc(k, t)−¯xc(k, t)| < ϵ, and the functions G, Gc
attain extremum on the set |x(kF )− ¯x(kF )| < ϵ, |xc(k, tF )− ¯xc(k, tF )| < ϵ.
Hence by virtue of Theorem 1, it follows that the functional I attains its
minimum on the element ¯m on the set Dϵ, i.e., a relative minimum. □
Thus Theorem 2 asserts that if in a neighborhood of ¯m of radius ϵ
there exist solutions to Eqs. (7), (8), (11)–(15), then ¯m provides a relative
minimum of the functional I .
4. Example
Consider the following two-stage system
1-st stage
x˙ c
1 = (xc
1)2(xc
2 − u)2, x˙ c
2 = xc
1xc
2 +
1
3
u3,
xc
1(0) = −1, xc
2(0) = −1, t ∈ [0, 2].
2-nd stage
x˙ c
1 = (xc
1 − t)2 + u2, t ∈ [2, 3].
Relative minimum for DCS 69
(a) xc
1
(b) xc
2
Figure 1. State variables
Table 1. Results of a first-order method
Iteration Functional I
0 1.98
1 1.85
The functional is I = xc
1(3) → min.
Construct the DCS system. It is easy to see that K = 0, 1, 2. Since xc
1
is a linking variable in the two periods under consideration, we can write a
process of the upper level in terms of this variable
x(0) = xc
1(0, 0) = −1, x(1) = xc
1(0, 2), x(2) = xc
1(1, 3),
I = x(2), xc
1(1, 2) = x(1), ξ = x(1).
The basic constructions have the form:
Hc(0, t, xc
1, xc
2, u, ψc1
, ψc2
) = ψc1
(︂
(xc
1)2(xc
2 − u)2
)︂
+ ψc2
(︃
xc
1xc
2 +
1
3
u3
)︃
,
Hc(1, t, xc
1, u, ψc1
) = ψc1
(︂
(xc
1 − t)2 + u2
)︂
.
In [11], we have found the solution using the first-order method with
initial uI = 0.5, II = 1.98. The results are given in Figires 1, 2 and Table 1.
70 Irina V. Rasina, Oles V. Fesko
Figure 2. Control u
(a) ψ1 (b) ψ2
Figure 3. Vector ψc
Let us verify that the solution obtained provides a relative minimum of
the functional I. Since the solution is numerical, the function Hc and
its first and second derivatives are also numerical. By virtue of the fact
that the equations of every stage do not depend on the variable x of the
upper level, we have λ = 0, β = 0, ω = 0. It is easy to see that at stage 1
ψcT = (ψc1
, ψc2)T and
σc =
(︄
σc
11 σc
12
σc
21 σc
22
)︄
.
The results displayed in Figures 3–4 show that the system of vectorRelative
minimum for DCS 71
(a) σ11 (b) σ12 = σ21
(c) σ22
Figure 4. Matrix σc
matrix differential equations for ψc, σc has the solution in both stages.
Therefore, ¯m provides a relative minimum of the functional.
Conclusions
In this paper, we derive sufficient relative minimum conditions for
discrete-continuous systems. It allows us to verify that the proposed
solution of the optimal control problem provides a local minimum for the
functional. An illustrative example is provided.
72 Irina V. Rasina, Oles V. Fesko
References
[1] S.V. Emelyanov. Theory of Systems with Variable Structures, Nauka,
Moscow, 1970 (Russian). ↑61
[2] V.I. Gurman. “Theory of Optimum Discrete Processes”, Autom. Remote
Control, 34:7 (1973), pp. 1082–1087 (English). ↑61
[3] S.N. Vassilyev. “Theory and Application of Logic-Based Controlled Systems”,
Proceedings of the International Conference Identification and Control
Problems, Moscow, Institute of control sciences, 2003, pp. 53–58 (Russian).
↑61
[4] A.S. Bortakovskii. “Sufficient Optimality Conditions for Control of Deterministic
Logical-Dynamic Systems”, Informatika, Ser. Computer Aided Design,
1992, no. 2-3, pp. 72–79 (Russian). ↑61
[5] B.M. Miller, E.Ya. Rubinovich. Optimization of the Dynamic Systems with
Pulse Controls, Nauka, Moscow, 2005 (Russian). ↑61
[6] V.I. Gurman, I.V. Rasina. “Discrete-Continuous Representations of Impulsive
Processes in the Controllable Systems”, Autom. Remote Control, 73:8
(2012), pp. 1290–1300 (English). ↑61,62,64
[7] I.V. Rasina. “Iterative Optimization Algorithms for Discrete-Continuous
Processes”, Autom. Remote Control, 73:10 (2012), pp. 1591–1603 (English).
↑61,62
[8] I.V. Rasina. Hierarchical Control Models for Systems with Inhomogeneous
Structures, Fizmatlit, Moscow, 2014 (Russian). ↑61,62,64
[9] V.F. Krotov. “Sufficient Optimality Conditions for the Discrete Controllable
Systems”, Dokl. Akad. Nauk SSSR, 172:1 (1967), pp. 18–21 (Russian). ↑62
[10] V.I. Gurman. Degenerate Problems of Optimal Control, Nauka, Moscow,
1977 (Russian). ↑66
[11] I.V. Rasina, O.V.Fesko. “First order control improvement method for discrete
continuous systems”, Program Systems: Theory and Applications, 9:38
(2018), pp. 65–76 (Russian). hUtRtpL::/D/psotai.p:sir1as0.ru./2re5ad2/p0st9a2/0128_037_695-–736.3pd1f 6-2018-9-3-65-76 ↑69
Received 15.03.2020
Revised 09.04.2020
Published 10.05.2020
Recommended by prof. A. M. Tsirlin
Relative minimum for DCS 73
Sample citation of this publication:
Irina V. Rasina, Oles V. Fesko. “Sufficient relative minimum conditions
for discrete-continuous control systems”. Program Systems: Theory and
Applications, 2020, 11:2(45), pp. 61–73.
DOI: 10.25209/2079-3316-2020-11-2-61-73
URL: http://psta.psiras.ru/read/psta2020_2_61-73.pdf
The same article in Russian: DOI: 10.25209/2079-3316-2020-11-2-51-64
About the authors:
Irina Viktorovna Rasina
Professor, Chief Researcher of the Research Center for System
Analysis in the Ailamazyan Program Systems Institute of
RAS. Specialist in the field of modeling and control of hybrid
systems, author and co-author of more than 100 articles and
5 monographs
iD: 0000-0001-8939-2968
e-mail: irinarasina@gmail.com
Oles Vladimirovich Fesko
Scientific Researcher of the Research Center for System
Analysis in the Ailamazyan Program Systems Institute of
RAS. Specialist in numerical experiments in mathematical
control theory
iD: 0000-0002-9329-5754
e-mail: oles.fesko@hotmail.com
Эта же статья по-русски: DOI: 10.25209/2079-3316-2020-11-2-51-64