SUFFICIENT RELATIVE MINIMUM CONDITIONS FOR DISCRETE-CONTINUOUS CONTROL SYSTEMS

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.

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

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.

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

(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),

γc = (tI , xcI

, tF , xc

F ) ∈ Γc(z),

Γc(z) =

γc : tI = τ (z), xcI

= ξ(z), (tF , xc

F ) ∈ Γc

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)

the discrete-continuous process. For each k ∈ K′, u(k) =

ud(k),mc(k)

where mc(k) ∈ Dc

is a continuous process

xc(k, t), uc(k, t)

. 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.

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

k, x(k), ud(k)

. The following generalized Lagrangian is constructed:

K\\K′\\kF

k, x(k), u(k)

z(k), γc (︁

)︁)︂

z(k), t, xc(k, t), uc(k, t)

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

− 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 )



R(k, x, u) : x ∈ X(k), u ∈ U(k, x)

, t ∈ K\\K′,

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

k, xs(k), us(k)

→ μ(k), k ∈ K;

zs, t, xc

s(t), uc

− μc(zs, t)

dt → 0, k ∈ K′ and t ∈

s) − lc(zs) → 0, k ∈ K′;

Then the sequence {ms} is a minimizing sequence for I on D.

The proof is given in [6,8].

Relative minimum for DCS 65

Suppose that xc(k, tI ) = ξ

, 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

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

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

From (6) one has

ψ(k) = Hx, Hu = 0,

H(k, ψ, x, u) = ψT (k + 1)f(k, x, u), k ∈ K\\K′,

c(k, t) = −Hcx

c , ˙λ(k, t) = Hcx

H(k, ψ, xcI

F ) = ψT (k + 1)θ(k, xcI

F ), k ∈ K′ \\ kF ,

Hc(k, t, x, xc, uc) = ψcT(k, t)fc(k, t, xc, uc), Hc = sup

The condition d2R < 0 or

d2R = ΔxTRxxΔx + 2ΔxTRxuΔu + ΔuTRuuΔu < 0

for k ∈ K\\K′, as in [10], can be replaced by the equivalent conditions

(9) d2R < 0,

(10) ΔuTRuuΔu < 0.

Assuming that the maximum is reached in a stationary point, we

xuΔx + RuuΔu = 0, from which we get Δu = −R−1

uuRxuΔx. Then

d2R = ΔxT (Rxx − RxuR−1

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:

We introduce the matrix

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),

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

σ(k + 1)θx+

+ θT

x σ(k + 1)θxcI

ξx + ξT

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) =

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

ψ(kF ) = −FxF , ψc(k, tF ) = −Hxc

(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

σ(k + 1)θxc

+ Hxc

ω(k, tF ) = 0, β(k, tF ) = 0.

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 +

(x − ¯x)T σ(k)(x − ¯x),

φc(z, t, xc) = λT (k, t)x + ψcT(k, t)xc+

(xc − ¯xc)T σc(k, t)(xc − ¯xc)+

(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 .

Consider the following two-stage system

Relative minimum for DCS 69

Figure 1. State variables

Table 1. Results of a first-order method

Iteration Functional I

The functional is I = xc

Construct the DCS system. It is easy to see that K = 0, 1, 2. Since xc

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

I = x(2), xc

The basic constructions have the form:

Hc(0, t, xc

) = ψc1

+ ψc2

Hc(1, t, xc

) = ψc1

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.

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

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.

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).

[4] A.S. Bortakovskii. “Sufficient Optimality Conditions for Control of Deterministic

Logical-Dynamic Systems”, Informatika, Ser. Computer Aided Design,

[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,

[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

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