Adv Syst Sci Appl 2023; 01:61–68
Published online at https://ijassa.ipu.ru.

The Problem of Controllability with the Phase Space Change

Irina Maximova

Peoples’ Friendship University of Russia (RUDN University), Moscow, Russia

Abstract: In this paper, we consider differential systems of the following structure: at two
consecutive time intervals the motion of the object is described by two different systems of
differential equations. We study the controllability of the object described by such system
from the initial set in one space to the given set in another space through so-called ”transition
hypersurface”. The transition of an object from one space to another one is given by a certain
reflection. Sufficient conditions of the controllability of such differential systems in the problem
with phase space change are obtained. Approaches to the study of both nonlinear and linear
systems are considered.

Keywords: controllability, reachability set, multiple-valued mapping, support function

1. INTRODUCTION

Problems with the change of the phase space are an important class of so-called hybrid
(composite) systems, they are characterized by the condition that at different time intervals
the motion of the object is described by different differential equations and some links
for trajectory mating. The initial source of such problems was the multistage processes
of space flight (see [16]). Problems with the phase space change arise in various applied
problems, including aircraft engineering, robotics, economics, etc. For example, the problem
of launching a missile from an underwater object: in this case the dimension of the space does
not change, but the environment and conditions of the motion do.

The change of the phase spaces can occur in the mathematical modeling of complex
dynamic systems, for example, large production complexes, multistage technological
processes. The process of controlling a system of chemical reactors, models of the dynamics
of metapopulations, and economic systems with a variable structure are also characterized
by a sequence of consecutive stages. The dimensionality of dynamic systems used when
modelling these processes depends on their state and can change over time, i.e. the
decomposition of the complex system is taking place.

The possibility of using systems with the variable dimensionality in modeling the
dynamics of biological communities was pointed out in the work [15]. During the
construction of realization theory, R. Kalman proposed to generalize the notion of dynamic
system that the dimensionality of its space of states could change over time (see [9]).
Problems of optimal control of composite systems were studied at different times by, for
example, V.G. Boltyansky, L.T. Ashchepkov, and V.N. Rozova.

Necessary optimality conditions were derived in the general problem of control with
intermediate constraints on the trajectory (see [2]). In the paper [7], the necessary optimality
conditions for the problem with a phase space change were obtained.

The paper [14] considers the problem of optimal control of several objects with sequential
mode of operation. The initial state of each subsequent object depends on the final state of

∗Corresponding author: maximova-is@rudn.ru



62 I. MAXIMOVA

the previous one. Each object is described by a system of ordinary differential equations on
the interval of its action. Necessary and sufficient optimality conditions were obtained for the
problem of optimal control with a criterion of sufficiently general form.

Necessary optimality conditions for the problem with a change of phase space were
obtained by Maximova in [11]. The results were generalized to the case of several spaces.

In many works devoted to problems of this kind, the optimization issue is mainly studied.
Meanwhile, typical theorems of the existence of optimal control assume the existence of
at least one admissible control that generates a trajectory that satisfies the given boundary
conditions, for example, a control that translates a trajectory from one given position to
another. The latter problem is the essence of the controllability problem. Thus, the problem
of controllability is important and relevant in solving optimal control problems.

In the paper [3], Bargsegyan V.R. considered a mathematical model of control of linear
composite systems described at different time intervals by different differential equations and
some finite relations for continuity of motion of composite systems. The analytical view of
motion of composite systems is constructed, the properties of motion and the geometrical
structure of the reachability region were investigated. Necessary and sufficient conditions for
the complete controllability were formulated. The method of solving the problem of control of
composite systems and the method of solving the problem of optimal control were proposed.

The monograph [4] is devoted to the control problems of composite linear dynamical
systems and systems with multipoint intermediate conditions. Particular attention is paid
to necessary and sufficient conditions for the complete controllability and observability
of composite linear systems, which in the stationary case are comparable to the Kalman
conditions by their completeness. Qualitative properties of controllability and observability
of composite systems are revealed. Constructive methods for solving control problems
of composite systems, systems with undivided multipoint intermediate conditions, with
constraints on the values of different parts of the phase vector coordinates at intermediate
points in time are proposed.

The issue of controllability for some classes of nonlinear systems with a phase space
change was studied by the author in [12]. The paper consists of two sections devoted to the
cases of nonlinear and linear systems, respectively. An example illustrating this approach is
considered.

2. NONLINEAR SYSTEMS

Let x and y be two phase variables:

x = (x1, . . . , xn) ∈ X = Rn, y = (y1, . . . , ym) ∈ Y = Rm.

The motion of the object is described by the following non-linear systems of differential
equations:

ẋ = f(t, x(t), u(t)), u(t) ∈ U, t ∈ [0, τ ], x ∈ X, (2.1)
ẏ = g(t, y(t), v(t)), v(t) ∈ V, t ∈ [τ, T ], y ∈ Y. (2.2)

The class of admissible controls consists of all functions

u(·) ∈ U = {u(t) ∈ Rn | u(·) ∈ L∞[0, τ ]; u(t) ∈ U1 ⊂ Rn, t ∈ [0, τ ]},
v(·) ∈ V = {v(t) ∈ Rm | v(·) ∈ L∞[τ, T ]; v(t) ∈ V1 ⊂ Rm, t ∈ [τ, T ]},

whereU1 ∈ Ω(Rn), V1 ∈ Ω(Rm). Here Ω(Rn) and Ω(Rm) are the sets of all nonempty convex
compact subsets of the spaces Rn and Rm, respectively.

The functions f(t, x(t), u(t)), g(t, y(t), v(t)) are such that for systems (2.1) and (2.2) the
existence and uniqueness theorem of the Cauchy problem is satisfied. Solutions of systems
(2.1) and (2.2) at t ∈ [0, τ ] and t ∈ [τ, T ] are absolutely continuous functions that satisfy
systems (2.1) and (2.2) almost everywhere on [0, τ ] and [τ, T ], respectively.

Copyright © 2023 ASSA. Adv Syst Sci Appl (2023)



THE PROBLEM OF CONTROLLABILITY 63

There is an initial set M0 ∈ Ω(Rn) and a non- intersecting convex ”transition
hypersurface” in the space X Γ. Let’s suppose that τ is the smallest time moment at which
the object reaches the hypersurface Γ. When an object moving according to the law (2.1)
reaches the hypersurface Γ, it moves to the space Y given by the mapping q : X → Y (it is
assumed that this mapping translates the convex set into the convex one), and further motion
is performed in the space Y according to the law (2.2). Finally, the terminal set M1 ∈ Ω(Rm)
(not overlapping with the set q(Γ)) is given in Y . A similar scheme of motion of an object
was considered in [7]. The problem is to find the conditions under which the object described
by the systems (2.1) and (2.2) will be controllable from M0 to M1.

Definition 2.1:
An object described by systems (2.1) and (2.2) is called controllable from M0 to M1, if there
exist allowable controls u(·) and v(·) such that the corresponding solutions of the systems
satisfy the boundary conditions x(0) ∈M0, x(τ) ∈ Γ and y(τ) = q(x(τ)), y(T ) ∈M1.

Remark 2.1:
For system (2.1), consider the set f(t, x(t), U) consisting of all vectors f(t, x(t), u(t)), where
u(t) ∈ U . If x(t) is a trajectory of system (2.1) corresponding to an admissible control u(t),
then for almost all t ∈ [0, τ ] the inclusion

ẋ(t) ∈ f(t, x(t), U) (2.3)

holds true. This leads us to the differential inclusion of

ẋ ∈ f(t, x, U). (2.4)

Solutions of the differential inclusion (2.4) are absolutely continuous functions x(t) defined
on the interval [0, τ ] that satisfy the inclusion (2.3) for almost all t ∈ [0, τ ].

So, under rather general assumptions, system (2.1) is equivalent to a differential inclusion
(2.4), i.e. for any solution x(·) of the inclusion (2.4) there exists a valid control u(·) such that
the function x(·) is the path of system (2.1) with the control u(·). This question is discussed
in detail in [8].

Taking into account previous remarks, we shall consider differential inclusion (2.4)
instead of nonlinear system (2.1). Let us denote f(t, x, U) by F (t, x), then the motion of
the controlled object is described by the differential inclusion

ẋ ∈ F (t, x), t ∈ [0, τ ]. (2.5)

Similarly, the motion of the controlled object in the space Y = Rm is described by the
differential inclusion

ẏ ∈ G(t, y), t ∈ [τ, T ]. (2.6)
The motion of an object from the space X to the space Y is described above.

Definition 2.2 ( [6]):
A multi-valued mapping F (t, x) is called concave by x on a set M ⊂ X if for any points
x1, x2 ∈M and any λ ∈ [0, 1] the condition

λF (t, x1) + (1− λ)F (t, x2) ⊂ F (t, λx1 + (1− λ)x2)

holds true.
We note that the above condition implies the set F (t, x) is convex for every x ∈M

(see [6]).
The reachability set K(t) for each t ∈ [0, τ ] consists of all points x(t) ∈ Rn, where x(t)

is the solution of the inclusion (2.5) with initial condition x(0) ∈M0.
Consider the motion of an object in the space X from the initial set M0 to the transition

hypersurface Γ. Let us suppose that the mapping F (t, x) is concave by x on the reachability

Copyright © 2023 ASSA. Adv Syst Sci Appl (2023)



64 I. MAXIMOVA

set K(τ) for all t ∈ [0, τ ]. It is known (see [6]) that in this case the family of all solutions on
the interval [0, τ ] with initial condition x(0) ∈M0 is a convex set in the space C[0, τ ]. The
convexity of the solution family implies that the reachability set K(τ) is convex (the reverse
is not true).

So, under the above assumptions, the reachability set K(τ) is convex. The intersection of
K(τ) with the transition hypersurface Γ, we obtain the set

K1(τ) = K(τ) ∩ Γ.

Suppose that there exists τ : K1(τ) ̸= ∅. Then K1(τ) is convex as intersection of two convex
sets. Let us transform the set K1(τ) as follows: K2(τ) = q(K1(τ)), where q : X → Y.

The resulting set K2(τ) is convex due to the properties of mapping q. The set K2(τ) is the
initial set when the object moves in the space Y into the set M1.

In the space Y we obtain the following controllability problem: whether the system (2.2)
is controllable from the set K2(τ) to the set M1 at the time interval [τ, T ]. Let us denote
the system reachability set (2.2) from K2(τ) at time T by K3(T ). Let us suppose that the
mapping G(t, y) is concave on y on the reachability set K3(T ) and K3(T ) is compact. Then,
for the controllability of the system (2.2) it is sufficient that K3(T ) ∩M1 ̸= ∅ or, in the terms
of support functions, the inequality

c(K3(T ), ψ) + c(M1,−ψ) ≥ 0

holds true for any ψ ∈ Rm (see [5]).
Thus, the controllability conditions from the set M0 ⊂ X to the set M1 ⊂ Y for systems

(2.1) and (2.2) can be expressed as the following statement:
Theorem 2.1:
Under the above assumptions, for the controllability of an object described by systems (2.1)
and (2.2) on the time interval [0, T ] it is sufficient that

c(K3(T ), ψ) + c(M1,−ψ) ≥ 0,

for any ψ ∈ Rm.

Remark 2.2:
For autonomous system (2.2), one can consider the motion of an object in Y space in
backward time and get the reachability set K4(T ) from the set M1 on the of the transition
hypersurface Γ. Then the controllability condition for the systems (2.1) and (2.2) on the time
interval [0, T ] is a non-empty intersection of the sets K4(T ) and K2(τ).

3. LINEAR SYSTEMS

In this part of the paper we consider the case where the motion of the controlled object is
described by a linear system of differential equations. Let us suppose that systems (2.1) and
(2.2) are linear, then we obtain the following problem.

3.1. Problem statement
Let x and y be two phase variables:

x = (x1, . . . , xn) ∈ X = Rn, y = (y1, . . . , ym) ∈ Y = Rm.

The motion of the object is described by the following linear systems of differential equations:

ẋ = Ax+ u, u(t) ∈ U, t ∈ [0, τ ], x ∈ X. (3.7)

ẏ = By + v, v(t) ∈ V, t ∈ [τ, T ], y ∈ Y. (3.8)

Copyright © 2023 ASSA. Adv Syst Sci Appl (2023)



THE PROBLEM OF CONTROLLABILITY 65

The class of admissible controls is the sets of functions

{u(·) ∈ L∞([0, τ ],Rn) | u(t) ∈ U, t ∈ [0, τ ]},

{v(·) ∈ L∞([τ, T ],Rm) | v(t) ∈ V, t ∈ [τ, T ]}.
In the space X there is an initial set M0 ∈ Ω(Rn) and a non-overlapping convex the

transition hypersurface Γ. Let τ be the smallest time moment at which the object reaches
the hypersurface Γ. The motion of the object from one space to another one occurs in the
same way as in the first part of the paper. The problem is to find the conditions under which
the object described by systems (3.7) and (3.8) is controllable from M0 to M1.

3.2. Main result
The reachability set K(τ) for system (3.7) is the set of endpoints of trajectories of system
(3.7) with initial set M0, corresponding to all possible admissible controls u(·) ∈ U and
considered at time τ. Due to linearity of system (3.7), the reachability set can be presented in
the explicit form:

K(τ) = eτAM0 +

τ∫
0

e(τ−s)AUds, (3.9)

Here eτAM0 is the image of the set M0 under the linear transformation eτA, and the integrand
is a multivalued mapping obtained for all s ∈ [0, τ ] as the image of the set U under the linear
transformation e(τ−s)A. To find the reachability set with the initial convex set M0, let us first
calculate its support function and then reconstruct the set K(τ) by its support function. Thus,
the support function of the reachability set has the form

c(K(τ), ψ) = c(M0, e
τA∗

ψ) +

τ∫
0

c(U, esA
∗
ψ)ds. (3.10)

Since the initial set is convex, the reachability set is also convex (see [5]). Then the
set K(τ) reconstructed from the support function is intersected at the time τ with the the
transition hypersurface Γ. By assumption, this yields the convex set

K1(τ) = K(τ) ∩ Γ.

Assume that there exists τ such thatK1(τ) ̸= ∅. Let us transform the setK1(τ) as follows:
K2(τ) = q(K1(τ)), where q : X → Y.

Then the set K2(τ) is convex due to properties of q. The set K2(τ) is the initial set for the
system (3.8) when the object moves in space Y to the set M1.

So, we have formulated the following controllability problem in the space Y : whether
system (3.8) is controllable from the set K2(τ) to the set M1 on the time interval [τ, T ]. Let
us define the controllability function φ : Rm → R1 by the relation

φ(ψ) = c(K2(τ), e
(T−τ)B∗

ψ) + c(M1,−ψ) +
T−τ∫
0

c(V, esB
∗
ψ)ds, (3.11)

see [5]). According to the controllability theorem [5], an object is controllable on the time
segment [τ, T ] from the set K2(τ) to the set M1 if and only if for the controllability function
is non-negative, i.e.

φ(ψ) ≥ 0 ∀ψ ∈ S,

Copyright © 2023 ASSA. Adv Syst Sci Appl (2023)



66 I. MAXIMOVA

where S is the unit sphere in Rn. In turn, this is equivalent to the condition

φ0 = min
ψ∈S

φ(ψ) ≥ 0.

Applying this theorem, we obtain the following result:
Theorem 3.1:
Under the above assumptions, for the controllability of an object described by systems (3.7)
and (3.8) on the interval [0, T ], it is sufficient that the controllability function

φ(ψ) = c(K2(τ), e
(T−τ)B∗

ψ) + c(M1,−ψ) +
T−τ∫
0

c(V, esB
∗
ψ) ds

is non-negative for any ψ ∈ S.

3.3. Example
Let X = R3 and Y = R2 and the motion of an object is described by the following systems
of equations: ẋ1 = x2 + u1,

ẋ2 = −x1 + u2, |u| ≤ 1, u = (u1, u2, u3) ∈ R3,
ẋ3 = u3, t ∈ [0, τ ],

(3.12)

{
ẏ1 = y1 + v1, |v| ≤ 1, v = (v1, v2) ∈ R2,
ẏ2 = v2, t ∈ [τ, T ].

(3.13)

In the space R3, consider the initial set M0 = {(0,−1, 0)} and the transition hypersurface

Γ = {(x1, x2, x3) ∈ R3|x2 = 0, x3 ≥ 0}.

By τ denote the smallest time moment at which the object reaches the hypersurface Γ. The
mapping that makes the transition from R3 to R2 has the form

q(x1, x2, x3) = (x1 + sin τ, x3) = (y1, y2).

In the space R2, consider the set M1 = S1(0,−3). The problem is to find conditions for an
object described by systems (3.12) and (3.13) be controllable from M0 to M1 on the time
interval [0, T ].

Denote by K(τ) the reachability set of system (3.12) from the set M0 at time τ . First,
consider the motion of the object in the space R3. The support function of the reachability set
K(τ) from (3.10) reads

c(K(τ), ψ) = −ψ1 sin τ − ψ2 cos τ + τ∥ψ∥,

where ∥ψ∥ is the standard Euclidean norm of the vector ψ. Restoring the set K(τ) by the its
support function c(K(τ), ψ), one can easily see that the reachability set K(τ) is the circle of
radius τ with the center at the point (− sin τ,− cos τ, 0). The intersection of the reachability
set with the transition hypersurface Γ occurs at τ > 1 and it is a segment with the endpoints

(− sin τ −
√
τ 2 − cos2 τ , 0, 0); (− sin τ +

√
τ 2 − cos2 τ , 0, 0).

We pass to the space R2 under the action of the mapping q. We obtain the set K1(τ),
which is the initial for system (3.13) when the object moves in the space R2. The set K1(τ)

Copyright © 2023 ASSA. Adv Syst Sci Appl (2023)



THE PROBLEM OF CONTROLLABILITY 67

is the segment with the endpoints

(−
√
τ 2 − cos2 τ , 0), (

√
τ 2 − cos2 τ , 0).

Thus, we have the following controllability problem in R2: find such conditions on
systems (3.12) and (3.13) that the object described by these systems is controllable from
the set K1(τ) to the set M1. The controllability function in this case has the form

φ(ψ) =
√
τ 2 − cos2 τ · |ψ1|+ ∥ψ∥+ 3ψ2 +

T−τ∫
0

√
ψ2
1 + (sψ1 + ψ2)2 ds, (3.14)

where ∥ψ∥ = 1, ψ = (ψ1, ψ2, ψ3). The controllability function (3.14) reaches its minimum at
ψ2 = −1, then due to the fact that ∥ψ∥ = 1, ψ1 = ψ3 = 0. This yields

min
ψ∈S

φ(ψ) = T − (2 + τ).

Therefore, when T > 2 + τ , the object described by systems (3.12) and (3.13) is controllable
on the time interval [0, T ] from the set M0 to the set M1.

4. CONCLUSION

A special class of controlled differential systems called hybrid or composite is considered.
Such systems are characterized by the condition that at different time intervals the motion of
the object is described by different differential equations and some links for trajectory mating.

Sufficient conditions of the controllability of such systems from the initial set in one space
to the terminal set of the other space are obtained. Sufficient controllability conditions for the
nonlinear case are obtained using the apparatus of convex analysis, the theory of multivalued
mappings, and the control theory. This class of composite systems has not been considered
before.

An actual application of the proposed approaches and results lies in the field of
mathematical biology (see [15]) and mathematical economics with control input, for instance,
dynamic market models with continuous, see, e.g., [1, 13].

REFERENCES

1. Arutyunov, A. V. & Pavlova, N. G. (2022). Equilibrium in market models described by
differential equations, Differ. Equ., 58(9), 1267–1276.

2. Ashepkov, L. T. (1981). Optimal system control with intermediate conditions, Applied
mathem. mech., 45(2), 215–222. [in Russian]

3. Bargsegayn, V. R. (2012). Constructive approach to the study of control problems of
linear composite systems, Control problems, 4, 11–17.

4. Bargsegayn, V. R. (2016). Controls of composite dynamical systems with multipoint
intermediate conditions, Moscow, Russia: Nauka [in Russian].

5. Blagodatskikh, V. I. (2001). Introduction to optimal control, Moscow, Russia: Visshaya
shkola, [in Russian].

6. Blagodatskih, V. I. & Filippov A. F. (1986). Differential inclusions and optimal control,
Proc. Steklov Inst. Math., 169, 199–259.

7. Boltyanskii, V. G. (1983). The optimization problem with phase space change, Differ.
Equ., 19(3), 518–521.

8. Borisovich, Yu. G., Gel’man, B. D., Myshkis, A. D. & Obukhovskii, V. V. (1999).
Introduction to the Theory of Multivalued Mappings and Differential Inclusions,
Dordrecht, Netherlands: Kluwer Academic Publishers.

Copyright © 2023 ASSA. Adv Syst Sci Appl (2023)



68 I. MAXIMOVA

9. Kalman, R., Falb, P., & Arbib, M. (1969). Topics in mathematical system theory, New
York, NY: McGraw-Hill.

10. Kramar, V., Alchakov, V., & Osadchenko, A. (2021). The Optimal Control of the
Vessel’s Automatic Dynamic Positioning System Under Deviation, Adv. Syst. Sci. Appl.,
21(1), 1–10.

11. Maksimova, I. S. (2012). Necessary conditions of optimality in the problem with
changing phase spaces, RUDN Reports. Series: Mathematics. Computer science.
Physics, 4, 5–14. [in Russian]

12. Maksimova, I. S. & Rozova, V. N. (2017). Local controllability in the problem with
phase space change, RUDN Reports. Series: Natural and Technical Sciences 25(4), 331–
338. [in Russian]

13. Pavlova, N. G. (2019). Study of the continuous-time open dynamic Leontief model as a
linear dynamical control system, Differ. Equ., 55(1), 113–119.

14. Rozova, V. N. (2006). Optimal control of step systems, RUDN Reports. Series: Natural
and Technical Sciences, 3, 15–23. [in Russian]

15. Svirezhev, Yu. M. (1987). Nonlinear Waves, Dissipative Structures, and Catastrophes
in Ecology, Moscow, Russia: Nauka. [in Russian]

16. Velichenko, V. V. (1966). On optimal control problems for equations with discontinuous
right sides, Automation and telemechanics, 7, 20–30. [in Russian]

Copyright © 2023 ASSA. Adv Syst Sci Appl (2023)


	Introduction
	Nonlinear systems
	Linear systems
	Problem statement
	Main result
	Example

	Conclusion