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Near-optimal control for a singularly perturbed linear stochastic singular system with Markovian jumping parametersR Guoqiang Ren a,b, Bin Liu a,b,∗ a b

School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, Hubei, PR China Hubei Key Laboratory of Engineering Modeling and Scientiﬁc Computing, Huazhong University of Science and Technology, Wuhan 430074, Hubei, PR China

a r t i c l e

i n f o

Article history: Received 2 January 2018 Revised 26 March 2019 Accepted 16 April 2019 Available online xxx Recommended by Eduardo Costa MSC: 93C70 49N10 93Exx 34H05 60J27 34A09

a b s t r a c t In this paper, we investigate the near-optimal control for a singularly perturbed linear stochastic singular system with Markovian jumping parameters. Firstly, we warrant the given system has an impulse-free solution. Secondly, by means of the singular value decomposition, the existence of solution are obtained for the singularly perturbed stochastic generalized coupled differential Riccati equation and establish an existence of solution to parameter-independent system for the stochastic generalized coupled differential Riccati equation. As an application, we apply the existence results to consider the near-optimal control of singularly perturbed linear stochastic singular system with Markovian jumping parameters, and obtain the desired explicit representation of the optimal controllers for the optimal control problem with the ﬁnite horizon. © 2019 European Control Association. Published by Elsevier Ltd. All rights reserved.

Keywords: Singularly perturbed Stochastic generalized coupled differential Riccati equation Optimal control Stochastic singular system

1. Introduction Theory of singular perturbations was introduced to control audience by Kokotovic by the end of the sixties. Due to the fact that many real physical systems are singularly perturbed, for example, aircrafts, robots, electrical circuits, power systems, nuclear reactors, chemical reactors, dc and induction motors, synchronous machines, distillation columns, ﬂexible structures, automobiles, this theory has become very popular in control system engineering [10,11]. The notion of singular perturbations in mathematics stands for systems of differential equations that have some derivatives multiplied by small positive parameters. Usually such small parameters are neglected, thus we may associate two subsystems of lower dimensions which are independent of the small parameters, namely the fast subsystem or the boundary layer subsystem and the reduced

R

This work was partially supported by NNSF of China (Grant No. 11571126). Corresponding author at: School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, Hubei, PR China. E-mail addresses: [email protected] (G. Ren), [email protected] (B. Liu). ∗

subsystem or the slow subsystem. The problem is to show under what conditions the solutions of the obtained subsystems provide a good approximation of the solutions of the given system. In the last ﬁfty years many journal papers and books were published by engineering and mathematics researchers on the subject of singularly perturbed control systems. In deterministic control problem framework, Nguyen and Gajic [14] study the ﬁnite time (horizon) optimal control problem for singularly perturbed systems. The solution is obtained in terms of the corresponding solution of the algebraic Riccati equation and the decomposition of the singularly perturbed differential Lyapunov equation into reduced-order differential Lyapunov/Sylvester equations. Glizer [8] consider singularly perturbed linear-quadratic optimal control problem in an inﬁnite dimensional Hilbert space, mainly exert boundary layer method an asymptotic solution of the corresponding operator Riccati equation is constructed, this result is illustrated by its application to the asymptotic solution of a set of integral-differential equations of the Riccati type associated with the control problem of a linear singularly perturbed integraldifferential equation of the second order. Wang et al. [20] for

https://doi.org/10.1016/j.ejcon.2019.04.002 0947-3580/© 2019 European Control Association. Published by Elsevier Ltd. All rights reserved.

Please cite this article as: G. Ren and B. Liu, Near-optimal control for a singularly perturbed linear stochastic singular system with Markovian jumping parameters, European Journal of Control, https://doi.org/10.1016/j.ejcon.2019.04.002

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the ﬁrst time utilize descriptor-system approach to study singular perturbation of linear regulator, and investigates the relationship between the regulator problem for a singularly perturbed system and the analogous one for a descriptor system, meanwhile, a decomposition approach is provided for solution of the linear quadratic regulator problem for singularly perturbed systems. Xu et al. [21] and Fridman [6] also use the descriptor-system approach to study singularly perturbed optimal control problem. Tan et al. [19] derives a set of ε -independent suﬃcient and necessary conditions for the H∞ control problem for singularly perturbed systems, a sub-optimal controller is explicitly constructed. In realities, the uncertainties are unavoidable. So over the past decades stochastic modeling has played an important role in many branches of science and engineering. The study of systems with stochastic disturbance has gained growing interest over the past few decades, and many research topics on stochastic systems have been investigated. Generally, uncertain terms involved in stochastic systems are classiﬁed mainly into two types, namely, the stateindependent on known as additive noise and the state-dependent one referred to as the multiplicative noise. Because many applications of stochastic systems are governed by an Itô differential equation, the theory of optimal control and the related area for singularly perturbed and multi-parameter stochastic systems has been well documented [1,2,4,5,15,16,18]. Dragan and Mukaidani [4] investigate the optimal control of a class of singularly perturbed linear stochastic systems with Markovian jumping parameters. After establishing an asymptotic structure for the stabilizing solution of the coupled stochastic algebraic Riccati equations, a parameter-independent composite controller is derived. Furthermore, the cost degradation in a reduced-order controller is discussed. Thus, the exactness of the proposed approximate control is discussed for the ﬁrst time. Dragan et al. [5] discusses an inﬁnite-horizon linear quadratic (LQ) optimal control problem involving state- and control-dependent noise in singularly perturbed stochastic systems, establish asymptotic structure along with a stabilizing solution for the stochastic algebraic Riccati equation (ARE) and also give the suﬃcient conditions for the existence of the stabilizing solution to the problem. A new sequential numerical algorithm for solving the reduced-order AREs is also described. Based √ on the asymptotic behavior of the ARE, obtain a class of O( ε ) approximate controller that stabilizes the system. Unlike the existing results in singularly perturbed deterministic systems, it is noteworthy that the resulting controller achieves an O(ε ) approximation to the optimal cost of the original LQ optimal control problem. Sagara et al. [17] addresses linear quadratic control with state-dependent noise for singularly perturbed stochastic systems (SPSS). First, establish the asymptotic structure of the stochastic algebraic Riccati equation (SARE). Second, a new iterative algorithm that combines Newtons method with the ﬁxed point algorithm is established. As a result, attain the quadratic convergence and the reduced-order computation in the same dimension of the subsystem. Meanwhile, a high-order state feedback controller that uses the obtained iterative solution is given and the degradation of the cost performance is investigated for the stochastic case for the ﬁrst time. Furthermore, the parameter independent controller is also given in case the singular perturbation is unknown. To the best of our knowledge, the theory of the linear stochastic singular system is quite fragmentary, and the theory of the singularly perturbed linear stochastic singular system is unknown, in order to make up for the gap between singularly perturbed linear stochastic system and singularly perturbed linear stochastic singular system, it is mainly motivation to this study. The main contributions of this paper is, by means of the singular value decomposition, converted singularly perturbed linear stochastic singular system with Markovian jumping into singularly perturbed linear stochastic system with Markovian jumping, and used the original

theory to establish existence of solution for singularly perturbed linear stochastic singular system with Markovian jumping parameters, we also obtained the desired explicit representation of the optimal controllers for the optimal control problem with the ﬁnite horizon. Inspired by the authors in Refs. [3,4,23], we consider the linear quadratic control problem for a singularly perturbed linear stochastic singular system with Markovian jumping parameters. Firstly, we warrant the given system has an impulse-free solution. Secondly, by means of the singular value decomposition, the existence of solution are obtained for the singularly perturbed stochastic generalized coupled differential Riccati equation and establish an existence of solution to parameter-independent system for the stochastic generalized coupled differential Riccati equation. As an application, we apply the existence results to consider the nearoptimal control of singularly perturbed linear stochastic singular system with Markovian jumping, and obtain the desired explicit representation of the optimal controllers for the optimal control problem with the ﬁnite horizon. The paper is organized as follows. In Section 2, we summarize some basic assumptions, deﬁnitions and some useful lemmas in order to prove the given system has a impulse-free solution. In Section 3, we establish the existence of the solution for singularly perturbed stochastic generalized coupled differential Riccati equation(SGCDRE) and establish an existence of solution to parameterindependent system for the stochastic generalized coupled differential Riccati equation. In Section 4, as an application, we apply the existence results to consider the near-optimal control of Markovian jump linear stochastic singular system, and obtain the desired explicit representation of the near-optimal controllers for the nearoptimal control problem with the ﬁnite horizon. We close paper by the concluding remark. Notation. Rn denotes the n-dimensional Euclidean space, Rm×n is the set of all m × n real matrices and R+ := (0, ∞ ). For symmetric matrices P, the notation P 0(respectively, P 0) means that matrix P is positive deﬁnite(respectively, positive semi-deﬁnite). I is an identity matrix of appropriate dimensions. The superscripts AT and A−1 stand for the transpose and the inverse of a matrix A, x is the Euclidean norm of the vector x.

2. Preliminaries In this section, consider the following singularly perturbed linear stochastic singular systems with Markovian jumps, modeled by

Edx(t ) = [A(ε , t, r (t ))x(t ) + B(ε , t, r (t ))u(t )]dt + [C (ε , μ, t, r (t ))x(t ) + D(ε , μ, t, r (t ))u(t )]dω (t ), x ( 0 ) = x0 .

(2.1)

where x(t ) = (xT1 (t ) xT2 (t ))T ∈ Rn1 × Rn2 , n = n1 + n2 is the system state vector, u(t ) ∈ Rm is the control input, and ω(t) is a one-dimensional standard Brownian motion that is deﬁned on the given complete probability space (, F, (Ft )0≤t≤T , P ). Deﬁne the set of all admissible controls Uad = L2F (0, T ; Rm ). The matrices E ∈ Rn×n is a singular constant matrix and we assume that rankE = r < n. x0 = (xT01 xT02 )T ∈ Rn1 × Rn2 , is the compatible initial condition which is deterministic. This form process r(t) is a continuoustime discrete-state Markov process taking values in a ﬁnite set S = {1, 2, . . . , N} with transition probability matrix P := { pi j } given by

pi j = P (r (t + ) = j, r (t ) = i ) =

i f i = j, λi j + 0(), λii + 0(), i f i = j,

1+

(2.2)

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where > 0. Here λij ≥ 0 is the form transition rate from i to j (i = j), and N

λi := −λii =

λi j .

Let the initial values x0 and r0 be independent random variables; x0 is also independent of the σ -algebra generated by {r(t), t ∈ (0, T]}. When the system operates in the ith mode (r (t ) = i ), for simplicity, let A(ε , t, r (t ))|r (t )=i = Ai (ε , t ), B(ε , t, r (t ))|r (t )=i = Bi (ε , t ), C (ε , μ, t, r (t ))|r (t )=i = Ci (ε , μ, t ), D(ε , μ, t, r (t ))|r (t )=i = Di (ε , μ, t ). In (2.1), the parameters ε > 0 and μ > 0 are typically small and are often unknown. The coeﬃcient matrices Ai (ε , t), Bi (ε , t), Ci (ε , μ, t), Di (ε, μ, t) ∈ L∞ (0, T) and have appropriate dimensions. We have used the following notations:

Ai ( ε , t ) =

Ai11 (t )

Ai12 (t )

ε −1 Ai21 (t )

ε −1 Ai22 (t )

Ci (ε , μ, t )

Ci11 (t )

, Bi ( ε , t ) =

με Ci21 (t ) με Ci22 (t ) Di1 (t ) D i ( ε , μ, t ) . με −1 Di2 (t )

Bi1 (t )

ε −1 Bi2 (t )

,

−1

J (0, x0 , r (0 ), u, T )

= xT (T )E T L(r (T ))Ex(T ) +

T

H (2.2 ) rank E

Ai ( ε , t )

Ci (ε , μ, t )

K K † K = K,

(s )−1 σ (s )dω (t ),

d (t ) = A(t ) (t )dt + C (t ) (t )dω (t ),

(0 ) = I,

Theorem 2.1. If assumptions H (2.1 ) and H (2.2 ) hold, then system (2.1) has a solution on [0, T], ∀i ∈ S, in which there is no impulse. Proof. Due to the constant-rank condition of matrix E, we can consider the singular value decomposition. Under assumption H (2.2 ), there exist orthogonal matrices Mi ∈ Rn×n , Ni ∈ Rn×n , ∀i ∈ S such that

Mi ENi =

r

0 0

0

Mi Ai (ε , t )Ni = (2.4)

= n + r,

Di (ε , μ, t ) = r,

∀i ∈ S.

∀i ∈ S.

K†KK† = K†,

(K K † )T = K K † , (K † K )T = K † K.

(2.5) Ci1 (ε , μ, t ) 0

0

dX (t ) = [A(t )X (t ) + b(t )]dt + [C (t )X (t ) +

Ci2 (ε , μ, t ) 0

(2.6)

D1i (ε , μ, t ) 0

(2.7)

Mi Bi ( ε , t ) =

A1i (ε , t ) A3i (ε , t )

A2i (ε , t ) A4i (ε , t )

(2.8)

B1i (ε , t ) B2i (ε , t )

(2.9)

where A1i (ε , t ), A2i (ε , t ), A3i (ε , t ), A4i (ε , t ), B1i (ε , t ), B2i (ε , t ) have appropriate dimensions, and let

ζ1 (t ) ζ (t ) = N x(t ) = ζ2 (t ) −1

where ζ 1 (t), ζ 2 (t) have appropriate dimensions. By above transformations, system (2.1) can be transformed into

⎧ r dζ1 (t ) = [A1i (ε , t )ζ1 (t ) + A2i (ε , t )ζ2 (t ) + B1i (ε , t )u(t )]dt ⎪ ⎪ ⎨ + [C 1 (ε , μ, t )ζ (t ) + C 2 (ε , μ, t )ζ (t + D1 (ε , μ, t )u(t )]dω (t ), 1 2 i i i ⎪0 = A3i (ε , t )ζ1 (t ) + A4i (ε , t )ζ2 (t ) + B2i (ε , t )u(t ), ⎪ ⎩ ζ1 (0 ) = (r−1 0 )Mx0 .

(2.10) On the other hand, under assumption H (2.1 ), the rank relation

rank(A4i (ε , t ) B2i (ε , t )) = rank(A3i (ε , t ) A4i (ε , t ) B2i (ε , t )) := p

Lemma 2.1 ([22] Variation of constants formula). For any ζ ∈ L2F (; Rn ), equation

(s )−1 [b(s ) − C (s )σ (s )]ds

where (·) is the unique solution of the following matrix-valued stochastic differential equation:

Deﬁnition 2.1 [15]. Let a matrix K ∈ Rm×n be given. Then the matrix K† is called the Moore–Penrose pseudoinverse of K if there exists a unique matrix K † ∈ Rn×m such that

0

where r is a nonsingular diagonal constant matrix and Ci1 (ε , μ, t ), Ci2 (ε , μ, t ), D1i (ε , μ, t ) have appropriate dimensions. Accordingly, deﬁne

0 Bi ( ε , t )

t

where denotes expectation, L(r (T )) ∈ L∞ (0, T ; Rn×n ), Qi (t ) ∈ L∞ (0, T ; Rn×n ), and Ri (t ) ∈ L∞ (0, T ; Rm×m ), i ∈ S, are symmetric matrices, and Li (t ) ∈ L∞ (0, T ; Rn×m ). For the existence of the impulse-free solution to the singularly perturbed linear stochastic singular systems with Markovian jumps (2.1), we impose the following assumptions:

E

0

M i D i ( ε , μ, t ) =

xT (t )Q (r (t ), t )x(t )

0

t

MiCi (ε , μ, t )Ni =

+ 2xT (t )L(r (t ), t )u(t ) + uT (t )R(r (t ), t )u(t )dt

0 E

,

Subject to (2.1)–(2.3), we consider the minimization of

+ (t )

Ci12 (t )

−1

H (2.1 ) rank

X (t ) = (t )ζ + (t )

(2.3)

j=1, j =i

3

σ (t )]dω (t ),

X (0 ) = ζ , where A(· ), C (· ) ∈ L∞ (0, T ; Rn×n ), b(· ), σ (· ) ∈ L2 (0, T ; Rn ) admits a unique solution X(·), which is represented by the following:

holds. In general, the matrix rank(A4i (ε , t ) B2i (ε , t )) does not have the full row rank, so there exists a nonsingular matrix Ui (t), ∀i ∈ S such that

Ui (t )(A4i (ε , t ) B2i (ε , t )) =

4

Ai ( ε , t )

2 (ε , t ) B i

0

0

4 (ε , t ) 2 (ε , t )) has full-row rank. Then, system where (A B i i (2.10) can be equivalently rewritten as

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⎧ r dζ1 (t ) = [A1i (ε , t )ζ1 (t ) + A2i (ε , t )ζ2 (t ) + B1i (ε , t )u(t )]dt ⎪ ⎪ ⎨ + [C 1 (ε , μ, t )ζ (t ) + C 2 (ε , μ, t )ζ (t ) + D1 (ε , μ, t )u(t )]dω (t ), 1

i

2

i

i

3 4 2 ⎪ ⎪0 = (Ai (ε , t )ζ1 (t ) + Ai (ε , t )ζ2 (t ) + Bi (ε , t )u(t ),

⎩ ζ1 (0 ) = (r−1 0 )Mx0 .

3. Existence of the solution for singularly perturbed SGCDRE

(2.11) where

3 (ε , t ) A i

0 )Ui (t )A3i (ε , t ),

= (I p

where Ip is denotes identity 4 (ε , t ) as matrix has p order. Because p ≤ n − r, we can write A i 4 ( 1 ) 4 ( 2 ) T (t ) T (t ))T , A2 (ε , t ) = (A (ε , t ) A (ε , t )), Let ζ2 (t ) = (ζ21 ζ22 i i i

(A2i (1) (ε , t ) A2i (2) (ε , t )), Ci2 (ε , μ, t ) = (Ci2(1) (ε , μ, t ) Ci2(2) (ε , μ, t )), T (t ), A (1 ) (t ) have appropriate dimensions. Then (2.11) can where ζ21 i2 be rewritten as

⎧ T r dζ1 (t ) = [A1i (ε , t )ζ1 (t ) + A2i (1) (ε , t )ζ21 (t ) ⎪ ⎪ ⎪ ⎪ 2 (2 ) T 1 ⎪ + Ai (ε , t ))ζ22 (t ) + Bi (ε , t )u(t )]dt ⎪ ⎪ ⎪ 2 (1 ) ⎪ 1 T ⎪ ⎨ + [Ci (ε , μ, t )ζ1 (t ) + Ci (ε , μ, t )ζ21 (t ) 2 (2 ) T 1 + Ci (ε , μ, t )ζ22 (t ) + Di (ε , μ, t )u(t )]dω (t ), ⎪ ⎪ ⎪ 3 4(1) ⎪ ⎪0 = (Ai (ε , t )ζ1 (t ) + Ai (ε , t )ζ21 (t ) ⎪ ⎪ ⎪ ⎪ + A4i (2) (ε , t )ζ22 (t ) + B2i (ε , t )u(t ), ⎪ ⎩ ζ1 (0 ) = (r−1 0 )Mx0 .

Ui (t )(A4i (ε , t ) B2i (ε , t ))Vi (t )

4(1) (ε , t ) A i

(2.12)

0

Vi (t ) =

Ip

0

0

0

0

0

4(2) (ε , t ) A i

2 (ε , t ) B i

0

0

1

2

(2.12) is equivalently transformed

⎧ 2 (ε , t )A 3 (ε , t ))ζ1 (t ) r dζ1 (t ) = [(A1i (ε , t ) − A ⎪ i i ⎪ ⎪ 1 (ε , t ) ⎪ +B u(t )]dt ⎪ i ⎪ ⎪ ⎨ + [C 1 (ε , μ, t ) − C2 (ε , μ, t )A3 (ε , t ))ζ (t ) 1 i i i 1 (ε , μ, t ) ⎪ + D u ( t ) ] d ω ( t ) , ⎪ i ⎪ ⎪ ⎪ 3 (ε , t )ζ1 (t ) + ζ2 (t ), ⎪ 0 = A ⎪ i ⎩ ζ1 (0 ) = (r−1 0 )Mx0 .

⎧ T −E P˙i (ε , μ, t ) = ATi (ε , t )Pi (ε , μ, t ) + PiT (ε , μ, t )Ai (ε , t ) ⎪ ⎪ ⎪ ⎪ −(PiT (ε , μ, t )Bi (ε , t ) + CiT (ε , μ, t ) ⎪ ⎪ ⎪ ⎪ ⎪ × (E E † )T Pi (ε , μ, t )E † Di (ε , μ, t ) ⎪ ⎪ ⎪ ⎪ + Li (t ))(Ri (t ) + DTi (ε , μ, t )(E † )T PiT (ε , μ, t ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎨×E E † Di (ε , μ, t ))−1 (BTi (ε , t )Pi (ε , μ, t ) + DTi (ε , μ, t )(E † )T PiT (ε , μ, t )E E † ⎪ ⎪ ⎪ × Ci (ε , μ, t ) + LTi (t )) ⎪ ⎪ ⎪ ⎪ ⎪ + CiT (ε , μ, t )(E † )T PiT (ε , μ, t )E E †Ci (ε , μ, t ), ⎪ ⎪ ⎪ ⎪ + j∈S λi j E T Pj (t ) + Qi (t ) (3.1a ) ⎪ ⎪ ⎪ T T ⎪ E P ( ε , μ , t ) = P ( ε , μ , t ) E, ⎪ i i ⎪ ⎩ (3.1b) E T Pi (T ) = E T L(r (T ))E. for each i ∈ S, t ∈ [0, T]. For notational simplicity, we deﬁned

4(1 ) (ε , t ) B 2 (ε , t )) Without loss of generality, we assume that (A i i has full-row rank. Otherwise, we can exchange some columns 4 ( 1 ) 4 ( 2 ) of A (ε , t ) for some columns of A (ε , t ), and then i i T (t ) and ζ T (t ). Let make the same exchanges between ζ21 22 T (t ) T (t ) (ζ2T (t ) uT1 (t ) uT2 (t ))T = V −1 (t )(ζ21 ζ22 uT (t )), where ζ (t ), uT (t ), u (t ) have appropriate dimensions, then the system 2

In this section, we establish the existence of the solution for a set of stochastic generalized coupled differential Riccati equations. And we impose the following assumptions: H (3.1 ) A3i (ε , t ) ≡ 0 and the matrix Bi (ε , t), ∃ t ∈ [0, T] is full of column rank. H (3.2 ) τi + τ j = 0, where τ i and τ j are arbitrary eigenvalues of A4i (ε , t ), ∀ t ∈ [0, T]. Theorem 3.1. Assume that H (3.1 ) and H (3.2 ) holds, consider L(r(T)) 0, 0, and Ri (ε , μ, t ) 0, there exist a solution Pi (ε , μ, t ) ∈ Rn×n , ∀i ∈ S, satisfying

Obviously, system (2.12) is equivalent to system (2.11). Since 4(1 ) (ε , t ) A 4(2 ) (ε , t ) B 2 (ε , t )) has full-row rank, then there ex(A i i i ists a nonsingular matrix Vi (t), ∀i ∈ S such that

=

Remark 2.3. When the diffusion term has ﬁnite state variables and control inputs, the discussion is similar.

Li (ε , μ, t ) = ATi (ε , t )Pi (ε , μ, t ) + PiT (ε , μ, t )Ai (ε , t ) + CiT (ε , μ, t )(E † )T PiT (ε , μ, t )E E †Ci (ε , μ, t ) + λi j E T Pj (ε , μ, t ), i ∈ S.

(3.2)

j∈S

Mi (ε , μ, t ) = BTi (ε , t )Pi (ε , μ, t ) + DTi (ε , μ, t )(E † )T PiT (ε , μ, t )E E †Ci (ε , μ, t ) + LTi (t ), i ∈ S.

(2.13)

2 (ε , t ) B 1 (ε , t )) = (A2(1 ) (ε , t ) A2(2 ) (ε , t ) B1 (ε , t ))V (t ), where (A i i i i i i 2 1 (Ci (ε , μ, t ) Di (ε , μ, t )) = (Ci2(1) (ε , μ, t ) Ci2(2) (ε , μ, t ) D1i (t ))Vi (t ), u(t ) = ( uT1 (t ) uT2 (t ))T . The ﬁrst equation of (2.13) is an ordinary stochastic differential equation, in which ζ 1 (t) is the state vector and u(t ) is the control vector. According to Lemma 2.1, the ﬁrst equation of (2.13) has a solution ζ 1 (t) on [0, T] under the initial condi (t )ζ (t ) exists. tion ζ1 (0 ) = (r−1 0 )Mx0 . Accordingly, ζ2 (t ) = −A 1 i3 Thus, system (2.13) has a impulse-free solution on [0, T], which implies that system (2.1) has a impulse-free solution on [0, T]. This is complete the proof. Remark 2.1. When Di (t) ≡ 0, S = {1}, and without have parameters ε, μ, the result is the same as in [23].

(3.3)

Ri (ε , μ, t ) = Ri (t ) + DTi (ε , μ, t )(E † )T PiT (ε , μ, t )E E † Di (ε , μ, t ), i ∈ S. (3.4) As in (3.1), for ﬁnite T ∈ R + arbitrarily ﬁxed, the set of SGCDRE is deﬁned as

⎧ T˙ −E Pi (ε , μ, t ) = Li (ε , μ, t ) ⎪ ⎪ ⎨−MT (ε , μ, t )R−1 (ε , μ, t )M (ε , μ, t ) + Q (t ), t ∈ [0, T ], i i i i T T ⎪ E P ( ε , μ , t ) = P ( ε , μ , t ) E, ⎪ i ⎩ T i E Pi (T ) = E T L(r (T ))E.

(3.5) Proof. We follows the matrix decomposition method of Theorem 2.1, there exist two orthogonal matrices Mi ∈ Rn×n , Ni ∈ Rn×n , ∀i ∈ S such that

Mi−T Pi (ε , μ, t )Ni =

Pi1 (ε , μ, t ) Pi3 (ε , μ, t )

Pi2 (ε , μ, t ) Pi4 (ε , μ, t )

(3.6)

Remark 2.2. When E = I, this system is considered in [4]. Please cite this article as: G. Ren and B. Liu, Near-optimal control for a singularly perturbed linear stochastic singular system with Markovian jumping parameters, European Journal of Control, https://doi.org/10.1016/j.ejcon.2019.04.002

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Qi1 (t ) QiT2 (t )

Qi2 (t ) Qi3 (t )

(t )Ni = Li1 (t ) T Ni Li (t ) = Li2 (t ) L(r (T ))11 L(r (T ))12 Mi−T L(r (T ))Mi−1 = LT (r (T ))12 L(r (T ))22 where Pi1 (ε , μ, t), Pi2 (ε , μ, t), Pi3 (ε , μ, t), Pi4 (ε , μ, t), Qi1 (t), NiT Qi

(3.7) (3.8) (3.9)

Qi2 (t), Qi3 (t), Li1 (t), Li2 (t), L(r(T))11 , L(r(T))12 , L(r(T))22 are all matrices with appropriate dimensions. By means of the relation (2.5), it is easy to obtain that the Moore–Penrose pseudo inverse of E is

E† = N

r−1

0 M. 0

0

(3.10)

we directly use the transformations (2.5)–(2.9) and (3.6)–(3.10) to SGCDRE (3.1). Then (3.1a) can be partitioned into

−

r

0

0

0

˙

=

A2i (ε , t )

PiT2 (ε , μ, t )

PiT4 (ε , μ, t )

A3i (ε , t )

A4i (ε , t )

A1i T (ε , t )

A3i T (ε , t )

Pi1 (ε , μ, t )

Pi2 (ε , μ, t )

A2i T (ε , t )

A4i T (ε , t )

Pi3 (ε , μ, t )

Pi4 (ε , μ, t )

Ci1T (ε , μ, t )

−1 0 r

Ci2T (ε , μ, t )

0

r

0

0

0

+

P˙i4 (ε , μ, t ) A1i (ε , t )

+

P˙i3 (ε , μ, t )

PiT3 (ε , μ, t )

+

×

P˙i2 (ε , μ, t )

PiT1 (ε , μ, t )

+

Pi1 (ε , μ, t )

Qi1 (t ) QiT2 (t )

j∈S

λi j

r

0

PiT1 (ε , μ, t )

PiT3 (ε , μ, t )

0

PiT2 (ε , μ, t )

PiT4 (ε , μ, t )

Ci2 (ε , μ, t )

0

0

0

Pj1 (ε , μ, t )

Pj2 (ε , μ, t )

0

Pj3 (ε , μ, t )

Pj4 (ε , μ, t )

0

0

0

0

Ci1 (ε , μ, t )

r−1

Qi2 (t ) Qi3 (t )

0

− MTi (ε , μ, t )R−1 ( ε , μ, t ) M i ( ε , μ, t ) i with the boundary condition r Pi1 (T ) =

(3.11)

rT L(r (T ))11 r ,

where

1 PiT1 (ε , μ, t ) PiT3 (ε , μ, t ) Bi ( ε , t ) Li1 (t ) MTi (ε , μ, t ) = + Li2 (t ) PiT2 (ε , μ, t ) PiT4 (ε , μ, t ) B2i (ε , t ) 1T −1 Ci (ε , μ, t ) 0 r 0 r 0 + 0 0 0 0 Ci2T (ε , μ, t ) 0 −1 Pi1 (ε , μ, t ) Pi2 (ε , μ, t ) r 0 × Pi3 (ε , μ, t ) Pi4 (ε , μ, t ) 0 0 1 D i ( ε , μ, t ) ×

,

0

Ri (ε , μ, t ) = Ri (t ) +

×

×

T −1 r

D1i (ε , μ, t ) 0

PiT1 (ε , μ, t ) PiT2 (ε , μ, t )

−1 r

0

0 0

0

PiT3 (ε , μ, t ) PiT4 (ε , μ, t ) D1i

( ε , μ, t ) 0

0 0

r 0

.

0 0

5

By Eq. (3.1b), we get r Pi1 (ε , μ, t ) = (r Pi1 (ε , μ, t ))T and Pi2 (ε , μ, t ) = 0. Then, from Eq. (3.11), we obtain three equations as follows:

⎧ −(r P˙i1 (ε , μ, t )) = (r Pi1 (ε , μ, t ))T (r−1 A1i (ε , t )) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ + (r−1 A1i (ε , t ))T (r Pi1 (ε , μ, t )) + ⎪ j∈S λi j r Pj1 (ε , μ, t ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ + PiT3 (ε , μ, t )A3i (ε , t ) + A3i T (ε , t )Pi3 (ε , μ, t ) + Qi1 (t ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ + (r−1Ci1 (ε , μ, t ))T (r Pi1 (ε , μ, t ))(r−1Ci1 (ε , μ, t )) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ −[(r Pi1 (ε , μ, t ))T (r−1 B1i (ε , t )) + PiT3 (ε , μ, t )B2i (ε , t )) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ + (r−1Ci1 (ε , μ, t ))T (r Pi1 (ε , μ, t ))T (r−1 D1i (ε , μ, t )) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ + Li1 (t )][Ri (t ) + (r−1 D1i (ε , μ, t ))T (r Pi1 (ε , μ, t )) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ × (r−1 D1i (ε , μ, t ))]−1 [(r Pi1 (ε , μ, t ))T (r−1 B1i (ε , t )) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ + PiT3 (ε , μ, t )B2i (ε , t )) + (r−1Ci1 (ε , μ, t )) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ × (r Pi1 (ε , μ, t ))T (r−1 D1i (ε , μ, t )) + Li1 (t )]T , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 = PiT4 (ε , μ, t )A3i (ε , t ) + (r−1 A2i (ε , t ))T (r Pi1 (ε , μ, t )) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ + A4i T (ε , t )Pi3 (ε , μ, t ) + QiT2 (t ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ + (r−1Ci2 (ε , μ, t ))T (r Pi1 (ε , μ, t ))T (r−1Ci1 (ε , μ, t )) ⎪ ⎪ ⎪ ⎨ −[(r Pi1 (ε , μ, t ))T (r−1 B1i (ε , t )) + PiT3 (ε , μ, t )B2i (ε , t )) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ + (r−1Ci1 (ε , μ, t ))T (r Pi1 (ε , μ, t )) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ × (r−1 D1i (ε , μ, t )) + Li1 (t )][Ri (t ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ + (r−1 D1i (ε , μ, t ))T (r Pi1 (ε , μ, t ))(r−1 D1i (ε , μ, t ))]−1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪× [PiT4 (ε , μ, t )B2i (ε , t )) ⎪ ⎪ ⎪ ⎪ ⎪ −1 2 T −1 1 T ⎪ ⎪ ⎪ + (r Ci (ε , μ, t )) (r Pi1 (ε , μ, t ))(r Di (ε , μ, t )) + Li2 (t )] , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 = PiT4 (ε , μ, t )A4i (ε , t ) + A4i T (ε , t )Pi4 (ε , μ, t ) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ + Qi4 (t ) + (r−1C 2 (ε , μ, t ))T (r Pi1 (ε , μ, t ))T (r−1C 2 (ε , μ, t )) ⎪ i i ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ −[PiT4 (ε , μ, t )B2i (ε , t )) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ + (r−1Ci2 (ε , μ, t ))T (r Pi1 (ε , μ, t ))(r−1 D1i (ε , μ, t )) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ + Li2 (t )][Ri (t ) + (r−1 D1i (ε , μ, t ))T (r Pi1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ × (ε , μ, t ))(r−1 D1i (ε , μ, t ))]−1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ × [PiT4 (ε , μ, t )B2i (ε , t )) + (r−1Ci2 (ε , μ, t ))T (r Pi1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ × (ε , μ, t ))(r−1 D1i (ε , μ, t )) + Li2 (t )]T . (3.12) A3i (ε , t )

By the assumption H (3.1 ) to be seen, ≡ 0 and the matrix Bi (ε , t) is full of column rank, without loss of generality, we can assume that B2i (ε , t ) ≡ 0, ∃ t ∈ [0, T]. Then Eq. (3.12) can be rewritten as

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⎧ −(r P˙i1 (ε , μ, t )) = (r Pi1 (ε , μ, t ))T (r−1 A1i (ε , t )) ⎪ ⎪ ⎪ ⎪ + (r−1 A1i (ε , t ))T (r Pi1 (ε , μ, t )) + ⎪ j∈S λi j r Pj1 (ε , μ, t ) ⎪ ⎪ ⎪+ Q (t ) + ( −1C 1 (ε , μ, t ))T ( P (ε , μ, t ))( −1C 1 (ε , μ, t )) ⎪ r i1 i1 ⎪ r r i i ⎪ ⎪ T −1 1 ⎪ −[ ( P ( ε , μ , t )) ( B ( ε , t )) ⎪ r i 1 r i ⎪ ⎪ ⎪ ⎪ + (r−1Ci1 (ε , μ, t ))T (r Pi1 (ε , μ, t ))T (r−1 D1i (ε , μ, t )) + Li1 (t )] ⎪ ⎪ ⎪ ⎪ t ) + (r−1 D1i (ε , μ, t ))T (r Pi1 (ε , μ, t )) ⎪ ⎪× [Ri (−1 ⎪ 1 −1 ⎪ × ( (3.13a ) ⎪ r Di (ε , μ, t ))] ⎪ ⎪ T −1 1 −1 1 ⎪ × [(r Pi1 (ε , μ, t )) (r Bi (ε , t )) + (r Ci (ε , μ, t )) ⎪ ⎪ ⎪ ⎪ ⎪ × (r Pi1 (ε , μ, t ))T (r−1 D1i (ε , μ, t )) + Li1 (t )]T , ⎪ ⎪ ⎪ −1 2 T 4T ⎪ ⎪ ⎪0 = (r Ai (ε , t )) (r Pi1 (ε , μ, t )) + Ai (ε , t )Pi3 (ε , μ, t ) ⎪ ⎨+ Q T (t ) + ( −1C 2 (ε , μ, t ))T (r Pi1 (ε , μ, t ))T ( −1C 1 (ε , μ, t )) r r i2 i i −[(r Pi1 (ε , μ, t ))T (r−1 B1i (ε , t )) ⎪ ⎪ ⎪ ⎪ −1 1 T ⎪ (3.13b ) ⎪+ (r Ci (ε , μ, t )) (r Pi1 (ε , μ, t )) ⎪ ⎪ −1 1 ⎪ × ( D ( ε , μ , t )) + L ( t ) ][ R ( t ) ⎪ i1 i r i ⎪ ⎪ ⎪ + (r−1 D1i (ε , μ, t ))T (r Pi1 (ε , μ, t ))(r−1 D1i (ε , μ, t ))]−1 ⎪ ⎪ ⎪ ⎪ ⎪× [(r−1Ci2 (ε , μ, t ))T (r Pi1 (ε , μ, t ))(r−1 D1i (ε , μ, t )) + Li2 (t )]T , ⎪ ⎪ ⎪ ⎪ 0 = PiT4 (ε , μ, t )A4i (ε , t ) + A4i T (ε , t )Pi4 (ε , μ, t ) + Qi4 (t ) ⎪ ⎪ ⎪ ⎪ ⎪+ (r−1Ci2 (ε , μ, t ))T (r Pi1 (ε , μ, t ))T (r−1 ⎪ ⎪ ⎪ ⎪ × Ci2 (ε , μ, t )) − [(r−1Ci2 (ε , μ, t ))T ⎪ ⎪ ⎪ ⎪ (3.13c) ⎪ × (r Pi1 (ε , μ, t ))(r−1 D1i (ε , μ, t )) + Li2 (t )][Ri (t ) ⎪ ⎪ ⎪ −1 1 T −1 1 −1 ⎪ + (r Di (ε , μ, t )) (r Pi1 (ε , μ, t ))(r Di (ε , μ, t ))] ⎪ ⎪ ⎩ × [(r−1Ci2 (ε , μ, t ))T (r Pi1 (ε , μ, t ))(r−1 D1i (ε , μ, t )) + Li2 (t )]T . By the conditions of Theorem 3.1, using the transformations (3.7) and (3.8), we can get that

Q11 (t )

L11 (t )

(t )

Ri (t )

LT11

=

Ni1

0

0

T

Li (t )

(t )

Ri (t )

LTi

I

Qi (t )

Ni1

0

0

I

,

where Ni = (Ni1 Ni2 ), Ni1 is full-column rank with appropriate dimension. Having a careful observation to the system (3.13), we can know that Eq. (3.13a) has a solution r Pi1 (ε , μ, t) on [0, T] with r Pi1 (ε , μ, t) 0, ∀t ∈ [0, T], guaranteed by Dragan et al. [3], then substituting it into (3.13b), we could get the solution Pi3 (ε , μ, t). By the assumption H(3.2 ), we can use the similar method in [23] to deal with Eq. (3.13c), so we can get a solution Pi4 (ε , μ, t) to Eq. (3.13c). From what has been discussed above, theorem is proved. This is complete the proof. Remark 3.1. In particular, when Di (t) ≡ 0, Li (t) ≡ 0, S = {1}, and without have parameters ε , μ in Theorem 3.1, we can see that the result can not be reduced to the result in [23]. So (3.1) can be regarded as an extension of the GDRE in [23].

It is shown above, suppose that

Pi1 (ε , μ, t ) Pi3 (ε , μ, t )

A1i (ε , t ) A3i (ε , t )

Pi2 (ε , μ, t ) Pi4 (ε , μ, t )

A2i (ε , t ) A4i (ε , t )

Ci1 (ε , μ, t ) 0

=

Pi1 (t ) ε Pi3 (t )

=

0

ε Pi4 (t ) 1

,

A2i (t ) Bi ( ε , t ) , −1 4 ε Ai (t ) B2i (ε , t )

A1i (t ) 0

B1i (t ) , 0

=

Ci2 (ε , μ, t ) 0

=

−1 1 με Ci (t ) ×

=

με −1Ci2 (t )

0

D1i (ε , μ, t ) 0

−1 1 με Di (t ) 0

,

0

,

then (3.13a)–(3.13c) can be rewritten as

⎧ −(r P˙i1 (t )) = (r Pi1 (t ))T (r−1 A1i (t )) + (r−1 A1i (t ))T (r Pi1 (t )) ⎪ ⎪ ⎪ ⎪ + j∈S λi j r Pj1 (t ) + Qi1 (t ) ⎪ ⎪ ⎪ ⎪ ⎪ + (r−1 με −1Ci1 (t ))T (r Pi1 (t ))(r−1 με −1Ci1 (t )) ⎪ ⎪ ⎪ ⎪ −[(r Pi1 (t ))T (r−1 B1i (t )) + (r−1 με −1Ci1 (t ))T (r Pi1 (t ))T ⎪ ⎪ ⎪ ⎪ × (r−1 με −1 D1i (t )) + Li1 (t )][Ri (t ) + (r−1 με −1 D1i (t ))T (3.14a ) ⎪ ⎪ ⎪ ⎪ × (r Pi1 (t ))(r−1 με −1 D1i (t ))]−1 [(r Pi1 (t ))T (r−1 B1i (t )) ⎪ ⎪ ⎪ ⎪ ⎪ + (r−1 με −1Ci1 (t ))(r Pi1 (t ))T (r−1 με −1 D1i (t )) + Li1 (t )]T , ⎪ ⎪ ⎪ ⎪ 0 = (r−1 A2i (t ))T (r Pi1 (t )) + ε −1 A4i T (t )ε Pi3 (t ) + QiT2 (t ) ⎪ ⎪ ⎪ ⎨ + ( −1 με −1C 2 (t ))T (r Pi1 (t ))T ( −1 με −1C 1 (t )) r r i i T −1 1 −1 −1 1 −[ ( P ( t )) ( B ( t )) + ( με Ci (t ))T (r Pi1 (t )) r i1 ⎪ r r i ⎪ ⎪ ⎪ ⎪ × (r−1 με −1 D1i (t )) + Li1 (t )][Ri (t ) ⎪ ⎪ ⎪ ⎪ + (r−1 με −1 D1i (t ))T (r Pi1 (t ))(r−1 με −1 D1i (t )]−1 (3.14b ) ⎪ ⎪ ⎪ −1 −1 2 T −1 −1 1 ⎪ × [ ( με C ( t )) ( P ( t ))( με D ( t )) + L ( t ) ]T , ⎪ r i1 i2 r r i i ⎪ ⎪ ⎪ T −1 4 −1 4 T ⎪0 = ε Pi4 (t )ε Ai (t ) + ε Ai (t )ε Pi4 (t ) + Qi4 (t ) ⎪ ⎪ ⎪ ⎪ + (r−1 με −1Ci2 (t ))T (r Pi1 (t ))T (r−1 με −1Ci2 (t )) (3.14c) ⎪ ⎪ ⎪ ⎪ −1 −1 2 T −1 −1 1 ⎪ −[(r με Ci (t )) (r Pi1 (t ))(r με Di (t )) + Li2 (t )][Ri (t ) ⎪ ⎪ ⎪ ⎪ + (r−1 με −1 D1i (t ))T (r Pi1 (t ))(r−1 με −1 D1i (t ))]−1 ⎪ ⎪ ⎩ × [(r−1 με −1Ci2 (t ))T (r Pi1 (t ))(r−1 με −1 D1i (t )) + Li2 (t )]T . One can see that in (3.14a)–(3.14c), the expression ρ = μ2 ε −2 occurs. Unfortunately, in the absence of additional assumptions, the fact that (μ, ε ) → (0, 0) does not guarantee the existence of lim(μ,ε )→(0,0 ) μ2 ε −2 . In order to overcome this inconvenience, in this paper, we make the following assumption: H (3.3 ) The small parameters ε > 0, μ > 0 satisfy the condition

lim

(μ,ε )→(0,0 )

ρ=

lim

(μ,ε )→(0,0 )

μ2 ε −2 = 1.

Under assumption H (3.3 ), taking formally ε → 0, μ → 0, and ρ = μ2 ε −2 → 1 in (3.14a)–(3.14c), we obtain the parameterindependent system

⎧ −(r P˙i1 (t )) = (r Pi1 (t ))T (r−1 A1i (t )) + (r−1 A1i (t ))T (r Pi1 (t )) ⎪ ⎪ ⎪ ⎪ + j∈S λi j r Pj1 (t ) + Qi1 (t ) + (r−1Ci1 (t ))T ⎪ ⎪ ⎪ ⎪ × (r Pi1 (t ))(r−1Ci1 (t )) − [(r Pi1 (t ))T (r−1 B1i (t )) ⎪ ⎪ ⎪ ⎪ + ( −1 με −1C 1 (t ))T ( P (t ))T ⎪ r i1 ⎪ r i ⎪ ⎪ −1 −1 1 ⎪ × ( με D ( t )) + L ⎪ i1 (t )][Ri (t ) r i ⎪ ⎪ ⎪ −1 1 T −1 1 −1 ⎪ + ( D ( t )) ( P ( t ))( (3.15a ) r i1 r r Di (t ))] ⎪ i ⎪ ⎪ T −1 1 ⎪ × [(r Pi1 (t )) (r Bi (t )) ⎪ ⎪ ⎪ ⎪ ⎪ + (r−1Ci1 (t ))(r Pi1 (t ))T (r−1 D1i (t )) + Li1 (t )]T , ⎪ ⎪ ⎨ 0 = (r−1 A2i (t ))T (r Pi1 (t )) + A4i T (t )Pi3 (t ) + QiT2 (t ) −1 2 T T −1 1 ⎪ ⎪ + (r Ci (t )) (r Pi1 (t )) (r Ci (t )) ⎪ ⎪ ⎪ ⎪−[(r Pi1 (t ))T (r−1 B1i (t )) + (r−1Ci1 (t ))T (r Pi1 (t ))(r−1 D1i (t )) ⎪ ⎪ ⎪ + Li1 (t )][Ri (t ) + (r−1 D1i (t ))T (r Pi1 (t ))(r−1 D1i (t )]−1 (3.15b ) ⎪ ⎪ ⎪ ⎪ ⎪ × [(r−1Ci2 (t ))T (r Pi1 (t ))(r−1 D1i (t )) + Li2 (t )]T , ⎪ ⎪ ⎪ ⎪ ⎪0 = PiT4 (t )A4i (t ) + A4i T (t )Pi4 (t ) + Qi4 (t ) ⎪ ⎪ ⎪ ⎪ + (r−1Ci2 (t ))T (r Pi1 (t ))T (r−1Ci2 (t )) − [(r−1Ci2 (t ))T (3.15c) ⎪ ⎪ ⎪ ⎪ × (r Pi1 (t ))(r−1 D1i (t )) + Li2 (t )][Ri (t ) ⎪ ⎪ ⎪ ⎪ ⎪ + (r−1 D1i (t ))T (r Pi1 (t ))(r−1 D1i (t ))]−1 ⎪ ⎩ × [(r−1Ci2 (t ))T (r Pi1 (t ))(r−1 D1i (t )) + Li2 (t )]T . we can know that Eq. (3.15a) has a solution r Pi1 (t) on [0, T] with r Pi1 (t) 0, ∀t ∈ [0, T], guaranteed by Dragan et al. [3], then substituting it into (3.15b), we could get the solution Pi3 (t). By the assumption H (3.2 ), we can use the similar method in [23] to deal with the equation (3.15c), so we can get a solution Pi4 (t) to Eq. (3.15c).

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4. Application to near-optimal control In section 2, we known the system (2.1) has a no-impulse solution on [0, T], ∀i ∈ S, inspired by the authors in Refs. [7,9,12,13], in this section, we apply the above existence results to study the near-optimal control of Markovian jump singularly perturbed linear stochastic singular system, and obtain the desired explicit representation of the near-optimal controllers for the near-optimal control problem with the ﬁnite horizon. First of all, we now give basic deﬁnition, assumption and lemmas before continuing our discussion, which will be used in the derivations of the main theorem. The objective of the near-optimal control in this paper is to ﬁnd the near-optimal control u∗ (· ) ∈ Uad that minimizes the performance index J(0, x0 , r(0), u, T). the optimal valued function is deﬁned as

V (0, x0 ) = min J (0, x0 , r (0 ), u, T ). u(· )∈Uad

Deﬁnition 4.1. The optimization problem is called well-posed if

−∞ < V (0, x0 ) < + ∞,

∀x0 ∈ Rn .

7

Proof. By the condition that E T Pi (ε , μ, t ) = PiT (ε , μ, t )E, we can apply generalized Itô’s formula to xT (t)ET Pi (ε , μ, t)x(t),

d[xT (t )E T Pi (ε , μ, t )x(t )]

= (Ai (ε , t )x(t ) + Bi (ε , t )u(t ))T Pi (ε , μ, t )x(t ) + xT (t )E T P˙i (ε , μ, t )x(t )

+ xT (t )PiT (ε , μ, t )(Ai (ε , t )x(t ) + Bi (ε , t )u(t )) +

λi j xT (t )E T

j∈S

× Pj (ε , μ, t )x(t ) + (Ci (ε , μ, t )x(t ) + Di (ε , μ, t )u(t ))T (E † )T

× PiT (ε , μ, t )E E † (Ci (ε , μ, t )x(t ) + Di (ε , μ, t )u(t )) dt + [· · · ]dω (t ), where [] does not affect the calculation result and can be omitted. Apply (3.1a) to the above equation, we get (3.2). This completes the proof. Lemma 4.3. For arbitrary u ∈ Uad , the cost functional deﬁned in (2.4) is given by

T

A well-posed problem is called attainable (with respect to x0 ) if there is a control u∗ (·) that achieves V(0, x0 ). In this case, the control u∗ (·) is called optimal (with respect to x0 ). Then, we give the controllability condition to ensure system (2.1) controllable and existence of optimal control. H (4.1 ) The triple (A4i , B2i , Ci2 , D1i ) is completely controllable and observable.

J (0, x0 , r (0 ), u, T ) = xT (0 )E T Pi x(0 ) +

Lemma 4.1 ([4] Generalized Itô’s formula). Let x(t) satisfy

i (ε , μ, t )(Mi (ε , μ, t )x(t ) + Ri (ε , μ, t )u(t )), with Pi (ε , μ, t)

dx(t ) = b(t , x(t ), r (t ))dt +

i

σ (t , x(t ), r (t ))dω (t ),

satisfying (3.1a) and (3.1b).

(ϕ (T , x(T ), r (T )) − ϕ (s, x(s ), r (s ))|r (s ) = i ) T = ϕt (t, x(t ), r (t )) + ϕ (t, x(t ), r (t )) dt r (s ) = i , (4.1)

J (0, x0 , r (0 ), u, T )

= xT (T )E T L(r (T ))Ex(T ) +

s

where

1 2

ϕ (t, x(t ), i ) = t r[σ T (t , x(t ), i )ϕxx (t , x(t ), i )σ (t , x(t ), i )] + bT (t, x(t ), i )ϕx (t, x(t ), i ) +

λi j ϕ (t, x(t ), i ).

j∈S

Lemma 4.2. Let f is a differentiable function be such that f (t, x, i ) = xT (t )E T Pi (ε , μ, t )x(t ), where Pi (ε , μ, t ) ∈ Rn×n satisﬁes the SGCDRE given by (3.1a) and (3.1b). Then, for system (2.1) with u ∈ Uad , the generalized Itô’s formula (4.1) can be written as

xT (t )E T Pi (ε , μ, t )x(t ) − xT (s )E T Pi (ε , μ, s )x(s ) t = xT (σ )(Pi (ε , μ, σ )Bi (σ )

0

xT (t )Q (r (t ), t )x(t )

+ DTi (ε , μ, σ )(E † )T PiT (ε , μ, σ )E E † Di (ε , μ, σ ))−1

− x (σ )Qi (σ )x(σ ) + u (σ )(

BTi

(ε , μ, σ )(E )

(ε , μ, t )(E E † )T Pi (ε , μ, t )E † Di (ε , μ, t ) + Li (t ))(Ri (t )

+ DTi (ε , μ, t )(E † )T PiT (ε , μ, t )E E † Di

× (BTi (ε , t )Pi (ε , μ, t ) + DTi (ε , μ, t ) × (E † )T PiT (ε , μ, t )E E †Ci (ε , μ, t ) + Li (t ))x(t )

+ Ri (t ))u(t ) dt + x (0 )E Pi x(0 ) T

(ε , μ, σ )E E Ci (ε , μ, σ ))u(σ ) + u (σ ) × (DTi (ε , μ, σ )(E † )T PiT (ε , μ, σ )E E † Di (ε , μ, σ ))u(σ ) dσ . †

+

(ε , σ )Pi (ε , μ, σ )

(ε , σ )Pi (ε , μ, σ ) PiT

0

CiT

+ uT (t )(DTi (ε , μ, t )(E † )T PiT (ε , μ, t )E E † Di (ε , μ, t )

+ DTi (ε , μ, σ )(E † )T PiT (ε , μ, σ )E E †Ci (ε , μ, σ ))x(σ ) † T

J (0, x0 , r (0 ), u, T ) T = xT (t )(PiT (ε , μ, t )Bi (ε , t )

+ DTi (ε , μ, t )(E † )T PiT (ε , μ, t )E E †Ci (ε , μ, t ) + LTi (t ))u(t )

× PiT (ε , μ, σ )E E † × Ci (ε , μ, σ ) + LTi (σ )))x(σ ) T

Now, from Lemma 4.1, setting s = 0 and t = T in (3.2), we get that

+ xT (t )(BTi (ε , t )Pi (ε , μ, t )

× (BTi (ε , σ )Pi (ε , μ, σ ) + DTi (ε , μ, σ )(E † )T T

T

× PiT (ε , μ, t )E E †Ci (ε , μ, t ) + LTi (t )))x(t ) + uT (t )

s

+

T

× (ε , μ, t ))−1 (BTi (ε , t )Pi (ε , μ, t ) + DTi (ε , μ, t )(E † )T

+ CiT (ε , μ, σ )(E E † )T Pi (ε , μ, σ )E † Di (ε , μ, σ ) + Li (σ ))(Ri (σ )

DTi

+ 2x (t )L(r (t ), t )u(t ) + u (t )R(r (t ), t )u(t )dt T

+ x (σ )(

where i (ε , μ, t ) = R−1 ( ε , μ, t ) , Mi (ε, μ, t )x(t ) + i Ri (ε , μ, t )u(t )2 (ε,μ,t ) := (Mi (ε , μ, t )x(t ) + Ri (ε , μ, t )u(t ))T

Proof. From (2.4) we have that

BTi

Mi (ε , μ, t )x(t )

+ Ri (ε , μ, t )u(t )2i (t ) dt

and ϕ (·, ·, i ) ∈ C 2 ([0, ∞ ) × Rn ), ∀i ∈ S, be given. Then

T

0

T

=

T 0

T

[xT (t )MTi (ε , μ, t )R−1 (ε , μ, t )Mi (ε , μ, t )x(t ) i

+ uT (t )Mi (ε , μ, t )x(t ) + xT (t )MTi (ε , μ, t )u(t )

Please cite this article as: G. Ren and B. Liu, Near-optimal (3.2) control for a singularly perturbed linear stochastic singular system with Markovian jumping parameters, European Journal of Control, https://doi.org/10.1016/j.ejcon.2019.04.002

ARTICLE IN PRESS

JID: EJCON 8

[m5G;May 2, 2019;22:18]

G. Ren and B. Liu / European Journal of Control xxx (xxxx) xxx

+ (Ri (ε , μ, t )u(t ))T R−1 (ε , μ, t )(Ri (ε , μ, t )u(t ))]dt i

turbed linear stochastic singular system, it motive us to this study. We not only warrant the given system has an impulse-free solution, and obtain the desired explicit representation of the optimal controllers for the optimal control problem with the ﬁnite horizon. The problem of inﬁnite horizon, we will leave in our future research.

+ xT (0 )E T Pi x(0 ) =

T

0

[xT (t )MTi (ε , μ, t )R−1 (ε , μ, t )Mi (ε , μ, t )x(t ) i

+ (Ri (ε , μ, t )u(t ))T R−1 (ε , μ, t )Mi (ε , μ, t )x(t ) i + x (t ) T

MTi

( ε , μ, t )

+ (Ri (ε , μ, t )u(t ))

R−1 i T

Acknowledgment

(ε , μ, t )(Ri (ε , μ, t )u(t ))

The authors express their gratitude to the anonymous reviewers and editors for their valuable comments and suggestions which led to the improvement of the original manuscript.

× R−1 (ε , μ, t )(Ri (t )u(t ))]dt + xT (0 )E T Pi x(0 ) i

= xT (0 )E T Pi x(0 ) + +

T 0

Supplementary material

[ηT i (ε , μ, t )η + yT i (ε , μ, t )y

Supplementary material associated with this article can be found, in the online version, at 10.1016/j.ejcon.2019.04.002.

ηT i (ε , μ, t )y + yT i (ε , μ, t )η]dt

= xT (0 )E T Pi x(0 ) +

= xT (0 )E T Pi x(0 ) +

= xT (0 )E T Pi x(0 ) +

T 0

T 0

T 0

References

[ (y +

η )T i (ε , μ, t )(y + η )]dt

y + η2i (ε,μ,t ) dt Mi (ε , μ, t )x(t )

+ Ri (ε , μ, t )u(t )i (ε,μ,t ) dt 2

where y = Mi (ε , μ, t )x(t ) and η = Ri (ε , μ, t )u(t ), which completes the proof. Theorem 4.1. Assume the SGCDRE admit a solution Pi (ε , μ, t ) ∈ Rn×n on t ∈ [0, T], the ﬁnite horizon LQ near-optimal control problem (2.1), (2.4) is well-posed. Then, the near-optimal control in the admissible class Uad is given by

uapp (t ) = −Ki (ε , μ, t )x(t ),

(4.3)

where Ki (ε , μ, t ) ∈ is given by Ki (ε , μ, t ) = R−1 (ε , μ, t )Mi (ε , μ, t ), i ∈ S. Furthermore the minimum cost is i given by Rm×n

V (0, x0 ) = min J (0, x0 , r (0 ), u, T ) = xT (0 )E T Pi (ε , μ, t )x(0 ). (4.4) u∈Uad

Proof. The proof is immediate from Lemma 4.3.

Remark 4.1. Compared with [3,23], although the method adopted here, to prove the suﬃciency of solvability of singularly perturbed SDCDRE for the well posedness of LQ problem, is the same, under the condition that Ri (t ) 0, we get a different result in the case of singularly perturbed linear stochastic singular systems with Markovian jumps. 5. Concluding remark In this paper, we consider the linear quadratic control problem for a singularly perturbed linear stochastic singular system with Markovian jumping parameters. To the best of our knowledge, the theory of the linear stochastic singular system is quite fragmentary, and the theory of the singularly perturbed linear stochastic singular system is unknown, in order to make up for the gap between singularly perturbed linear stochastic system and singularly per-

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Please cite this article as: G. Ren and B. Liu, Near-optimal control for a singularly perturbed linear stochastic singular system with Markovian jumping parameters, European Journal of Control, https://doi.org/10.1016/j.ejcon.2019.04.002