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To appear in: Journal of the Franklin Institute Received date: 22 January 2016 Revised date: 25 July 2016 Accepted date: 21 November 2016 Cite this article as: Weihai Zhang, Yaning Lin and Lingrong Xue, Linear quadratic pareto optimal control problem of stochastic singular systems, Journal of the Franklin Institute, http://dx.doi.org/10.1016/j.jfranklin.2016.11.021 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Linear Quadratic Pareto Optimal Control Problem of Stochastic Singular Systems

∗

Weihai Zhang †, Yaning Lin , Lingrong Xue College of Electrical Engineering and Automation, Shandong University of Science and Technology, Qingdao 266590, Shandong Province, P. R. China

November 24, 2016

1. College of Electrical Engineering and Automation, Shandong University of Science and Technology, Qingdao 266590, China; 2. College of Science, Shandong University of Technology, Zibo 255000, China.

Abstract: This paper is concerned with the ﬁnite horizon linear quadratic (LQ) Pareto optimal control problem of stochastic singular systems. By means of the square completion technique, for the ﬁnite horizon LQ optimal control of stochastic singular systems, we establish a new kind of generalized diﬀerential Riccati equations (GDREs) and present the existence condition of the solution of the GDREs. Then, for the ﬁnite horizon LQ Pareto optimal control, it is shown that under the solvability of the corresponding GDREs, all Pareto candidates can be obtained by solving a weighting sum optimal control. Finally, an example is provided to show the eﬀectiveness of our main results. Keywords: Pareto optimality; stochastic singular systems; ﬁnite horizon LQ optimal control; generalized diﬀerential Riccati equations. ∗ This

research was supported by NSF of China (Nos.61573227, 61633014), the Research Fund for the Taishan

Scholar Project of Shandong Province of China, the SDUST Research Fund (No.2015TDJH105) and the State Key Laboratory of Alternate Electrical Power System with Renewable Energy Sources(Grant No. LAPS16011) † Corresponding

author. E-mail: w [email protected] (W.Zhang).

1

1. Introduction Singular systems, which are also referred to as descriptor systems, generalized state-space systems, diﬀerential algebraic systems, or implicit systems, have extensive applications in many practical systems, for example, circuit boundary control systems, chemical processes, electrical networks, economy systems and other areas [7]. Hence a great number of fundamental notions and theories for singular systems have been researched, such as stability and stabilization [29, 31, 38], LQ optimal control [28, 30, 37], H∞ control [19, 26] and so on. As it is well known, environmental noise exists and cannot be neglected in many dynamical systems [5, 14, 20, 25]. So, singular systems with stochastic noise are more realistic mathematical models; see Example 5.2 in [33] and system (5) in [36]. There have been some reports on the study of stochastic singular systems, but most of which are concerned with the stability and stabilization. In [5], the author developed stochastic stability and stochastic stabilization criteria for stochastic singular systems, and designed two kinds of state feedback controllers to stochastically stabilize the systems. In [13], the authors gave a complete proof of mean-square exponential stability for singular stochastic Itˆo systems, and then investigated observer-based controller designs. The reference [35] discussed stability criteria for stochastic singular systems with state-dependent noise in both continuous-time and discrete-time cases. In [27], the authors studied the stochastic stabilization problem for a class of singular Markovian jump systems, and proposed a new kind of stochastic controller design such that the closed-loop system has a unique solution and is almost surely exponentially admissible. However, there are few articles talking about the optimal control problem of stochastic singular systems. In [33], under some assumptions, the authors developed a suﬃcient condition for mean-square admissibility in terms of LMI, investigated ﬁnite-time horizon and inﬁnite-time horizon LQ control problems using square completion technique, and discussed some results involving new stochastic generalized Riccati equation. The dynamic games of continuous-time and discrete-time systems have been extensively investigated by many researchers (see, e.g. [3] and the reference therein). Among various dynamic games, the Pareto game arises when multiple players, aﬀecting a dynamic system, coordinate their actions/controls in order to optimize their objectives, which has been widely used

2

in economic theory such as optimal economic growth, environmental economics and engineering [1, 3, 8, 9, 10, 22, 23, 24]. The set of Pareto eﬃcient equilibria for the indeﬁnite cooperative LQ control problem of linear aﬃne systems was determined in [9]. Necessary and suﬃcient conditions for the existence of a Pareto optimum of the ﬁnite horizon cooperative diﬀerential games of nonlinear systems were presented in [10]. The reference [23] derived conditions for the existence of Pareto optimal solutions for the LQ inﬁnite horizon cooperative diﬀerential games. In [24], the necessary and suﬃcient conditions for the existence of Pareto solutions in inﬁnite horizon cooperative diﬀerential games with open loop information structure were discussed. However, most of the reports are about the deterministic systems. To the best of our knowledge, the LQ pareto optimal control of stochastic singular systems is still unsolved. In this paper, we concentrate our attention on the ﬁnite horizon LQ Pareto optimality problem for a class of continuous-time stochastic Itˆ o singular systems. A well known way to obtain Pareto optimal controls is to solve a weighting sum optimal control problem [16, 32]. Every control minimizing a weighting sum of the cost functionals of all players (where all weights are in the unit simplex) is Pareto optimal. So, we ﬁrst discuss the general ﬁnite horizon LQ optimal control problem in Section 2. Using the square completion technique, we give a suﬃcient condition for the well-posedness by introducing a new type of GDREs. Furthermore, we put forward the existence condition of the solution of the GDREs. Next, in Section 3, we consider the ﬁnite horizon stochastic singular LQ Pareto optimal control problem. Via the discussion of convexity of the cost functional, under some assumptions, it is revealed that the solvability of the GDREs gives a suﬃcient condition under which we can ﬁnd all Pareto eﬃcient solutions by the weighting sum optimization method. Finally, an illustrative example is given to show the eﬀectiveness of the proposed results. Notations:

Rn : the set of all real n-dimensional vectors. Rm×n : the set of all m × n real

matrices. A > 0 (resp. A ≥ 0): A is a real symmetric positive deﬁnite (resp. positive semideﬁnite) matrix. In : n-dimensional identity matrix. AT : the transpose of matrix A. E(x): the mathematical expectation of x. tr(A): the trace of a square matrix A. A := {α = (α1 , α2 )|αi ≥ 0 and

2

i=1

αi = 1}. L2 [0, T ] := {u(t)|u(t) is a square integrable function on [0, T ]}. C 1,2 (R+ × Rn ):

the class of functions V (t, x) once continuously diﬀerential with respect to t ∈ R+ and twice continuously diﬀerential with respect to x ∈ Rn , except possibly at the point x = 0. 3

2. Finite Horizon Singular LQ Optimal Control In this section, we consider the following stochastic singular system ⎧ ⎪ ⎪ ⎨ Edx(t) = Ax(t) + Bu(t) dt + A x(t)dw(t), p

⎪ ⎪ ⎩ Ex(0) = x0 ,

(1)

where x(t) ∈ Rn is the system state vector, u(t) ∈ Rm is the control vector, w(t) is a onedimensional standard Wiener process that is deﬁned on the complete probability space (Ω, F , P). E, A, B, Ap are known matrices of appropriate dimensions and rank(E) = r ≤ n; x0 ∈ Rn is the compatible initial condition which is deterministic. Deﬁne the set of all admissible controls Uad = {u(·) ∈ L2 [0, T ]| the corresponding solution x(·) of (1) satisﬁes x(·) ∈ L2 [0, T ]}. For each x0 and u(·) ∈ Uad , we consider the following cost functional J(0, T, u, x0 ) E

0

T

T x (t)Qx(t) + uT (t)Ru(t) dt + xT (T )E T QT Ex(T ) ,

(2)

where Q = QT , QT = QTT and R > 0. The objective of the optimal control problem in this section is to ﬁnd the optimal control u∗ (t) ∈ Uad that minimizes the cost functional J(0, T, u, x0 ). The optimal value function is deﬁned as V (0, T, x0 ) min J(0, T, u, x0 ). u∈Uad

Definition 2.1. The optimization problem (1)-(2) is called well-posed if V (0, T, x0 ) > −∞. u∗ (t) is said to be optimal if u∗ (t) achieves the minimum of J(0, T, u, x0 ). Next, we give some assumptions which are necessary for the proofs of our main results. Assumption 2.1. rank(E ⎡ Ap ) = rank(E). ⎤ ⎢ E Assumption 2.2. rank ⎢ ⎣ 0

A E

B ⎥ ⎥ = n + rankE. ⎦ 0

Remark 2.1. Assumption 2.2 is a basic requirement for singular controlled systems [37]. Under Assumption 2.2, we can choose an appropriate control to eliminate the impulse phenomena

4

in system (1). Furthermore, Assumption 2.1 and Assumption 2.2 can guarantee the existence and uniqueness of the impulse-free solution of the stochastic singular system (1). When B = 0, Assumption 2.1 and Assumption 2.2 reduce to Assumption 1 in [13]. On this point, we have the following conclusion. Lemma 2.1. If Assumption 2.1 and Assumption 2.2 hold, then system (1) has a unique solution on [0, +∞), in which there is no impulse. Proof. Considering the singular value decomposition (SVD) of matrix E, under Assumption 2.1, there exist two orthogonal matrices M, N ∈ Rn×n such that ⎡ ⎤ ⎡ ⎢ Σr M EN = ⎢ ⎣ 0

0 ⎥ ⎥, ⎦ 0

⎢ Ap1 M Ap N = ⎢ ⎣ 0

⎤

Ap2 ⎥ ⎥, ⎦ 0

(3)

where Σr ∈ Rr×r is a nonsingular diagonal matrix and Ap1 ∈ Rr×r , Ap2 ∈ Rr×(n−r) . Accordingly, deﬁne

⎡

⎤

⎢ A11 M AN = ⎢ ⎣ A21

⎤

⎡

A12 ⎥ ⎥, ⎦ A22

⎢ B1 ⎥ ⎥, MB = ⎢ ⎦ ⎣ B2

(4)

where A11 ∈ Rr×r , A12 ∈ Rr×(n−r) , A21 ∈ R(n−r)×r , A22 ∈ R(n−r)×(n−r) , B1 ∈ Rr×m , B2 ∈ R(n−r)×m and let

⎡

⎤

⎢ ξ1 (t) ⎥ ⎥, ξ(t) = N −1 x(t) = ⎢ ⎣ ⎦ ξ2 (t)

(5)

where ξ1 (t) ∈ Rr , ξ2 (t) ∈ Rn−r . By (3)-(5), system (1) can be transformed into ⎧ ⎪ ⎪ ⎪ Σr dξ1 (t) = [A11 ξ1 (t) + A12 ξ2 (t) + B1 u(t)]dt ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ + [Ap1 ξ1 (t) + Ap2 ξ2 (t)]dw(t), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

(6)

ξ1 (0) = [Σ−1 r 0]M x0 , 0 = A21 ξ1 (t) + A22 ξ2 (t) + B2 u(t).

By Assumption 2.2, the matrix A22

B2 has full row rank, so there exists a nonsingular matrix

X ∈ R(n−r+m)×(n−r+m) such that

A22

B2

X=

5

In−r

0

.

Denoting

⎡

⎤

⎢ X11 X ⎢ ⎣ X21

X12 ⎥ ⎥, ⎦ X22

⎡

⎤

B1 ⎥ ⎢ A¯12 ⎥X ⎢ ⎦ ⎣ A¯p2 0

⎢ A12 ⎢ ⎣ Ap2

⎤

⎡

¯1 ⎥ B ⎥, ⎦ ¯ D

where X11 ∈ R(n−r)×(n−r) , X12 ∈ R(n−r)×m , X21 ∈ Rm×(n−r) , X22 ∈ Rm×m , A¯12 ∈ Rr×(n−r) , ¯1 ∈ Rr×m , A¯p2 ∈ R(n−r)×(n−r) , D ¯ ∈ R(n−r)×m . Taking the nonsingular transformation B ⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎢ ξ1 (t) ⎥ ⎢ Ir ⎢ ⎥ ⎢ ⎢ ⎥ ⎢ ⎢ ξ (t) ⎥ = ⎢ 0 ⎢ 2 ⎥ ⎢ ⎢ ⎥ ⎢ ⎣ ⎦ ⎣ 0 u(t)

0

0

X11

X12

X21

X22

⎥ ⎢ ξ1 (t) ⎥ ⎥⎢ ⎥ ⎥⎢ ⎥ ⎥ ⎢ ξ¯ (t) ⎥ , ⎥⎢ 2 ⎥ ⎥⎢ ⎥ ⎦⎣ ⎦ u ¯(t)

where ξ¯2 (t) ∈ Rn−r , u¯(t) ∈ Rm . By (7), system (6) is transformed as ⎧ ⎪ ⎪ ¯1 u ⎪ Σr dξ1 (t) = (A11 − A¯12 A21 )ξ1 (t) + B ¯(t) dt ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ¯ u¯(t) dw(t), ⎨ + (Ap1 − A¯p2 A21 )ξ1 (t) + D ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

(7)

(8)

ξ1 (0) = [Σ−1 r 0]M x0 , ξ¯2 (t) = −A21 ξ1 (t).

The ﬁrst equation of (8) is an ordinary stochastic diﬀerential equation, in which ξ1 (t) is the state vector and u ¯(t) is the control vector. According to Theorem 3.1 in [21], the ﬁrst equation of (8) has 0]M x0 . Accordingly, a unique solution ξ1 (t) on [0, +∞) under the initial condition ξ1 (0) = [Σ−1 r ξ¯2 (t) = −A21 ξ1 (t) exists and is unique. Thus, system (8) has a unique impulse-free solution on [0, +∞), which implies that system (1) has a unique impulse-free solution on [0, +∞). Now, we are in a position to give the main results of the LQ optimal control problem of stochastic singular systems. First, we give the following useful lemmas. Lemma 2.2. [15] Let V t, x(t) ∈ C 1,2 (R+ × Rn ), x(t) satisﬁes the following stochastic diﬀerential equation dx(t) = f (t)dt + g(t)dw(t), then V t, x(t) is an Itˆ o’s process and dV t, x(t) = LV t, x(t) dt + Vx t, x(t) g(t)dw(t), where 1 LV t, x(t) = Vt t, x(t) + Vx t, x(t) f (t) + tr g T (t)Vxx t, x(t) g(t) . 2 6

Lemma 2.3. [2] Let a matrix M ∈ Rm×n be given, then there exists a unique matrix M + ∈ Rn×m , which is called the Moore-Penrose pseudo inverse of M , such that ⎧ ⎪ ⎪ ⎨ M M + M = M, M + M M + = M + , ⎪ ⎪ ⎩

(M M + )T = M M + , (M + M )T = M + M.

The following GDREs for matrix P (t) play an important role in the study of the LQ optimal control of stochastic singular system ⎧ −E T P˙ (t) = AT P (t) + P T (t)A + Q + ATp (E + )T P T (t)EE + Ap ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ − P T (t)BR−1 B T P (t), ⎨

(9a)

⎪ ⎪ ⎪ E T P (T ) = E T QT E, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ E T P (t) = P T (t)E.

(9b) (9c)

Remark 2.2. (i)When E = I, system (1) degenerates to the deterministic stochastic system. (9) can be regarded as an extension of the GDRE in [34]. (ii) When Ap = 0, system (1) reduces to the deterministic singular system. In this case, (9) is consistent with the diﬀerential Riccati equations (DREs) of [30]. Make a transformation for P (t),

⎤

⎡

⎢ P11 (t) M −T P (t)N = ⎢ ⎣ P21 (t)

P12 (t) ⎥ ⎥, ⎦ P22 (t)

(10)

where P11 (t) ∈ Rr×r , P12 (t) ∈ Rr×(n−r) , P21 (t) ∈ R(n−r)×r , P22 (t) ∈ R(n−r)×(n−r) . Accordingly, denote

⎡

⎡

⎤

⎢ Q11 ˆ N T QN = ⎢ Q ⎣ QT12

Q12 ⎥ ⎥, ⎦ Q22

⎤

⎢ QT 11 M −T QT M −1 = ⎢ ⎣ QTT 12

QT 12 ⎥ ⎥, ⎦ QT 22

(11)

where Q11 , QT 11 ∈ Rr×r , Q12 , QT 12 ∈ Rr×(n−r) , Q22 , QT 22 ∈ R(n−r)×(n−r) . In order to discuss the existence condition of the solution for the GDREs (9), we need the following assumptions. Assumption 2.3. Q ≥ 0 and QT ≥ 0. ˆ ≥ 0 and QT 11 ≥ 0, so there exist two matrices C1 ∈ Rn×r and Under Assumption 2.3, Q C2 ∈ Rn×(n−r) such that

⎡ ⎢ Q11 ˆ=⎢ Q ⎣ QT12

⎤

⎤

⎡

Q12 ⎥ ⎢ ⎥ ⎥=⎢ ⎥ C 1 ⎦ ⎣ ⎦ Q22 C2T C1T

7

C2

.

Assumption 2.4. The triple A22

B2

C2 is completely controllable and observable.

Assumption 2.5. λi + λj = 0, where λi and λj are arbitrary eigenvalues of A22 . On the existence condition of the solution of GDREs (9), we have the following theorem. Theorem 2.1. If one of the following two conditions is satisﬁed: (1) Under Assumption 2.3 and Assumption 2.4, Ap2 = 0; (2) Under Assumption 2.3 and Assumption 2.5, B2 = 0, then the GDREs (9) has a solution P (t) with Σr P11 (t) ≥ 0 and P12 (t) = 0. Proof. Using the relation (3), the Moore-Penrose pseudo inverse of E is ⎡ ⎤ ⎢ Σ−1 r E+ = N ⎢ ⎣ 0

0 ⎥ ⎥ M. ⎦ 0

By relations (3), (4), (10) and (11), equation (9a) can be partitioned into ⎤ ⎡ ⎤⎡ ˙ ˙ ⎢ Σr 0 ⎥ ⎢ P11 (t) P12 (t) ⎥ ⎥ ⎥⎢ − ⎢ ⎦ ⎣ ⎦⎣ ˙ ˙ P21 (t) P22 (t) 0 0 ⎡ ⎤⎡ ⎤ ⎢ AT11 = ⎢ ⎣ AT12 ⎡

AT21 ⎥ ⎢ P11 (t) ⎥⎢ ⎦⎣ T P21 (t) A22 ⎤⎡

⎢ +⎢ ⎣

T (t) P11

⎢ +⎢ ⎣

ATp1

P12 (t) ⎥ ⎥ ⎦ P22 (t)

T P21 (t)

⎥ ⎢ A11 ⎥⎢ ⎦⎣ T T A21 P12 (t) P22 (t) ⎡ ⎤⎡ ⎤⎡

⎡

ATp2

0 ⎥ ⎢ Σr ⎥⎢ ⎦⎣ 0 0

T (t) ⎢ P11 −⎢ ⎣ T (t) P12

⎤

⎡

⎤

A12 ⎥ ⎢ Q11 Q12 ⎥ ⎥+⎢ ⎥ ⎦ ⎣ ⎦ T A22 Q12 Q22 ⎤⎡ ⎤⎡ T (t) P11

T P21 (t)

0 ⎥⎢ ⎥ ⎢ Σr ⎥⎢ ⎥⎢ ⎦⎣ ⎦⎣ T T 0 P12 (t) P22 (t) 0 ⎤⎡ ⎤⎡

T P21 (t) ⎥ ⎢ S11 ⎥⎢ ⎦⎣ T T P22 (t) S12

S12 ⎥ ⎢ P11 (t) ⎥⎢ ⎦⎣ S22 P21 (t)

0 ⎥⎢ ⎥⎢ ⎦⎣ 0 ⎤

(12) ⎤⎡ Σ−1 r 0

0 ⎥ ⎢ Ap1 ⎥⎢ ⎦⎣ 0 0

⎤ Ap2 ⎥ ⎥ ⎦ 0

P12 (t) ⎥ ⎥ ⎦ P22 (t)

with the boundary condition Σr P11 (T ) = Σr QT 11 Σr ≥ 0,

(13)

where S11 = B1 R−1 B1T , S12 = B1 R−1 B2T , S22 = B2 R−1 B2T . By (9c), we get Σr P11 (t) =

8

(Σr P11 (t))T and P12 (t) = 0. Then, from (12), we obtain three equations as follows: ⎧ T −1 T T ⎪ (Σr P˙11 (t)) = −Q11 − (Σr P11 (t))T (Σ−1 r A11 ) − P21 (t)A21 − (Σr A11 ) (Σr P11 (t)) − A21 P21 (t) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ T −1 T T ⎪ ⎪ + (Σr P11 (t))T (Σ−1 ⎪ r S12 )P21 (t) + P21 (t)(Σr S12 ) (Σr P11 (t)) + P21 (t)S22 P21 (t) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ −1 −1 T −1 ⎪ ⎪ + (Σr P11 (t))T (Σ−1 ⎪ r S11 Σr )(Σr P11 (t)) − (Σr Ap1 ) (Σr P11 (t))(Σr Ap1 ), (14a) ⎪ ⎪ ⎪ ⎨ T T T T (t)A21 − (Σ−1 0 = −QT12 − P22 r A12 ) (Σr P11 (t)) − A22 P21 (t) + P22 (t)S22 P21 (t) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ T T −1 T −1 ⎪ ⎪ (t)(Σ−1 (14b) + P22 r S12 ) (Σr P11 (t)) − (Σr Ap2 ) (Σr P11 (t))(Σr Ap1 ), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ T T ⎪ ⎪ (t)A22 − AT22 P22 (t) + P22 (t)S22 P22 (t) 0 = −Q22 − P22 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ T −1 (14c) − (Σ−1 r Ap2 ) (Σr P11 (t))(Σr Ap2 ). (1) If Ap2 = 0, (14) can be rewritten as ⎧ T −1 T T (Σr P˙11 (t)) = −Q11 − (Σr P11 (t))T (Σ−1 ⎪ r A11 ) − P21 (t)A21 − (Σr A11 ) (Σr P11 (t)) − A21 P21 (t) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ T −1 T T ⎪ + (Σr P11 (t))T (Σ−1 ⎪ r S12 )P21 (t) + P21 (t)(Σr S12 ) (Σr P11 (t)) + P21 (t)S22 P21 (t) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ −1 −1 T −1 ⎪ + (Σr P11 (t))T (Σ−1 ⎨ r S11 Σr )(Σr P11 (t)) − (Σr Ap1 ) (Σr P11 (t))(Σr Ap1 ), (15a) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

T T T T (t)A21 − (Σ−1 0 = −QT12 − P22 r A12 ) (Σr P11 (t)) − A22 P21 (t) + P22 (t)S22 P21 (t) T T (t)(Σ−1 + P22 r S12 ) (Σr P11 (t)),

(15b)

T T (t)A22 − AT22 P22 (t) + P22 (t)S22 P22 (t). 0 = −Q22 − P22

(15c)

(15c) is an algebraic Riccati equation. Considering Q22 = C2T C2 ≥ 0 and S22 = B2 R−1 B2T with R > 0, under Assumption 2.4, (15c) has a unique positive deﬁnite solution P22 (t). Substituting P22 (t) into (15b), we get P21 (t) = L2 (t) + L1 (t)(Σr P11 (t)), where −T −1 T (Σr A12 ) − (Σ−1 L1 (t) = − A22 − S22 P22 (t) r S12 )P22 (t) , −T T Q12 + AT21 P22 (t) . L2 (t) = − A22 − S22 P22 (t) Furthermore, substituting P21 (t) into (15a), and making some calculations, we get (Σr P˙11 (t)) = −Q0 − (Σr P11 (t))T A0 − AT0 (Σr P11 (t)) + (Σr P11 (t))T S0 (Σr P11 (t)) T −1 − (Σ−1 r Ap1 ) (Σr P11 (t))(Σr Ap1 )

9

(16)

with the boundary condition (13), where Q0 = Q11 + LT2 (t)A21 + AT21 L2 (t) − LT2 (t)S22 L2 (t), T −1 T A0 = Σ−1 r A11 + L1 (t)A21 − (Σr S12 )L2 (t) − L1 (t)S22 L2 (t), T −1 T −1 T S0 = Σ−1 r S11 + L1 (t)(Σr S12 ) + (Σr S12 )L1 (t) + L1 (t)S22 L1 (t).

Similar to the proof of Theorem 1 in [28], we can prove that there exist B0 and C0 such that S0 = B0 R−1 B0T , Q0 = C0T C0 . By [4], there exists a unique symmetric positive semi-deﬁnite solution Σr P11 (t) ≥ 0 to the DRE (16). Therefore, ⎡ ⎢ P11 (t) P (t) = M T ⎢ ⎣ P21 (t)

⎤ P12 (t) ⎥ ⎥ N −1 ⎦ P22 (t)

is the solution of (9), which satisﬁes Σr P11 (t) ≥ 0 and P12 (t) = 0. (2) If B2 = 0, then S12 = S22 = 0. and Considering P12 (t) = 0, pre-multiplying and post-multiplying (12) by Ir − AT21 A−T 22

Ir − AT21 A−T 22

T

respectively, we obtain

−1 −1 (Σr P˙11 (t)) = −(Σr P11 (t))T (Σ−1 r A11 ) − (Σr A12 )A22 A21 T −1 −1 − (Σ−1 (Σr P11 (t)) r A11 ) − (Σr A12 )A22 A21 −T T T −1 ˆ + (Σr P11 (t))T (Σ−1 − Ir − AT21 A−T r S11 Σr )(Σr P11 (t)) 22 Q Ir − A21 A22

(17)

T −1 −1 −1 −1 (Σr P11 (t)) (Σ−1 − (Σ−1 r Ap1 ) − (Σr Ap2 )A22 A21 r Ap1 ) − (Σr Ap2 )A22 A21 ˆ ≥ 0, and S11 = B1 R−1 B T with R > 0, according with the boundary condition (13). Considering Q 1 to [4], the DRE (17) has a unique symmetric positive semi-deﬁnite solution Σr P11 (t) ≥ 0 on [0, T ]. Substituting Σr P11 (t) and S22 = 0 into (14c), and considering Q22 = C2T C2 , we have ⎡ ⎤T ⎡ ⎤ ⎢ T AT22 P22 (t) + P22 (t)A22 + ⎢ ⎣

C2 1

(Σr P11 (t)) 2 (Σ−1 r Ap2 )

⎥ ⎢ ⎥ ⎢ ⎦ ⎣

C2 1

(Σr P11 (t)) 2 (Σ−1 r Ap2 )

⎥ ⎥ = 0. ⎦

(18)

Under Assumption 2.5, (18) has a unique solution P22 (t). Substituting Σr P11 (t) and P22 (t) into (14b), and considering S12 = S22 = 0, we get T T −1 T −1 − QT12 − P22 P21 (t) = A−T (t)A21 − (Σ−1 r A12 ) (Σr P11 (t)) − (Σr Ap2 ) (Σr P11 (t))(Σr Ap1 ) . 22 Therefore, the GDREs (9) have a solution ⎡

⎤

⎢ P11 (t) P12 (t) ⎥ −1 ⎥N , P (t) = M T ⎢ ⎦ ⎣ P21 (t) P22 (t) 10

in which Σr P11 (t) ≥ 0 and P12 (t) = 0. The conclusion is proved. Remark 2.3. We give a suﬃcient but not necessary condition for the existence of the solution to GDREs (9). That is to say, even if the conditions in Theorem 2.1 are not established, the solution to GDREs (9) may also exist. Please refer to the example in Section 4. Remark 2.4. When Ap1 = Ap2 = 0, i.e. Ap = 0, GDREs (9) degenerates to DREs of deterministic singular systems of [30]. In this case, Assumption 2.3 and Assumption 2.4 are consistent with the existence conditions of the solution in [30]. For this case, our results extended the existing results in [30]. Based on the condition for the unique solvability of Sylvester equation [12], Assumption 2.5 can guarantee the existence and uniqueness of the solution of (18). For the well-posedness of the ﬁnite horizon LQ optimal control problem (1)-(2), we have the following conclusion. Theorem 2.2. If the GDREs (9) admit a solution on [0, T ], then the ﬁnite horizon LQ optimal control problem (1)-(2) admits uncountably many feedback optimal control strategies given by u∗ (t) = −R−1 B T K(t)x(t), where

(19) ⎤

⎡ ⎢ K(t) = M T ⎢ ⎣ ⎡

P11 (t)

0

P21 (t) + P22 (t)L(t) − F (t)L(t) F (t) ⎤

⎥ −1 ⎥N ⎦

⎢ P11 (t) P12 (t) ⎥ −1 ⎥ N being a solution of (9), L(t) = − A22 −S22 P22 (t) −1 A21 − with P (t) = M T ⎢ ⎦ ⎣ P21 (t) P22 (t) T S12 P11 (t) − S22 P21 (t) and F (t) is any time-varying matrix making A22 − S22 F (t) invertible within the interval [0, T ]. Additionally, the minimum value of the cost functional (2) is given by J(0, T, u∗ , x0 ) = xT (0)E T P (0)x(0). Proof. Under the condition E T P (t) = P T (t)E, applying Itˆo’s formula, we obtain T d xT (t)E T P (t)x(t) = d Ex(t) P (t)x(t) + xT (t)E T dP (t) x(t) + xT (t)P T (t)d Ex(t) T + d Ex(t) (E + )T P T (t)EE + d Ex(t) T = Ax(t) + Bu(t) P (t)x(t) + xT (t)E T P˙ (t)x(t) + xT (t)P T (t) Ax(t) + Bu(t)

+ xT (t)ATp (E + )T P T (t)EE + Ap x(t) dt + · · · dw(t), 11

(20)

where

···

does not aﬀect the calculation results and can be omitted.

Integrating (20) from 0 to T , taking expectations on the both sides, one gets E xT (T )E T P (T )x(T ) − xT (0)E T P (0)x(0) T T =E Ax(t) + Bu(t) P (t)x(t) + xT (t)E T P˙ (t)x(t)

(21)

0

+ xT (t)P T (t) Ax(t) + Bu(t) + xT (t)ATp (E + )T P T (t)EE + Ap x(t) dt . Adding (21) to (2), and using the square completion technique, we have J(0, T, u, x0 ) T xT (t) Q + AT P (t) + P T (t)A + E T P˙ (t) =E 0

+

ATp (E + )T P T (t)EE + Ap T −1 T

+E

0

u(t) + R

− P T (t)BR−1 B T P (t) x(t)dt

T B P (t)x(t) R u(t) + R−1 B T P (t)x(t) dt

+ E x (T )E T QT Ex(T ) − E xT (T )E T P (T )x(T ) + xT (0)E T P (0)x(0)

T

= xT (0)E T P (0)x(0) T T +E u(t) + R−1 B T P (t)x(t) R u(t) + R−1 B T P (t)x(t) dt . 0

Obviously, when u(t) = −R−1 B T P (t)x(t),

(22)

J(0, T, u, x0 ) attains its minimum xT (0)E T P (0)x(0). Now, substituting (22) into (6), we obtain ⎧ Σr dξ1 (t) = A11 ξ1 (t) + A12 ξ2 (t) − S11 P11 (t)ξ1 (t) − S12 P21 (t)ξ1 (t) − S12 P22 (t)ξ2 (t) dt ⎪ ⎪ ⎪ ⎪ ⎨ + Ap1 ξ1 (t) + Ap2 ξ2 (t) dw(t), (23a) ⎪ ⎪ ⎪ ⎪ ⎩ T P11 (t)ξ1 (t) − S22 P21 (t)ξ1 (t) − S22 P22 (t)ξ2 (t). (23b) 0 = A21 ξ1 (t) + A22 ξ2 (t) − S12 From (23b), we have ξ2 (t) = L(t)ξ1 (t).

(24)

Using the relation (24), after some calculations, we obtain u∗ (t) = −R−1 B1T P11 (t) + B2T P21 (t) + P22 (t)L(t) ξ1 (t) −R

−1

B2T F (t)

Equation (25) can be rewritten as ⎡ ⎢ u∗ (t) = −R−1 B1T B2T ⎢ ⎣

(25)

ξ2 (t) − L(t)ξ1 (t) . ⎤⎡ P11 (t)

P21 (t) + P22 (t)L(t) − F (t)L(t) 12

0

⎤

⎥ ⎢ ξ1 (t) ⎥ ⎥⎢ ⎥. ⎦⎣ ⎦ F (t) ξ2 (t)

(26)

Making an inverse transformation to (26) yields (19). By the same reason in the implementation of the optimal control as [6], we conclude that the two trajectories obtained by applying (22) and (26) respectively coincide exactly since A22 − S22 F (t) is invertible within the interval [0, T ]. Therefore, (19) constitutes the optimal linear feedback strategies for the problem (1)-(2) and their gains are not unique. Remark 2.5. Compared with [33], although the square completion technique adopted to prove the suﬃcient condition of well-posedness is the same, we chose a diﬀerent Lyapunov function and obtained a new type of GDREs. Furthermore, we presented the existence condition of the solution of the GDREs. So, our results improved the known results in [33]. Remark 2.6. Unlike the standard stochastic systems, the optimal control strategy of stochastic singular systems is not unique.

3. LQ Pareto Optimal Control of Stochastic Singular Systems In this section, we consider the cooperative diﬀerential game that two players decide to coordinate their actions with an intent to minimize their cost functionals. For player i, i = 1, 2, the cost functional

xT (t)Qi x(t) + uT1 (t)Ri1 u1 (t) + uT2 (t)Ri2 u2 (t) dt 0 + xT (T )E T QiT Ex(T ) ,

Ji (0, T, u1 , u2 , x0 ) = E

T

(27)

T T > 0, Rij = Rij ≥ 0, for j = i, j = 1, 2 and x(t) ∈ Rn is the where Qi = QTi , QiT = QTiT , Rii = Rii

state vector of the following stochastic singular equation ⎧ ⎪ ⎪ ⎨ Edx(t) = Ax(t) + B1 u1 (t) + B2 u2 (t) dt + Ap x(t)dw(t), ⎪ ⎪ ⎩ Ex(0) = x0 ,

(28)

where ui (t) ∈ Rmi is the control vector of player i. E, A, B1 , B2 , Ap are known matrices of appropriate dimensions and rank(E) = r ≤ n; x0 ∈ Rn is the compatible initial condition which is deterministic. Since the players coordinate their actions, we denote the joint action by u(t) (u1 (t), u2 (t)) ∈ Rm with m =

2

i=1

mi . we assume an open-loop information structure.

The set of all admissible controls is denoted by U. In this section, we consider U = {u(·) (u1 (·), u2 (·)) ∈ L2 [0, T ]| the corresponding solution x(·) of (28) satisﬁes x(·) ∈ L2 [0, T ]}. Since 13

we are interested in the joint minimization of the objectives of the players, the cost incurred by a single player cannot be minimized without increasing the cost incurred by other players. So, we consider solutions which cannot be improved upon by all the players simultaneously, i.e., the so-called Pareto optimal solutions. Definition 3.1. [9] Let U denote the set of admissible controls. Then uˆ ∈ U is called Pareto eﬃcient if the set of inequalities ˆ, x0 ), i = 1, 2, Ji (0, T, u, x0 ) ≤ Ji (0, T, u does not allow for any solution u ∈ U, where at least one of the inequalities is strict. The corresponding point (J1 (0, T, u ˆ, x0 ), J2 (0, T, u ˆ, x0 )) ∈ R2 is called a pareto solution. The set of all Pareto solutions is called the Pareto frontier. The objective of this section is to ﬁnd the set of Pareto eﬃcient solutions of the ﬁnite horizon LQ diﬀerential game of the stochastic singular systems. First, we give the following assumption and lemmas.

⎤

⎡

⎢ E Assumption 3.1. rank ⎢ ⎣ 0

A

B1

E

0

B2 ⎥ ⎥ = n + rankE. ⎦ 0

Remark 3.1. Assumption 3.1 is in accordance with Assumption 2.2 when B = [B1 B2 ]. Likewise, Assumption 2.1 and Assumption 3.1 can guarantee the existence and uniqueness of the impulse-free solution of system (28). Under transformation (3), we can denote ⎡ ⎤ ⎢ B11 ⎥ ⎥, M B1 = ⎢ ⎣ ⎦ B12

⎡

⎤

⎢ B21 ⎥ ⎥, M B2 = ⎢ ⎣ ⎦ B22

then Assumption 3.1 is equivalent to that the matrix [A22

B12

(29)

B22 ] has full row rank, which

means that the system (28) is impulsive controllable. A well known way to ﬁnd Pareto optimal controls is to solve a parameterized optimal control problem [16, 32]. Lemma 3.1, given below, states that every control minimizing a weighting sum of the cost functionals of all players (where all weights are in the unit simplex) is Pareto optimal. So, varying the weights over the unit simplex, one obtains, in principle, diﬀerent Pareto optimal controls. A proof of the lemma can be found in [16].

14

Lemma 3.1. [16] Let α = (α1 , α2 ) ∈ A. Assume u ˆ ∈ U is such that u ˆ ∈ arg min

2

u∈U

αi Ji (0, T, u, x0 ) ,

(30)

i=1

then u ˆ is Pareto eﬃcient. So, we consider the weighting sum optimal control problem. For an arbitrary α = (α1 , α2 ) ∈ A, we deﬁne the weighting sum objective cost functional as Jα (0, T, u, x0 ) = E

T

0

¯ T Ex(T ) , ¯ ¯ xT (t)Qx(t) + uT (t)Ru(t) dt + xT (T )E T Q

(31)

¯ = 2 αi Qi , R ¯ = diag{α1 R11 + α2 R21 , α1 R12 + α2 R22 }, Q ¯T = where u(t) = [uT1 (t) uT2 (t)]T , Q i=1 2

i=1

αi QiT .

Accordingly, system (28) can be rewritten as ⎧ ⎪ ⎪ ¯ ⎨ Edx(t) = Ax(t) + Bu(t) dt + Ap x(t)dw(t),

(32)

⎪ ⎪ ⎩ Ex(0) = x0 ¯ = [B1 B2 ]. By (29), we have with B ¯= MB

M B1

M B2

⎤

⎡ ⎢ B11 =⎢ ⎣ B12

⎡

⎤

¯1 ⎥ B21 ⎥ ⎢ B ⎥⎢ ⎥. ⎦ ⎣ ⎦ ¯2 B B22

(33)

Being a suﬃcient condition, it is unclear whether we obtain all Pareto optimal controls in this way. In fact, the above procedure may yield no Pareto optimal controls, while an inﬁnite number of them can exist. The next example illustrates this point. Example 3.1. Consider the following stochastic singular system ⎧ ⎪ ⎪ ⎪ ⎪ dx1 (t) = x1 (t) + 2x3 (t) + u1 (t) dt + 3x2 (t) + x3 (t) dw(t), x1 (0) = 1, ⎪ ⎪ ⎨ dx2 (t) = u1 (t) − u2 (t) dt, x2 (0) = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 0 = x2 (t) + x3 (t)

(34)

with the cost functionals

J1 (0, 1, u1 , u2 ) =

0

J2 (0, 1, u1 , u2 ) =

1

1

0

[u1 (t) − u2 (t)]dt, x2 (t)[u2 (t) − u1 (t)]dt.

Then, by calculations, J2 (0, 1, u1 , u2 ) = − 12 x22 (1) = − 12 J12 (0, 1, u1 , u2 ) for all (u1 , u2 ). Obviously, by choosing diﬀerent values of (u1 , u2 ), every point in curve J2 = − 21 J12 can be attained. Furthermore, by Deﬁnition 3.1, it is clear that every point on this curve is Pareto optimal. 15

Next, for 0 < α < 1, we consider the minimization of J(α) αJ1 (0, 1, u1 , u2 ) + (1 − α)J2 (0, 1, u1 , u2 ) subject to (34). If we choose u1 (t) = c and u2 (t) = 0, straightforward calculations 2 yield that J(α) = αc − 1−α 2 c . By choosing c arbitrarily large, J(α) can be made arbitrarily small,

which means J(α) has not a minimum. Lemma 3.2 mentioned below points out that if the action spaces as well as the players’ objective functionals are convex, then minimization of the weighting sum of the objectives results in all Pareto solutions. Lemma 3.2. [10] Assume that the control space U is convex and the costs Ji (0, T, u, x0 ), i = 1, 2 are convex in u. Then, if u ˆ is Pareto eﬃcient, there exist α ∈ A, such that u ˆ ∈ arg min u∈U

2

αi Ji (0, T, u, x0 ) .

i=1

This property can be used to obtain both necessary and suﬃcient conditions for the existence of the Pareto optimal solutions. That is to say, under convexity conditions on Ji s and convexity assumption regarding the strategy space, all Pareto eﬃcient strategies can be obtained by considering the minimization problem (30). To this end, we ﬁrst derive a preliminary result. Lemma 3.3. Under Assumption 2.1 and Assumption 2.2, considering cost functional (2), where Q = QT , QT = QTT , R = RT and x(t) solves (1), J(0, T, u, x0 ) is convex as a function of u ∈ U, where U is any convex subset of Rm , if and only if J(0, T, u, 0) ≥ 0 for all u ∈ U. Proof. Let xu (t) denote the state trajectory of (1) driving by u. Assume v, w ∈ U. By convexity of strategy set U, for all λ ∈ [0, 1], we have λv + (1 − λ)w ∈ U. Under Assumption 2.1 and Assumption 2.2, the solution of the system (1) is unique. According to linearity property and uniqueness of the solution, we have xλv+(1−λ)w (t) = λxv (t) + (1 − λ)xw (t).

16

Using this relation, we obtain J(0, T, λv + (1 − λ)w, x0 ) T xTλv+(1−λ)w Qxλv+(1−λ)w + (λv + (1 − λ)w)T R(λv + (1 − λ)w) dt =E 0 + xTλv+(1−λ)w (T )E T QT Exλv+(1−λ)w (T ) = λ2 J(0, T, v, x0 ) + (1 − λ)2 J(0, T, w, x0 ) T + 2λ(1 − λ)E xTv Qxw + v T Rw dt + xTv (T )E T QT Exw (T ) . 0

Therefore, λJ(0, T, v, x0 ) + (1 − λ)J(0, T, w, x0 ) − J(0, T, λv + (1 − λ)w, x0 ) T (xv − xw )T Q(xv − xw ) + (v − w)T R(v − w) dt = λ(1 − λ)E 0 T + (xv − xw ) (T )E T QT E(xv − xw )(T ) . Again, by the linearity property and the uniqueness of the solution, we identify the expression on the right side as λ(1 − λ)J(0, T, v − w, 0). The conclusion is proved. Combining Theorem 2.2 and Lemmas 3.1, 3.2 and 3.3, we can derive straightforwardly the existence and computational algorithms for the ﬁnite horizon Pareto optimal control problem. Lemma 3.4. Under Assumption 2.1 and Assumption 3.1, considering the cooperative game ¯ T and R = R, ¯ have a ¯ Q = Q, ¯ QT = Q (27) and (28). Assume that GDREs (9) with B = B, solution Pα (t) on [0, T ] for all α ∈ A. Then, all Pareto eﬃcient solutions are obtained by u∗α := arg min(α1 J1 + α2 J2 ), subject to (32) u∈U

via determining α ∈ A. The corresponding Pareto solutions are J1 (u∗α ), J2 (u∗α ) . Proof. Under the assumption that the GDREs (9) admit a solution Pα (t) on [0, T ] with ¯ T and R = R, ¯ according to Theorem 2.2, the ﬁnite horizon weighting sum ¯ Q = Q, ¯ QT = Q B = B, optimal control problem (31) and (32) is well-posed, and the cost functional (31) attains a minimum Jα (0, T, u∗ , x0 ) = xT (0)E T Pα (0)x(0). So, Jα (0, T, u∗, 0) = 0. By Lemma 3.3, under Assumption 2.1 and Assumption 3.1, Jα (0, T, u, x0 ) is convex in u for all α ∈ A. Therefore, by Lemmas 3.1 and 3.2, all Pareto eﬃcient solutions can be obtained by the weighting sum optimization method. This yields the next procedure to calculate all Pareto eﬃcient solutions for this game. Theorem 3.1. Under Assumption 2.1 and Assumption 3.1, considering the cooperative game

17

(27) and (28). Assume that (35) below admits a solution on [0, T ] for arbitrary α ∈ A. ⎧ ¯ + AT (E + )T P T (t)EE + Ap −E T P˙ (t) = AT P (t) + P T (t)A + Q ⎪ p ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ¯ T P (t), ¯R ¯ −1 B ⎪ − P T (t)B ⎨ ⎪ ⎪ ¯ T E, ⎪ E T P (T ) = E T Q ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ E T P (t) = P T (t)E.

(35b) (35c)

Then, the set of all cooperative Pareto solutions is given by family of the Pareto eﬃcient strategies is ⎡ ⎢ u∗α (t) = ⎢ ⎣ where

u∗1α (t) u∗2α (t)

{(J1 (u∗α ), J2 (u∗α ))|α

∈ A}. Here, the

⎤ ⎥ ¯ T Kα (t)x(t), ¯ −1 B ⎥ = −R ⎦

⎤

⎡ ⎢ Kα (t) = M T ⎢ ⎣

(35a)

Pα11 (t)

0

Pα21 (t) + Pα22 (t)Lα (t) − F¯ (t)Lα (t) F¯ (t)

⎡

⎥ −1 ⎥N ⎦

⎤

⎢ Pα11 (t) with Pα (t) = M T ⎢ ⎣ Pα21 (t)

Pα12 (t) ⎥ ⎥ N −1 being a solution of (35). Denoting ⎦ Pα22 (t) ⎡

⎤ ⎡ ¯ ⎢ B1 ⎥ −1 ⎢ ¯ ⎢ ⎥R S¯ = ⎢ ⎣ ⎦ ⎣ ¯ B2

⎤T ⎡ ¯ B1 ⎥ ⎢ S¯11 ⎥ ⎢ ⎦ ⎣ ¯2 B S¯21

⎤ ¯ S12 ⎥ ⎥, ⎦ ¯ S22

T then Lα (t) = −[A22 − S¯22 Pα22 (t)]−1 [A21 − S¯12 Pα11 (t) − S¯22 Pα21 (t)] and F¯ (t) is any time-varying

¯ α (t), the matrix making A22 − S¯22 F¯ (t) invertible within the interval [0, T ]. With Acl (t) = A − SK closed-loop system is Edx(t) = Acl (t)x(t)dt + Ap x(t)dω(t), Ex(0) = x0 . The corresponding cost is Ji (u∗α ) = xT (0)E T P˜iα (0)x(0), where P˜iα (t) is a solution of the following GDREs ⎧ ⎪ ⎪ T T ⎪ −E T P˜˙iα (t) = ATcl (t)P˜iα (t) + P˜iα (t)Acl (t) + Qi + ATp (E + )T P˜iα (t)EE + Ap ⎪ ⎪ ⎪ ⎡ ⎤ ⎪ ⎪ ⎪ ⎪ ⎪ 0 ⎥ ⎪ ⎢ Ri1 ⎪ ⎪ ⎥R ¯ −1 B ¯ T Kα (t), ¯R ¯ −1 ⎢ + KαT (t)B ⎨ ⎣ ⎦ 0 Ri2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ E T P˜iα (T ) = E T QiT E, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ T ⎩ E T P˜iα (t) = P˜iα (t)E. The proof is similar to Theorem 2.2, so we omit it due to the limitation of space. Remark 3.2. Although the basic idea (under convexity conditions on cost functionals and convexity assumption regarding the strategy space, all Pareto eﬃcient strategies can be obtained 18

by considering the weighting sum minimization problem) is the same with [9], our system is more complicated due to the presence of singular matrix E and Brownian motion w(t). So, our conclusion generalized the known result in [9], which mainly studied the Pareto optimal control of deterministic systems. Besides, the theory of the related LQ optimal control of stochastic singular systems is immature, which increased the research challenge.

4. Examples In order to demonstrate the eﬃciency of the proposed approach, we consider the following ecological ﬁshing model (see [11], Section 4), which is described by the following dynamic equations: f˙(t)

=

βf (t) − p(t) − u1 (t), f (0) = f0 ,

(36)

εp(t) ˙

=

γp(t) + u2 (t),

(37)

where f is the ﬁsh stock, p is the amount of pollution, u1 is the ﬁshing by the ranger, u2 is the waste dropped into the river, β and γ are nonzero constant coeﬃcients. The revenue functionals to be maximized are

J1

= =

τf f 2 (t) − u21 (t) dt,

T

2 τp p (t) − u22 (t) dt.

0

J2

T

0

Letting ε = 0 in (37), we get an algebraic equation, it means that the negative eﬀect of pollution on the growth of the ﬁsh stock is proportional to p. Introducing xT (t) := [f T (t) pT (t)]T , we can rewrite the above model (36)-(37) as Ex(t) = Ax(t) + B1 u1 (t) + B2 u2 (t), which is a deterministic singular system, where ⎡ ⎤ ⎡ ⎤ ⎢ 1 0 ⎥ ⎥, E=⎢ ⎣ ⎦ 0 0

⎢ β A=⎢ ⎣ 0

⎡

−1 ⎥ ⎥, ⎦ γ

⎤

⎢ −1 ⎥ ⎥, B1 = ⎢ ⎣ ⎦ 0

⎡

⎤

⎢ 0 ⎥ ⎥. B2 = ⎢ ⎣ ⎦ 1

However, the white noise always exists and can not be omitted. As a result, we introduce stochastic perturbation to the ﬁsh stock f and obtain the following stochastic singular system: Edx(t) = Ax(t) + B1 u1 (t) + B2 u2 (t) dt + Ap x(t)dw(t), 19

⎡

⎤

⎢ σ where Ap = ⎢ ⎣ 0

0 ⎥ ⎥ with σ representing the intensity of the white noise, which is in the form ⎦ 0

of system (28). Obviously, Assumption 2.1 and Assumption 3.1 are satisﬁed in this example. Accordingly, the revenue functions can be rewritten as the form in (27) with ⎡ ⎡ ⎡ ⎤ ⎤ ⎤ ⎢ −τf Q1 = ⎢ ⎣ 0

0 ⎥ ⎥, ⎦ 0

⎢ 0 Q2 = ⎢ ⎣ 0

0 −τp

R11 = R22 = 1, So

⎤

⎢ −α1 τf ⎢ −1 0 ⎥ ⎥, Q ¯=⎢ ¯ =⎢ B ⎣ ⎦ ⎣ 0 1 0

⎢ 0 Q1T = Q2T = ⎢ ⎣ 0

0 ⎥ ⎥, ⎦ 0

R12 = R21 = 0.

⎡

⎤

⎡

⎥ ⎥, ⎦

⎡

⎢ α1 ⎥ ⎥, R ¯=⎢ ⎣ ⎦ −α2 τp 0 0

⎤

⎡

⎤

0 ⎥ ⎢ 0 ⎥, Q ¯T = ⎢ ⎣ ⎦ 0 α2

0 ⎥ ⎥ ⎦ 0

in the cooperative diﬀerential game (31)-(32). In what follows, we will verify the eﬀectiveness of Theorem 3.1. For any α = (α1 , α2 ) ∈ A, although Assumption 2.3 is not satisﬁed, when γ 2 > τp and b2 − 4ac > 0, where a =

1 α1

+

1 α2

1 γ 2 −τp ,

·

b = σ 2 + 2β, c = α1 τf , there exists a solution to

the GDREs (35). By straightforward calculations, we obtain the following solution of the involved GDREs (35) ⎤

⎡ ⎢ Pα11 Pα = ⎢ ⎣ Pα21 where d1 = b +

⎡

Pα12 ⎥ ⎢ ⎥=⎢ ⎦ ⎣ −√ Pα22

d1 a(t−T ) ] 2a [1+e d1 a(t−T ) 1+ d e 2

1 γ 2 −τp

·

d1 a(t−T ) ] 2a [1+e d 1+ d1 ea(t−T ) 2

⎤ 0 α2 γ + α2

γ 2 − τp

⎥ ⎥, ⎦

√ √ b2 − 4ac and d2 = b − b2 − 4ac. Elementary manipulations show then that the

family of the Pareto eﬃcient strategies is u∗1α

=

u∗2α

=

1 Pα11 f ∗ , α1 1 − [(Pα21 + Pα22 Lα − F¯ Lα )f ∗ + F¯ p∗ ], α2

where Lα = (α2 γ−Pα22 )−1 Pα21 and F¯ = α2 γ is a constant, f ∗ solves df ∗ = (β−Lα − α11 Pα11 )f ∗ dt+ σf ∗ dw, f ∗ (0) = f0 and p∗ = Lα f ∗ . When F¯ = 0, we plotted the Pareto frontier in Fig.1 for the parameter values β = 3, γ = 2, σ = 0.2, τf = 1, τp = 1, T = 5 and f0 = 1. If the weight of player 1 in the cooperative cost functional would be α1 = 0.6, the state and control trajectories are shown in Fig.2 and Fig.3, respectively. 20

1.4

1.2

1

* α

J2(u )

0.8

0.6

0.4

0.2

0 −0.15

−0.1

−0.05

0

J1(u* ) α

Fig.1. Pareto frontier when F¯ = 0.

1.6 f* *

p

1.4

1

*

Optimal states f and p

*

1.2

0.8 0.6 0.4 0.2 0

0

0.5

1

1.5 Time(sec)

2

Fig.2. State trajectories if α1 = 0.6, when F¯ = 0.

21

2.5

3

5 *

u1α 4

*

u2α

*

*

Optimal controls u1α and u2α

3 2 1 0 −1 −2 −3 −4

0

0.5

1

1.5 Time(sec)

2

2.5

3

Fig.3. Control trajectories if α1 = 0.6, when F¯ = 0.

5. Conclusions In this paper, we have investigated the ﬁnite horizon LQ Pareto optimal control of stochastic singular systems. Firstly, ﬁnite horizon stochastic singular LQ optimal control has been discussed by means of the square completion technique, and a new type of GDREs have also been established. The existence condition of the solution of the GDREs has been put forward. Secondly, for ﬁnite horizon LQ Pareto optimal control, it is shown that the solvability of the corresponding GDREs presents a suﬃcient condition, under which all Pareto candidates can be obtained by solving a weighting sum optimal control. A numerical example has also been provided to show the eﬀectiveness of our main results. The event-triggering mechanism has the advantage of saving the communication and computation resources. In our future research, the event-triggering mechanism [17, 18] will be taken into account in the analysis of optimal control of stochastic singular systems.

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