H∞ control with random jumps

Acta Mathematica Scientia 2015,35B(2):348–358 http://actams.wipm.ac.cn

A MAXIMUM PRINCIPLE APPROACH TO STOCHASTIC H2 /H∞ CONTROL WITH RANDOM JUMPS∗

Üé )

Qixia ZHANG (

School of Mathematical Sciences, University of Jinan, Jinan 250022, China E-mail : [email protected]

šéû)

Qiliang SUN (

School of Mathematical Sciences, University of Jinan, Jinan 250022, China E-mail : [email protected] Abstract A necessary maximum principle is given for nonzero-sum stochastic differential games with random jumps. The result is applied to solve the H2 /H∞ control problem of stochastic systems with random jumps. A necessary and sufficient condition for the existence of a unique solution to the H2 /H∞ control problem is derived. The resulting solution is given by the solution of an uncontrolled forward backward stochastic differential equation with random jumps. Key words

Nonzero-sum stochastic differential games; maximum principle; Poisson process; stochastic H2 /H∞ control; forward backward stochastic differential equations

2010 MR Subject Classification

1

91A15; 93B36; 93E20

Introduction

In practice, when the exogenous disturbance enters the system, an H∞ control design is often first considered to efficiently eliminate the effect of the disturbance. We refer the reader to [1, 2] and the references therein. If the purpose is to select control not only to restrain the exogenous disturbance, but also to minimize a cost function when the worst case disturbance v ∗ is implemented, this is the so-called mixed H2 /H∞ control problem. Mixed H2 /H∞ control problem attracted much attention and was widely applied to various fields; see [3, 4] and the references therein. Up to now, most of the work on stochastic H2 /H∞ control is constrict in Itˆo type or Markovian jump systems. Yet there are still many systems which contain Poisson jumps in economics and natural science. For example, in financial mathematics, stock price is classically considered as geometric Brown motion, however, in practice, the price of stocks can be made ∗ Received January 5, 2013; revised May 27, 2014. The first author is supported by the Doctoral foundation of University of Jinan (XBS1213) and the National Natural Science Foundation of China (11101242).

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a sudden shift by the exogenous disturbance such as wars, decisions of large banks or the corporations, and national policy, etc. In order to describe such phenomenon, Poisson process is usually inserted in the well-known geometric Brownian motion model, and the new model is driven not only by Brownian motion but also by Poisson process. In 1975, Rishel [5] firstly considered the optimal control problem with random Poisson jumps. From then on, many scholars and economists also study such systems and their applications; for further reference, we refer to [6–9]. However, those results mostly concentrate on optimal control problems and their applications in financial market or their corresponding theories. Of course, such systems can be disturbed by exogenous disturbance and their H2 /H∞ control problems are also important problems. The objective of this article is to develop an H2 /H∞ theory for the disturbance attenuation of stochastic systems with Poisson process. In next section, we establish a necessary condition in the form of maximum principle for the existence of the Nash equilibrium point of the nonzero-sum stochastic differential games with random jumps. The sufficient condition of maximum principle can be seen in [6]. There are already a lot of literatures on the maximum principle [10–13]. In Section 3, the stochastic H2 /H∞ control problem with random jumps is put into the framework of the nonzero-sum stochastic differential game problem, then, we use the necessary maximum principle to find the candidate solution in the form of the solution to an uncontrolled forward backward stochastic differential equation (FBSDE) with random jumps. As the sufficient condition of maximum principle does not hold, then, we can not use the verification theorem to check whether the candidate solution is optimal or not. In Section 4, we find that the stochastic H2 /H∞ control problem is associated with the corresponding uncontrolled perturbed system and give a necessary and sufficient condition for the candidate solution to be the unique solution. Finally, Section 5 ends this article with some remarks. For convenience, we adopt the following notations. Rn : the n-dimensional Euclidean space; S n : the set of symmetric n × n matrices with real elements; AT : the transpose of the matrix A; A ≥ 0 (A > 0): A is positive semidefinite (positive definite) real matrix; I: identity matrix; h·, ·i: the scalar product; n P |x|2 := hx, xi = xT x = x2i for n-dimensional vector x = (x1 , · · · , xn )T ; i=1

C(0, T ; X): the set of a given Hilbert space X-valued continuous functions; N (·) >(≥)0: N (·) ∈ C(0, T ; S n ) and N (t) >(≥)0 for a.s. t ∈ [0, T ]; RT L2F (0, T ; Rn ): ={ft , 0 ≤ t ≤ T , is a Rn -valued Ft adapted process s.t. E[ 0 |ft |2 dt] < ∞}; M 2 (0, T ; Rn ): ={gt , 0 ≤ t ≤ T , is a Rn -valued progressively measurable process s.t. RT E[ 0 |gt |2 dt] < ∞}.

2

The Necessary Maximum Principle for Nonzero-sum Games

Let (Ω, F , {Ft }t≥0 , P ) be a stochastic basis such that F0 contains all P -null elements of T F and Ft+ = Ft+ε = Ft , t ≥ 0. Suppose that the filtration {Ft }t≥0 is generated by the following two mutually independent processes:

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i) A d-dimensional standard Brownian motion {B(t)}t≥0 , ii) A Poisson random measure N on R+ ×Z, where Z ⊂ Rl is a nonempty open set equipped ˆ (dz, dt) = λ(dz)dt, such that N ˜ (A × [0, t]) = with its Borel field B(Z), with compensator N ˆ (N − N )(A × [0, t])t≥0 is a martingale for all A ∈ B(Z) satisfying λ(A) < ∞. λ is assumed to be a σ-finite measure on (Z, B(Z)) and is called the characteristic measure. For notational simplification, we assume that d = 1. We consider the case when two Players I and II intervene on the dynamic of a system and their advantages are not necessarily antagonistic but each one acts such as to save its own interest. This situation is a nonzero-sum game. Suppose that the system is described by a stochastic differential equation on a complete filtered probability space (Ω, F , {Ft }t≥0 , P ) of the form    dX(t) = b(t, X(t), u(t), v(t))dt + σ(t, X(t), u(t), v(t))dB(t)  Z   ˜ (dz, dt), + g(t, X(t− ), u(t), v(t), z)N (2.1)  Z     X(0) = X0 ∈ Rn , t ∈ [0, T ].

u(t) = u(t, ω) and v(t) = v(t, ω), ω ∈ Ω are control processes of Player I and II, respectively. We assume that u(t) and v(t) have values in given closed sets K1 and K2 for a.a. t, respectively, and that u(t), v(t) are adapted to the given filtration {Ft }t≥0 . b : [0, T ] × Rn × K1 × K2 → Rn , σ : [0, T ] × Rn × K1 × K2 → Rn , g : [0, T ] × Rn × K1 × K2 × Z → Rn are given continuous functions. The Players act on the system (2.1) with strategy (u, v), then, the costs associated with I and II are, respectively, J1 (u, v) and J2 (u, v) of the form "Z # T Ji (u, v) = E fi (t, X(t), u(t), v(t))dt + Φi (X(T )) , i = 1, 2, 0

where fi : [0, T ] × Rn × K1 × K2 −→ R, i=1, 2 are continuous functions, namely the profit rate, and Φi : Rn −→ R, i = 1, 2, are concave functions, namely the bequest function. We call (u, v) an admissible control if (2.1) has a unique strong solution and "Z # T E |fi (t, X(t), u(t), v(t))|dt + |Φi (X(T ))| < ∞, i = 1, 2. 0

Now, we introduce the admissible control sets Θ : ={ut , 0 ≤ t ≤ T , is a K1 -valued Ft adapted process} and Π : ={vt , 0 ≤ t ≤ T , is a K2 -valued Ft adapted process} of admissible controls u and v, respectively. The problem is to find a control (u∗ , v ∗ ) ∈ Θ × Π such that J1 (u, v ∗ ) ≤ J1 (u∗ , v ∗ )

for any

u ∈ Θ,

J2 (u , v) ≤ J2 (u , v ) for any

v ∈ Π.

∗

∗

∗

The pair of control (u∗ , v ∗ ) is called a Nash equilibrium for the game because when Player I (resp. II) acts with the strategy u∗ (resp. v ∗ ), the best that has to do II (resp. I) is to act with v ∗ (resp. u∗ ). Let us introduce the Hamiltonian functions associated with this game, namely H1 and H2 , from [0, T ] × Rn × K1 × K2 × Rn × Rn × R to R, which are defined by Hi (t, x, u, v, pi , qi , γi ) = fi (t, x, u, v) + bT (t, x, u, v)pi + σ T (t, x, u, v)qi

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+

Z

Z



 g T (t, x, u, v, z)γi (t, z) λ(dz),

i = 1, 2,

(2.2)

where R is the set of functions γ : [0, T ] × Z → Rn such that the integral in (2.2) converges. From now on, we assume that Hi is differentiable with respect to x, i = 1, 2. The adjoint equation in the unknown adapted processes pi (t) ∈ Rn , qi (t) ∈ Rn and γi (t, z) ∈ Rn is the backward stochastic differential equation (BSDE)  Z  ˜ (dz, dt), t ∈ [0, T ],  dpi (t) = −∇x Hi (t, X, u, v, pi , qi , γi )dt + qi (t)dB(t) + γi (t− , z)N Z

  p (T ) = ∇Φ (x(T )), i i

i = 1, 2,

(2.3) ∂ϕ ∂ϕ ∂ϕ T where ∇y ϕ(·) = ( ∂y , , · · · , ) is the gradient of ϕ : R → R with respect to y = n ∂yn 1 ∂y2 (y1 , y2 , · · · , yn ).

T. T. K. An and B. Øksendal [6] give the following sufficient maximum principle for nonzerosum differential game problem. Theorem 2.1 Let (u∗ , v ∗ ) ∈ Θ × Π with corresponding state process X ∗ (t) = X (u ,v ) (t). Suppose that there exists a solution (p∗i (t), qi∗ (t), γi∗ (t, z)), i = 1, 2, of the corresponding adjoint equation (2.3) such that for all u ∈ Θ and v ∈ Π, we have ∗

∗

H1 (t, X ∗ (t), u∗ (t), v ∗ (t), p∗1 (t), q1∗ (t), γ1∗ (t, z)) ≥ H1 (t, X ∗ (t), u(t), v ∗ (t), p∗1 (t), q1∗ (t), γ1∗ (t, z)) and H2 (t, X ∗ (t), u∗ (t), v ∗ (t), p∗2 (t), q2∗ (t), γ2∗ (t, z)) ≥ H2 (t, X ∗ (t), u∗ (t), v(t), p∗2 (t), q2∗ (t), γ2∗ (t, z)). Moreover, suppose that for any t ∈ [0, T ], Hi (t, x, u, v, p∗i (t), qi∗ (t), γi∗ (t, z)), i = 1, 2, is concave in x, u, v and Φi (x), i = 1, 2, is concave in x. Then, (u∗ , v ∗ ) is an equilibrium point for the game and J1 (u∗ , v ∗ ) = sup J1 (u, v ∗ ), u∈Θ

J2 (u∗ , v ∗ ) = sup J2 (u∗ , v). v∈Π

Following the idea developed by T. T. K. An and B. Øksendal [6], it is not difficult to analyze the necessary maximum principle of the game problem. Thus, we omit the detailed deduction and only state the main result for simplicity. In addition to the assumptions above, we assume the following: (A1) For any (α, ρ) ∈ Θ × Π, define β(s) = α(s)χ[t,t+h] (s) and η(s) = ρ(s)χ[t,t+h] (s) for all t, h such that 0 ≤ t ≤ t + h ≤ T , then, there exists a δ > 0 such that u + yβ ∈ Θ, v + zη ∈ Π, where y, z ∈ (−δ, δ).

(A2) Hi is differentiable with respect to u and v, i = 1, 2. We now state

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Theorem 2.2 Let (A1) and (A2) hold. Suppose that (u∗ , v ∗ ) is an equilibrium point and X ∗ is the corresponding state trajectory. Then, we have ∇u H1 (t, X ∗ (t), u∗ (t), v ∗ (t), p∗1 (t), q1∗ (t), γ1∗ (t, z))

= ∇v H2 (t, X ∗ (t), u∗ (t), v ∗ (t), p∗2 (t), q2∗ (t), γ2∗ (t, z)) = 0, where (p∗i , qi∗ , γi∗ ) (i = 1, 2) is the solution of the adjoint equation (2.3).

3

Applications to Stochastic H2 /H∞ Control With Random Jumps

This section is devoted to study the stochastic H2 /H∞ control problem with random jumps, which naturally motivates the study of the nonzero-sum stochastic differential game. Consider the following linear stochastic system driven by both the Brownian motion and Poisson process:       dxt = At (ω)xt + Bt (ω)ut + Ct (ω)vt dt + Dt (ω)xt + Et (ω)ut + Ft (ω)vt dB(t)    Z    ˜ (dz, dt), (3.1) + Lt (z)xt− + Mt (z)ut + Nt (z)vt N   Z    x(0) = x0 ∈ Rn ,

with the controlled output being a vector as  Zt = 

St (ω)xt ut



,

(3.2)

where x ∈ Rn is the system state, Z ∈ RnZ is the penalty output, u ∈ L2F (0, T ; Rnu ) and v ∈ L2F (0, T ; Rnv ) stand for the control input and exogenous disturbance signal, respectively. The coefficients A(·), B(·), C(·), D(·), E(·), F (·), L(·), M (·), N (·), and S(·) are bounded progressively measurable processes of suitable dimensions. Now, we define the stochastic H2 /H∞ control with random jumps as follows. Definition 3.1 For any given r > 0, 0 < T < ∞, and v ∈ L2F (0, T ; Rnv ), find, if possible, a control law u = u∗ ∈ L2F (0, T ; Rnu ), such that (i) The trajectory of the closed-loop system (3.1) staring from x0 = 0 satisfies Z T Z T   2 E |Z(t)| dt = E x(t)T S(t)T S(t)x(t) + |u∗ (t)|2 dt 0

0

< r2 E

Z

T

0

|v(t)|2 dt

for ∀v 6= 0 ∈ L2F (0, T ; Rnv ). (ii) When the worst case disturbance [4] v ∗ ∈ L2F (0, T ; Rnv ), if existing, is implemented in (3.1), u∗ minimizes the quadratic performance Z T Z T   2 E |Z(t)| dt = E x(t)T S(t)T S(t)x(t) + |u(t)|2 dt 0

0

simultaneously. If an admissible control u ∈ L2F (0, T ; Rnu ) only satisfies (i), then, this u eliminates the effect of the disturbance and is a solution to H∞ control of system (3.1). Obviously, there may

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be more than one solution to the H∞ control problem. If the previous (u∗ , v ∗ ) exist, u∗ not only restrains the exogenous disturbance, but also minimizes the cost function when the worst case disturbance v ∗ is implemented, then, we say that the stochastic H2 /H∞ control admits a solution (u∗ , v ∗ ). If we define Z T Z T  T T  J1 (u, v) = −E |Z(t)|2 dt = −E x S Sx + uT u dt 0

0

and

Z

T

J2 (u, v) = E

T

=E

Z

0

0



 |Z(t)|2 − r2 |v(t)|2 dt



 xT S T Sx + uT u − γ 2 |v|2 dt,

then, it can be seen that ([3, 4]) the stochastic H2 /H∞ control problem is equivalent to find the Nash equilibria (u∗ , v ∗ ) defined as J1 (u∗ , v ∗ ) ≥ J1 (u, v ∗ ),

J2 (u∗ , v ∗ ) ≥ J2 (u∗ , v),

∀u ∈ L2F (0, T ; Rnu ),

∀v ∈ L2F (0, T ; Rnv ),

∀v 6= 0 ∈ L2F (0, T ; Rnv ), x0 = 0.

J2 (u∗ , v) < 0,

Throughout this article, we take Q(t) = S(t)T S(t) for convenience, then, Q(·) ≥ 0. Now, we will apply the necessary maximum principle in Section 2 to solve the stochastic H2 /H∞ control problem. The Hamiltonian functions are in the forms of    H1 (t, x, u, v, p1 , q1 , γ1 ) = −xT Qx − uT u + pT1 (Ax + Bu + Cv) + q1T (Dx + Eu + F v)    Z    T     + γ1 (Lx + M u + N v) (z)λ(dz),  Z

  H2 (t, x, u, v, p2 , q2 , γ2 ) = xT Qx + uT u − r2 v T v + pT2 (Ax + Bu + Cv) + q2T (Dx + Eu + F v)     Z    T     + γ2 (Lx + M u + N v) (z)λ(dz). Z

(3.3)

Here, the adjoint processes (pi , qi , γi ) (i = 1, 2) satisfy  Z Z    ˜ (dz, dt),  dp1 = 2Qx − AT p1 − DT q1 − (LT γ1 )(z)λ(dz) dt + q1 dB(t) + γ1 (t− , z)N Z

Z

  p (T ) = 0, 1

(3.4) and  Z Z    ˜ (dz, dt),  dp2 = − 2Qx + AT p2 + DT q2 + (LT γ2 )(z)λ(dz) dt + q2 dB(t) + γ2 (t− , z)N Z

  p (T ) = 0. 2

Z

(3.5) Obviously, p2 (t) = −p1 (t), q2 (t) = −q1 (t), and γ2 (t, z) = −γ1 (t, z). As the Hamiltonian functions H1 and H2 are not concave in x, u, v, the sufficient maximum principle does not hold in this case, we can formally use the result of the necessary maximum

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principle (Theorem 2.2), ∇u H1 (t, x, u, v, p1 , q1 , γ1 ) = −2u + B T p1 + E T q1 +

Z

(M T γ1 )(z)λ(dz) = 0, Z 2 T T ∇v H2 (t, x, u, v, p2 , q2 , γ2 ) = −2r v + C p2 + F q2 + (N T γ2 )(z)λ(dz) = 0, Z

Z

to choose a candidate solution R  B T p1 + E T q1 + Z (M T γ1 )(z)λ(dz)  ∗  ,  u (t) = 2 R T T T    v ∗ (t) = − C p1 + F q1 + Z (N γ1 )(z)λ(dz) . 2r2 In next section, we will give the necessary and sufficient condition for the candidate solution to be the unique solution.

4

The Necessary and Sufficient Condition

The aim of this section is to give a necessary and sufficient condition for the candidate solution to be the unique solution. We begin our presentation with some preliminaries. Consider the following uncontrolled stochastic perturbed system:       dxt = At (ω)xt + Ct (ω)vt dt + Dt (ω)xt + Ft (ω)vt dB(t)    Z       ˜ (dz, dt), + Lt (z)xt− + Nt (z)vt N (4.1) Z     x(0) = x0 ∈ Rn ,     Zt = St (ω)xt ,

For any 0 < T < ∞, define the perturbation operator L : L2F (0, T ; Rnv ) → L2F (0, T ; RnZ ) as L(v) = Z|x0 =0 ,

t ≥ 0,

v ∈ L2F (0, T ; Rnv ),

with its norm kLk2 : =

=

sup v∈L2F (0,T ;Rnv ),v6=0,x0 =0

sup v∈L2F (0,T ;Rnv ),v6=0,x0 =0

Obviously, L is a nonlinear operator.

kL(v)k2 kvk2 RT

1

(xT Qx)dt} 2 . RT 1 {E 0 |v|2 dt} 2

{E

0

Definition 4.1 Letting r > 0, system (4.1) is said to have L2 -gain less than r if for any nonzero v ∈ L2F (0, T ; Rnv ), kLk2 < r.

The following theorem is a necessary and sufficient condition for the candidate solution to be the unique solution.

Theorem 4.2 For system (3.1), the finite horizon stochastic H2 /H∞ control problem admits a solution if and only if the corresponding uncontrolled system (4.1) has L2 -gain less than r. Moreover, if the control problem admits a solution, then, the solution is unique with R B T p∗1 + E T q1∗ + Z (M T γ1∗ )(z)λ(dz) ∗ u = , 2

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v =−

C T p∗1 + F T q1∗ +

R

Z (N 2r2

T

γ1∗ )(z)λ(dz)

355

,

where (x∗ , p∗1 , q1∗ , γ1∗ ) is the solution of the following FBSDE:  R  BB T p∗1 + BE T q1∗ + B Z (M T γ1∗ )(z)λ(dz))  ∗ ∗   dx = Ax +   2    R   T ∗ T ∗  CC p1 + CF q1 + C Z (N T γ1∗ )(z)λ(dz)    − dt   2r2   R     EB T p∗1 + EE T q1∗ + E Z (M T γ1∗ )(z)λ(dz)  ∗   + Dx +   2   R    T ∗ T ∗ T ∗  F C p + F F q + F (N γ1 )(z)λ(dz)  1 1 Z  − dB(t)    2r2  R Z  M B T p∗1 + M E T q1∗ + M Z (M T γ1∗ )(z)λ(dz)) ∗  + Lx +    2 Z   R   T ∗ T ∗ T ∗   N C p1 + N F q1 + N Z (N γ1 )(z)λ(dz) ˜   N (dz, dt) −   2r2      Z    ∗ ∗ T ∗ T ∗   dp = 2Qx − A p − D q − (LT γ1∗ )(z)λ(dz) dt 1 1  1   Z  Z    ∗  ˜ (dz, dt),  +q1 dB(t) + γ1∗ (t− , z)N    Z    x∗ (0) = x , p∗ (T ) = 0. 0 1

(4.2)

Proof 1) The Sufficient Condition: To show that the finite horizon stochastic H2 /H∞ control problem admits a unique solution (u∗ , v ∗ ) if the corresponding uncontrolled system (4.1) has L2 -gain less than r, we will show that (u∗ , v ∗ ) is a solution firstly. Looking at the above FBSDE, from [14–16], the FBSDE (4.2) admits a unique solution (x∗ , p∗1 , q1∗ , γ1∗ ) ∈ M 2 (0, T ; Rn+n+n+n ). Now, we try to prove that v ∗ is the worst case disturbance. For any given v 6= v ∗ ∈ L2F (0, T ; Rnv ), suppose that xv is the trajectory corresponding to (u∗ , v) ∈ L2F (0, T ; Rnu ) × L2F (0, T ; Rnv ). It is seen that the trajectory corresponding to (0, v ∗ − v) is x∗ − xv with initial state 0. Hence, x∗ − xv is the trajectory corresponding to v ∗ − v for system (4.1). Then, Z T   E − (x∗ − xv )T Q(x∗ − xv ) + r2 (v ∗ − v)T (v ∗ − v) dt > 0 (4.3) 0

and

J2 (u∗ , v ∗ ) − J2 (u∗ , v) Z T  ∗T  =E x Qx∗ − xvT Qxv − r2 v ∗T v ∗ + r2 v T v dt 0

=E

Z

T

0

 ∗  (x − xv )T Q(x∗ − xv ) − 2xvT Q(xv − x∗ ) − r2 v ∗T v ∗ + r2 v T v dt.

v ∗ We use Itˆo’s formula to p∗T 1 (x − x ) and obtain Z T   v ∗ 0=E d p∗T 1 (x − x ) 0

(4.4)

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=E

Z

=E

Z

T 0

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Z h i  T ∗ T ∗T v ∗ T ∗ 2x Q(x − x ) + C p1 + F q1 + (N T γ1∗ )(z)λ(dz) (v − v ∗ ) dt Z

T 0

Then, we have

 ∗T  2x Q(xv − x∗ ) − 2r2 v ∗T (v − v ∗ ) dt. J2 (u∗ , v ∗ ) − J2 (u∗ , v) Z T   =E − (x∗ − xv )T Q(x∗ − xv ) + r2 (v ∗ − v)T (v ∗ − v) dt. 0

From (4.3), we obtain

J2 (u∗ , v ∗ ) − J2 (u∗ , v) > 0. Thus, v ∗ is the worst case disturbance. Moreover, for x0 = 0, the FBSDE (4.2) admits a unique solution (x∗ , p∗1 , q1∗ , γ1∗ ) = (0, 0, 0, 0), then, J2 (u∗ , v) < J2 (u∗ , v ∗ ) Z T  ∗T  =E x Qx∗ + u∗T u∗ − r2 v ∗T v ∗ dt 0

= 0.

Now, we try to prove that u∗ minimizes the cost function J1 when the worst case disturbance v is implemented into system (3.1). For any given u ∈ L2F (0, T ; Rnu ), let xu be the trajectory corresponding to (u, v ∗ ), then, ∗

J1 (u∗ , v ∗ ) − J1 (u, v ∗ ) Z T  ∗T  = −E x Qx∗ + u∗T u∗ − xuT Qxu − uT u dt =E

Z

0

0 T h

H1 (t, x∗ , u∗ , v ∗ , p∗1 , q1∗ , γ1∗ ) − H1 (t, xu , u, v ∗ , p∗1 , q1∗ , γ1∗ )

−hAx∗ + Bu∗ + Cv ∗ , p∗1 i + hAxu + Bu + Cv ∗ , p∗1 i

−hDx∗ + Eu∗ + F v ∗ , q1∗ i + hDxu + Eu + F v ∗ , q1∗ i Z Z i

∗ 

∗  − γ1 , Lx∗ + M u∗ + N v ∗ (z)λ(dz) + γ1 , Lxu + M u + N v ∗ (z)λ(dz) dt. Z

Z

∗

∗

∗

As H1 is concave in x and u and ∇u H(t, x , u , v , p∗1 , q1∗ , γ1∗ ) = 0, then, we obtain J1 (u∗ , v ∗ ) − J1 (u, v ∗ ) Z Th ≥E ∇x H1 (t, x∗ , u∗ , v ∗ , p∗1 , q1∗ , γ1∗ )(x∗ − xu ) 0

−hAx∗ + Bu∗ + Cv ∗ , p∗1 i + hAxu + Bu + Cv ∗ , p∗1 i

−hDx∗ + Eu∗ + F v ∗ , q1∗ i + hDxu + Eu + F v ∗ , q1∗ i Z Z i

∗ 

∗  ∗ ∗ ∗ − γ1 , Lx + M u + N v (z)λ(dz) + γ1 , Lxu + M u + N v ∗ (z)λ(dz) dt. Z

Z

On the other hand, we have Z T   ∗ u 0=E d p∗T 1 (x − x ) 0

=E

Z

0

T

h

− ∇x H1 (t, x∗ , u∗ , v ∗ , p∗1 , q1∗ )T (x∗ − xu )

No.2

Q.X. Zhang & Q.L. Sun: STOCHASTIC H2 /H∞ CONTROL WITH RANDOM JUMPS

357

+hAx∗ + Bu∗ , p∗1 i − hAxu + Bu, p∗1 i + hDx∗ + Eu∗ , q1∗ i − hDxu + Eu, q1∗ i Z Z i 

∗ 

∗ γ1 , Lxu + M u (z)λ(dz) dt, + γ1 , Lx∗ + M u∗ (z)λ(dz) − Z

Z

then, J1 (u∗ , v ∗ ) − J1 (u, v ∗ ) = I1 ≥ 0. B T p∗ +E T q∗ +

R

(M T γ ∗ )(z)λ(dz)

C T p∗ +F T q∗ +

R

(N T γ ∗ )(z)λ(dz)

1 1 1 1 1 1 Z Z So, (u∗ , v ∗ ) = ( ,− ) is a solution of 2 2r 2 the stochastic H2 /H∞ control problem. We are now in a position to prove the uniqueness of the solution. Assume that the stochastic H2 /H∞ control has a solution (u1 , v 1 ), x1 is the corresponding trajectory, and (p1 , q 1 , γ 1 ) is the solution of the following BSDE: Z 
1 1 T 1 T 1   dp (t) = [2Q(t)x (t) − A(t) p (t) − D(t) q (t) − LT γ 1 (z)λ(dz)]dt    Z  Z ˜ (dz, dt), +q 1 (t)dB(t) + γ 1 (t− , z)N    Z    1 p (T ) = 0,

Implementing v 1 , as

inf J1 (u, v 1 ) is a standard LQ optimal control problem, then, by u∈L2F (0,T ;Rnu ) R B T p1 +E T q1 + Z (M T γ 1 )(z)λ(dz) uniqueness, u1 = . 2 R C T p1 +F T q1 + Z (N T γ 1 )(z)λ(dz) Let x be the trajectory corresponding to (u1 , v) = (u1 , − ), then, 2r 2 0 ≥ J2 (u1 , v) − J2 (u1 , v 1 ) Z T  T  =E x Qx − x1T Qx1 − r2 v T v + r2 v 1T v 1 dt 0

=E

T

Z

0



 (x − x1 )T Q(x − x1 ) + 2x1T Q(x − x1 ) − r2 v T v + r2 v 1T v 1 dt.

Applying Itˆo’s formula to p1T (x − x1 ), we have Z T 0=E d[p1T (x − x1 )] 0

T

=E

Z

T

=E

Z

0

0

h 

1T

2x

1



T 1

T 1

Q(x − x ) + C p + F q +

Z

Z

i T (N T γ 1 )(z)λ(dz) (v − v 1 ) dt

2x1T Q(x − x1 ) − 2r2 v T (v − v 1 )]dt,

then, 0 ≥ J2 (u1 , v) − J2 (u1 , v 1 ) Z T   =E (x − x1 )T Q(x − x1 ) + r2 (v − v 1 )T (v − v 1 ) dt. 0

C T p1 +F T q1 +

R

(N T γ 1 )(z)λ(dz)

Z . Therefore, Because of Q being nonnegative, we obtain v 1 = v = − 2r 2 we have (u1 , v 1 ) = (u∗ , v ∗ ). 2) The Necessary Condition: Here, we assume that a solution exists, then, from the uniqueness of the solution, we obtain the fact that (u∗ , v ∗ ) is the solution, and we will show that system (4.1) has L2 -gain less than r.

358

ACTA MATHEMATICA SCIENTIA

Vol.35 Ser.B

For x0 = 0, FBSDE (4.2) has a unique solution (x∗ , p∗1 , q1∗ , γ1∗ ) = (0, 0, 0, 0), then, (u∗ , v ∗ ) = (0, 0) and J2 (u∗ , v) < J2 (u∗ , v ∗ ) = 0, ∀v 6= 0 ∈ L2F (0, T ; Rnv ). Therefore, system (4.1) has L2 -gain less than r.

5



Conclusion

In this article, we study the necessary maximum principle for nonzero-sum stochastic differential games with random jumps, and the result is applied to solve the H2 /H∞ control problem of stochastic systems described by stochastic differential equations with Brownian motion and Poisson process. To obtain the H2 /H∞ control of the stochastic system, one should solve an uncontrolled FBSDE with random jumps. Because the stochastic systems are very complicated, searching for an application example which has an analytical solution or a numerical solution is without doubt a valuable work. References [1] Hinrichsen D, Pritchard A J. Stochastic H∞ . SIAM Journal on Control and Optimization, 1998, 36: 1504– 1538 [2] Zames G. Feedback and optimal sensitivity: Model reference transformation, multiplicative seminorms and approximative inverses. IEEE Transactions on Automatic Control, 1981, 26(2): 301–320 [3] Limbeer D J, Anderson B D, Hendel B. A Nash Game Approach to Mixed H2 /H∞ Control. IEEE Transactions on Automatic Control, 1994, 39(1): 69–82 [4] Zhang W H, Zhang H S, Chen B X. Stochastic H2 /H∞ control with (x, u, v)-dependent noise: Finite horizon case. Automatica, 2006, 42(11): 1891–1898 [5] Rishel R. A minimum principle for controlled jump processes. Lecture Notes in Economics and Mathematical Systems, 1975, 107: 493–508 [6] An T T T, Øksendal B. A Maximum Principle for Stochastic Differential Games with Partial Information. Journal of Optimization Theory and Applications, 2008, 139(3): 463–483 [7] Boel R K, Varaiya D. Optimal control of jump processes. SIAM Journal on Control and Optimization, 1977, 15: 92–119 [8] Tang S J, Li X J. Necessary condition for optimal control of stochastic systems with random jumps. SIAM Journal on Control and Optimization, 1994, 32: 1447–1475 [9] Wu Z, Yu Z Y. Linear quadratic nonzero-sum differential games with random jumps. Applied Mathematics and Mechanics, English Edition, 2005, 26(8): 1034–1039 [10] Peng S G. Backward stochastic differential equations and applications to optimal control. Applied Mathematics and Optimization, 1993, 27: 125–144 [11] Shi J T, Wu Z. A Risk-sensitive Stochastic Maximum Principle for Optimal Control of Jump Diffusions and its Applications. Acta Mathematica Scientia, 2011, 31B(2): 419–433 [12] Wang G C, Yu Z Y. A partial information non-zero sum differential game of backward stochastic differential equations with applications. Automatica, 2012, 48: 342–352 [13] Yong J M, Zhou X Y. Stochastic Controls: Hamiltonian Systems and HJB Equations. New York: SpringerVerlag, 1999 [14] Zhang D T. Backward Linear-Quadratic Stochastic Optimal Control and Nonzero-sum Differential Game Problem with Random Jumps. Journal of Systems Science and Complexity, 2011, 24(4): 647–662 [15] Peng S G, Wu Z. Fully Coupled Forward-Backward Stochastic Differential Equations and Applications to Optimal Control. SIAM Journal on Control and Optimization, 1999, 37(3): 825–843 [16] Wu Z. Fully coupled FBSDE with Brownian motion and Poisson process in stopping time duration. Journal of the Australian Mathematical Society, 2003, 74: 249–266