H2/H∞ control for jump-diffusions with Markovian ...

H2/H∞ control for jump-diffusions with Markovian switching

∗

Meijiao Wanga , Qingxin Meng†b , and Yang Shenc a

Business School, University of Shanghai for Science and Technology, Shanghai 200093, China b Department of Mathematical Sciences, Huzhou University, Zhejiang 313000, China c Department of Mathematics and Statistics, York University, Toronto, Ontario, M3P 1P3, Canada

March 16, 2018

Abstract: In this paper, a stochastic H2 /H∞ control problem is considered for Poisson jump-diffusion systems with Markovian switching, which are driven by a Brownian motion and a Poisson random measure with the system parameters modulated by a continuous-time, finite-state Markov chain. A stochastic jump bounded real lemma is proved that the perturbation operator kLk < γ is equivalent to the global solvability of a system of Riccati type differential equations parameterized by γ. Based on this, necessary and sufficient conditions for the existence of H2 /H∞ control are presented in terms of two sets of systems of interconnected Riccati type differential equations. Keywords: Jump-diffusion systems; Markovian switching; Stochastic H2 /H∞ control; Jump bounded real lemma; System of Riccati type differential equations AMS Subject Classification: 93E20, 60H10, 93B36, 49N10

1

Introduction

This work is concerned with a stochastic H2 /H∞ control problem for Poisson random jump diffusion systems with Markovian switching, which can be considered as Itˆo jump-diffusion processes with all the system parameters modulated by a continuous-time, finite-state Markov chain. The H∞ control and H2 /H∞ control are important robust control design methods in modern control theory. Compared with the sole H∞ control, the H2 /H∞ control may take into consideration of both the average performance and disturbance attenuation index of the closed-loop system, and therefore appears to be more attractive in practice. The mixed H2 /H∞ control for deterministic systems has attracted considerable attention and is widely applied to various practical fields, see [6, 22, 24, 32] and the references therein. As for the stochastic case, the research of stochastic H∞ control can be traced back at least to Hinrichsen and Pritchard [18], in which an infinite horizon H∞ control problem was tackled for a class of stochastic Itˆo systems with state- and control-dependent multiplicative noises and a stochastic bounded real lemma was presented for the first time in the form of linear matrix inequalities. Inspired by this work, Chen and Zhang [2] addressed the stochastic H2 /H∞ control problem for stochastic Itˆ o systems with state-dependent noise and settled it for both the finite and infinite horizon case, which, to some extent, extended the deterministic H2 /H∞ control results of Limebeer et al. [24] to the stochastic setting. Since then, the H∞ and H2 /H∞ control theories have been well developed for continuous- and discrete-time linear and nonlinear stochastic Itˆo systems with multiplicative noise. One may refer to [15, 16, 40, 41, 43, 45], to name but a few. More recently, Wang [35] discussed the H2 /H∞ control for ∗

This work was supported by the Natural Science Foundation of Zhejiang Province for Distinguished Young Scholar (No.LR15A010001), and the National Natural Science Foundation of China (No.11471079, 11301177) † Corresponding author. E-mail address: mjiao [email protected] (M. Wang), [email protected] (Q. Meng), [email protected] (Y. Shen)

1

stochastic Itˆo systems with random coefficients, where sufficient and necessary conditions for the existence of the H2 /H∞ control were given by a pair of coupled backward stochastic Riccati equations. This result extended the work of Chen and Zhang [2] to the case of random coefficients setting. In two recent papers [27, 28], H∞ and H2 /H∞ control problems have been extended to mean-field stochastic Itˆo systems. In recent years, many extensions to the stochastic Itˆo systems have been introduced in the literature to provide more realistic descriptions for system dynamics. Among them, the most popular extensions include Markovian regime-switching and Poisson jump-diffusion systems. One way to introduce additional randomness into the stochastic Itˆ o systems is to incorporate a Markov chain, resulting in the so-called Markovian regimeswitching systems. Such systems can represent a large variety of plants subject to abrupt changes in their structures and are often used to describe some typical hybrid systems that arise in many applications of systems with multiple modes or failure modes, such as fault-tolerant control systems, multiple target tracking and manufacturing systems (see, e.g., [8, 9, 29, 31]). Due to the broad applications of Markovian switching systems, the relevant research has become a very active theme in the control theory (refer to [3, 11, 29, 31]). The stochastic H∞ and H2 /H∞ control for Markovian regime-switching systems have become a popular research field. For instance, in a recent monograph [7], the H∞ control has been elaborately addressed for discretetime Markov jump systems with multiplicative noise. The state and output feedback H∞ control of nonlinear stochastic Markov jump systems with state and disturbance dependent noise has been tackled in [25] by using coupled Hamilton-Jacobi inequalities. The H2 /H∞ control problem for Markov jump systems with (x, u, v)dependent noise was investigated in [14] and [19]. For other relevant developments in this regard, interested readers can refer to the recent monograph and articles, such as [8, 9], [33, 34], [44], etc. Another important direction of extensions to stochastic Itˆo systems is to augment the Itˆo stochastic differential equations with Poisson random jumps, leading to the so-called jump-diffusion systems. Such dynamic systems are exposed to sudden, infrequent, highly localized changes that occur in a short period of time, and are frequently encountered in engineering, manufacturing, economics and biosystem applications (see, e.g., [4, 13, 30, 38]). Lin [26] studied the H∞ control problem for a class of Poisson jump-diffusion stochastic linear systems with constant coefficients. Chen [5] investigated the H∞ robust control designs for nonlinear stochastic systems with external disturbance and Poisson noise, where a fuzzy approach was proposed to solve the Hamilton-Jacobi inequality. Wang [36] conducted the H2 /H∞ control for Poisson jump-diffusion systems, and presented sufficient and necessary conditions for the existence of a state feedback H2 /H∞ control by four coupled matrix-valued Riccati equations. Roughly speaking, the stochastic systems that we will consider combine a diffusion component, Poisson random jumps and Markovian regime switches. In such systems, the Poisson random jumps describe their state discontinuous changes, while Markovian regime switches characterize their discrete shifts in different modes. They have been widely used in various fields, such as communication network, flexible manufacturing system, air traffic management, complex power system, mathematical finance and risk management (see [1, 20, 23, 37, 39, 42] and the references therein). A careful examination of the existing works reveals that although fruitful results have been obtained in H2 /H∞ control designs, the H2 /H∞ control for Poisson jump-diffusion systems with Markovian switching has not been investigated. This paper aims to fill in this gap. In this paper, sufficient and necessary conditions are derived for the existence of a state feedback H2 /H∞ control in terms of two sets of systems of interconnected Riccati type differential equations. The rest of this paper is organized as follows. In Section 2, we present problem formulation including notations, useful lemmas and some preliminaries on the stochastic H∞ and H2 /H∞ control. In Section 3, inspired by the ideas in [18, 9, 33], we develop the stochastic bounded real lemma for jump-diffusion systems with Markovian switching. This enables us to show necessary and sufficient conditions for the existence of a stabilizing controller which keeps the effect of the perturbation on the to-be-controlled output below a given disturbance attenuation level. That is, based on this result, we then derive necessary and sufficient conditions for the existence of the stochastic H2 /H∞ control in Section 4. Concluding remarks are presented in Section 5.

2

2

Problem formulation

In this section, we first introduce basic notations and some preliminary results and then formulate the stochastic H∞ and H2 /H∞ control problems.

2.1

Notations

The following notations will be used throughout this paper. (i) Rn is the real n-dimensional space, E is some nonempty subset of R1 , Rn×m is the set of all real n × m matrices, and In is the identity matrix in Rn×n . If A is a matrix (or a vector), A0 is the transpose of A, and kAk is the square root of the summarized squares of all the components of the vector or matrix A. T r(A) is the trace of A. A−1 is the inverse of a nonsingular square matrix A. A > 0 means that A is a positive semi-definite square matrix. hA1 , A2 i is the inner product of two vectors A1 and A2 . χM denotes the indicator function of a set M . (ii) Sn denotes the space of all n × n symmetric matrices, Sn+ (resp. Sn− ) the subset of all nonnegative (resp. nonpositive) definite matrices of Sn , and Snd (resp. Snd− ) denotes the space of all H = (H1 , · · · , Hd ) with Hi ∈ Sn (resp. Sn− ). Mdn,m stands for the space of A = (A1 , A2 , ·, Ad ) where Ai ∈ Rn×m . Particularly, Mdn,n is denoted by Mdn for short. On Mdn,m , we introduce the norm kAk = maxi∈D kAi k. (iii) Let T be a fixed, strictly positive real number and (Ω, F, P ) be a given probability space, on which a Brownian motion, a Markov chain and a Poisson random meaure are defined and assumed to be mutually independent. More specifically, (A1) W = {W (t); 0 ≤ t ≤ T } is a one-dimensional standard Brownian motion. (A2) r = {rt ; 0 6 t 6 T } is a right-continuous, homogeneous Markov chain with the state space D = {1, 2, · · · , d} and the probability transition matrix  qij ∆t + o(∆t), i 6= j, P {rt+∆t = j | rt = i} = 1 + qii ∆t + o(∆t), i = j, P with dj=1 qij = 0, for all i ∈ D and qij > 0, for i, j ∈ D where i 6= j. We further assume that P {r0 = i} > 0, for all i ∈ D. Then for any τ ∈ [0, T ], we have P {rτ = i} > 0. (A3) η = {ηt : Ω × Dη → E } is an Ft -adapted stationary Poisson point process with characteristic measure ν, where Dη is a countable subset of (0, ∞). Let (E , B(E ), ν) be a measurable space with ν(E ) < ∞. Then the counting measure induced by η is µ((0, t] × M ) := ]{s ∈ Dη ; s 6 t, ηs ∈ M },

t > 0, M ∈ B(E ),

and µ ˜(dθ, dt) := µ(dθ, dt) − ν(dθ)dt is a compensated Poisson random martingale measure. Moreover, we assume that {F t }06t6T is the P -completed natural filtration generated by (r,W,˜ µ)

Ft

:= σ[rs ; 0 6 s 6 t] ∨ σ[W (s); 0 6 s 6 t] # "Z Z ∨σ µ ˜(dθ, du); 0 6 s 6 t , A ∈ B(E ), A×(0,s]

where σ1 ∨ σ2 ∨ σ3 denotes the smallest σ-algebra generated by σ1

3

S

σ2

S

σ3 .

(iv) Let H be a finite-dimensional vector space, C(τ, T ; H) the space of H-valued continuous functions on [τ, T ] endowed with the maximum norm, SF2 (τ, T ; H) the space of H-valued Ft -adapted c` adl` ag processes f on [τ, T ] satisfying kf k = E{supτ 6t6T |f (t)|2 } < ∞, Lν,2 (E ; H) the space of H-valued measurqR 2 able functions f defined on the measurable space (E , B(E ), ν) satisfying kf k = E |f (θ)|H ν(dθ) < ν,2 ν,2 ∞, F ([0, T ] × E ; H) the space of L (E ; H)-valued and Ft -predictable processes f satisfying kf k = q LRR E{ E ×(0,T ] |f (t, θ)|2H ν(dθ)dt} < ∞, L2 ([τ, T ], H) the space of H-valued measurable functions f on qR T k 2 [τ, T ] satisfying kf k = τ |f (t)|H dt < ∞, and U [τ, T ] the space of all measurable and Ft -adapted R T processes f : [τ, T ] × Ω → Rk with E{ τ |f (t)|2 dt | rτ = i} < ∞, i ∈ D, τ ∈ [0, T ]. Obviously, U k [τ, T ] is a Hilbert space with the inner product

Z d X hf, gi = E

T

 f (t)g(t)dt | rτ = i , 0

kf k2 = hf, f i.

τ

i=1

Sometimes we may write P for a deterministic function P (t), omitting the variable t, whenever no confusion arises. Under this convention, P > (>)0 means P (t) > (>)0, for all t ∈ [0, T ].

2.2

Two useful lemmas

Now we recall two lemmas about stochastic differential equations (SDEs) driven by the Brownian motion and the Poisson random jump with Markovian switching parameters. The former is a theorem on the existence and uniqueness of a solution, the latter is a generalized Itˆo’s formula. Lemma 2.1. [17] Let x(0) be an F0 -measurable random variable, and b : [0, T ] × Rn × D −→ Rn , σ : [0, T ] × Rn × D −→ Rn , π : [0, T ] × E × Rn × D −→ Rn be given mappings satisfying (i) b(·, 0, i) ∈ L2 (0, T ; Rn ), σ(·, 0, i) ∈ L2 (0, T ; Rn ), and π(·, ·, 0, i) ∈ Lν,2 ([0, T ] × E ; Rn ); (ii) for some positive constant C > 0, and for all (t, x, x ¯, i) ∈ [0, T ] × Rn × Rn × D, the Lipschitz continuity condition is satisfied in the sense |b(t, x, i) − b(t, x ¯, i)|2 + |σ(t, x, i) − σ(t, x ¯, i)|2 Z + |π(t, θ, x, i) − π(t, θ, x ¯, i)|2 ν(dθ) 6 C|x − x ¯|2 . E

Then the stochastic differential equation with the Poisson random jump and Markovian switching parameters Z t Z t ZZ x(t) = x(0) + b(s, x(s), rs )ds + σ(s, x(s), rs )dW (s) + π(s, θ, x(s−), rs )˜ µ(dθ, ds) 0

E ×(0,t]

0

has a unique solution x ∈ SF2 (0, T ; Rn ). Moreover, a priori estimate holds:    Z T Z E sup |x(t)|2 | r0 = i 6 KE |x(0)|2 + |b(t, 0, rt )|2 dt + 06t6T

ZZ + E ×(0,T ]

T

|σ(t, 0, rt )|2 dt 0 0  2 |π(t, θ, 0, rt )| ν(dθ)dt | r0 = i ,

where K is a positive constant depending only on the Lipschitz constant C and the time index T . Lemma 2.2. (Generalized Itˆ o’s formula. See [8] and [21]) Let x(t) satisfy Z dx(t) = b(t, x(t), rt )dt + σ(t, x(t), rt )dW (t) + c(t, θ, x(t−), rt )˜ µ(dθ, dt), E

4

(2.1)

and φ(·, ·, i) ∈ C 1,2 ([0, T ] × Rn ), for any i ∈ D, be given. Then, dφ(t, x(t), rt ) = A φ(t, x(t), rt )dt + hφx (t, x(t), rt ), σ(t, x(t), rt )idW (t) Z + [φ(t, x(t−) + c(t, θ, x(t−), rt ), rt ) − φ(t, x(t−), rt )]˜ µ(dθ, dt),

(2.2)

E

where X 1 qij φ(t, x, j) A φ(t, x, i) := φt (t, x, i) + hφx (t, x, i), b(t, x, i)i + tr{σ 0 (t, x, i)φxx (t, x, i)σ(t, x, i)} + 2 j∈D Z + [φ(t, x + c(t, θ, x, i), i) − φ(t, x, i) − hφx (t, x, i), c(t, θ, x, i)i]ν(dθ), E

and φt and φx denote the partial derivatives of φ with respect to t and x respectively, and φxx the second-order partial derivative of φ with respect to x.

2.3

Stochastic H∞ and H2 /H∞ control

Consider the following Poisson jump-diffusion system with Markovian switching parameters:  dx(t) = [Art (t)x(t) + Br1t (t)v(t) + Br2t (t)u(t)]dt + [A0rt (t)x(t) + Br01t (t)v(t) + Br02t (t)u(t)]dW (t)    Z  + [Ert (t, θ)x(t−) + Fr1t (t, θ)v(t) + Fr2t (t, θ)u(t)]˜ µ(dθ, dt), x(0) = x0 ,  E    zt = Crt (t)x(t) + Dr1t (t)v(t) + Dr2t (t)u(t).

(2.3)

We assume that (H1 ) the coefficient matrices Ai , A0i : [0, T ] → Rn×n , Bi1 , Bi01 : [0, T ] → Rn×m , Bi2 , Bi02 : [0, T ] → Rn×s , Ci : [0, T ] → Rq×n , Di1 : [0, T ] → Rq×m , Di2 : [0, T ] → Rq×s , Ei : [0, T ] → Lν,2 (E ; Rn×n ), Fi1 : [0, T ] → Lν,2 (E ; Rn×m ) and Fi2 : [0, T ] → Lν,2 (E ; Rn×s ) are continuous matrix-valued functions on [0, T ], for all i ∈ D; (H2 ) (Di1 (t))0 Di1 (t) > δIm , (Di2 (t))0 Di2 (t) = Is , Ci0 (t)Dij (t) = 0, j = 1, 2 and (Di1 (t))0 Di2 (t) = 0, for all i ∈ D and t ∈ [0, T ]. Then from Lemma 2.1, for all (u, v, x0 ) ∈ U s [0, T ] × U m [0, T ] × Rn , there exists a unique solution x = x(·, u, v, x0 ) ∈ SF2 (0, T ; Rn ) to the state equation of system (2.3). Note that we assume the Brownian Motion to be one-dimensional just for simplicity, and there is no essential difficulty in the analysis below for the multidimensional cases. We view v as an external disturbance which adversely affects the to-be-controlled output z ∈ Rq (whose desired value is represented by 0). The disturbing effect is to be ameliorated via control input u ∈ U s [0, T ]. The effect of the disturbance on the to-be-controlled output z of system (2.3) is then described by the perturbation operator Lcl : v 7→ z which (for zero initial state) maps finite disturbance signals v into the corresponding finite output signals z of the closed-loop system. The size of this linear operator is measured by the induced norm. The larger this norm is, the larger is the effect of the unknown disturbance v on the to-be-controlled output z in the worst case. Then the H∞ control problem is to determine whether or not for each γ > 0 there exists a stabilizing controller u∗ achieving kLu∗ k < γ, where Lu∗ : U m [0, T ] → U q [0, T ] can be defined as Lu∗ (v) = Crt x(·, u∗ , v, 0) + Dr1t v + Dr2t u∗ . Obviously, there may be more than one solution satisfying the required condition. We require a controller not only to attenuate external disturbances efficiently, but also to minimize the output when the worst case disturbance is applied to the system; this is the so-called H2 /H∞ control problem. That is, we wish to find a feedback control u∗ ∈ U s [0, T ] achieving kLu∗ k < γ, and meanwhile, minimizing the output z when the worst case disturbance v ∗ ∈ U m [0, T ] is applied to the system (2.3). 5

By constructing two performance indice as follows: Z T  2 2 2 J1 (u, v, i) = E [γ |v(t)| − |z(t)| ]dt|r0 = i ,

i ∈ D,

(2.4)

0

and Z

T



2

i ∈ D,

|z(t)| dt|r0 = i ,

J2 (u, v, i) = E

(2.5)

0

where the first reflects an H∞ constraint, while the second is an H2 optimality requirement, we can convert the H2 /H∞ control into finding a pair of equilibrium points (u∗ , v ∗ ), such that J1 (u∗ , v ∗ , i) 6 J1 (u∗ , v, i),

∀v ∈ U m [0, T ], i ∈ D,

J2 (u∗ , v ∗ , i) 6 J2 (u, v ∗ , i),

∀u ∈ U s [0, T ], i ∈ D.

and Pd

If i=1 J1 (u∗ , v ∗ , i) > 0 with x0 = 0, we certainly have kzk2 6 γ 2 kvk2 , for all v ∈ U m [0, T ], which ensures that kLu∗ k 6 γ. The second Nash inequality shows that u∗ minimizes output energy z when the external disturbance is at its worst and given by v ∗ . Clearly, if the Nash equilibria (u∗ , v ∗ ) exist, then u∗ is our desired H2 /H∞ controller, and v ∗ is the corresponding worst case disturbance. Then the H2 /H∞ control problem can be converted into finding the Nash equilibria (u∗ , v ∗ ). We approach this problem as a linear quadratic (LQ) optimal control problem and obtain the solution by studying the associated stochastic Riccati equations. In this paper, we will provide and prove sufficient and necessary conditions for the existence of linear state feedback pairs (u∗ , v ∗ ).

3

Stochastic jump bounded real lemma

In this section, we establish one of the main results of this paper, namely, a stochastic jump bounded real lemma (SJBRL). As is well known, the SJBRL plays a central role in investigating the H∞ control and estimation problems for stochastic systems. Roughly speaking, this result provides necessary and sufficient conditions for a given stochastic system to be stable with kLk < γ. The SJBRL is of theoretical interest on its own, because it allows one to determine kLk which measures the influence of the disturbances in the worst case scenario. To this end, we consider the following linear stochastic system:  dx(t) = [Art (t)x(t) + Brt (t)v(t)]dt + [A0rt (t)x(t) + Br0t (t)v(t)]dW (t)    Z  + [Ert (t, θ)x(t−) + Frt (t, θ)v(t)]˜ µ(dθ, dt), x(τ ) = ξ, (3.1)  E    z(t) = Crt (t)x(t) + Drt (t)v(t), where τ ∈ [0, T ] and ξ ∈ Rn are the initial time and state, respectively. We denote v ∈ U m [τ, T ] as the external disturbance and z ∈ U q [0, T ] the controlled output. Furthermore, we assume (H3 ) all the coefficients Ai , A0i : [0, T ] → Rn×n , Bi , Bi0 : [0, T ] → Rn×m , Ci : [0, T ] → Rq×n , Di : [0, T ] → Rq×m , Ei : [0, T ] → Lν,2 (E ; Rn×n ) and Fi : [0, T ] → Lν,2 (E ; Rn×m ) are continuous matrix-valued functions, for all i ∈ D; (H4 ) (Di (t))0 Di (t) > δIm and (Di (t))0 Ci (t) = 0, for all i ∈ D, t ∈ [0, T ]. For all (v, ξ) ∈ U m [τ, T ]×Rn , there exists a unique solution x = x(·, v; τ, ξ) ∈ SF2 (τ, T ; Rn ) to the state equation in system (3.1). Moreover, using Lemma 2.1 we obtain that there exists c > 0 (depending upon T ) such that Z d X E i=1

T 2



|x(t, v; τ, 0)| dt | rτ = i

τ

Z d X 6c E i=1

6

τ

T

 |v(t)| dt | rτ = i , 2

for any i ∈ D and τ ∈ [0, T ]. It follows that the system (3.1) defines a linear bounded perturbation operator L : U m [0, T ] → U q [0, T ] by (t, v) ∈ [0, T ] × U m [0, T ].

(Lv)(t) = Crt (t)x(t, v; 0, 0) + Drt (t)v(t),

It is clear that, z(·) = (Lv)(·), when x(0) = 0. The H∞ norm of this perturbation operator, i.e., kLk, is then defined as follows RT P ( di=1 E{ 0 z(t)0 z(t)dt | r0 = i})1/2 k(Lv)k kLk = sup = sup . (3.2) RT Pd v∈U m [0,T ] kvk v∈U m [0,T ] ( i=1 E{ 0 v(t)0 v(t)dt | r0 = i})1/2 v6=0,x(0)=0

v6=0,x(0)=0

The larger this norm is, the larger is the effect of the unknown disturbance v on the output z, i.e., kLk measures the influence of the disturbance in the worst case scenario. Therefore, a method enabling us to compute the size of kLk is rather important. In the control literature, the bounded real lemma refers to a tailor-made technique to evaluate the size of kLk. So, in view of (3.2), the norm of L is the minimal γ > 0 such that kzk 6 γkvk, for all v ∈ U m [0, T ]. It is very helpful and common in the H∞ -control literature to associate a finite-time quadratic cost functional with the problem. Therefore, our approach relies on a detailed study of the following cost functionals: Z T  2 2 2 J(v; τ, ξ, i) = E (γ |v(t)| − |z(t)| )dt|rτ = i τ Z T  2 0 0 =E [h(γ I − Drt (t)Drt (t))v(t), v(t)i − hCrt (t)Crt (t)x(t), x(t)i]dt | rτ = i , (3.3) τ

and J(v; τ, ξ) =

d X

J(v; τ, ξ, i),

(3.4)

i=1

which depend on the initial condition (τ, ξ, i) ∈ [0, T ]×Rn ×D, the disturbance v ∈ U m [τ, T ] and some prescribed disturbance attenuation level γ > 0. We are especially interested in the linear quadratic (LQ) control problem of minimizing the above cost functional J(v; τ, ξ, i) over v ∈ U m [τ, T ]. In our development in this section, we will employ the convention in LQ theory and refer to the disturbance v as a “control”. We introduce the following system of Riccati type differential equations (SRDEs) parametrized by γ, which is obtained by using the dynamic programming principle and the generalized Itˆo’s formula (2.2): Z  0 0 0 0  ˙  P (t) + A (t)P (t) + P (t)A (t) + (A (t)) P (t)A (t) + Ei0 (t, θ)Pi (t)Ei (t, θ)ν(dθ) i i i i i  i i i   E   d X γ (3.5) 0 0 −1 − Ci (t)Ci (t) − Φi (t, P (t))Λi (t, P (t)) Φi (t, P (t)) + qij Pj (t) = 0, Pi (T ) = 0,     j=1    γ Λi (t, P (t)) > 0, i ∈ D, with Φi (t, P ) := Pi Bi (t) + (A0i (t))0 Pi Bi0 (t) +

Z E

Ei0 (t, θ)Pi Fi (t, θ)ν(dθ)

and Λγi (t, P ) := γ 2 I − Di0 (t)Di (t) + (Bi0 (t))0 Pi Bi0 (t) +

Z E

Fi0 (t, θ)Pi Fi (t, θ)ν(dθ).

(3.6)

(3.7)

In the next lemma, we rewrite the cost functional (3.3) in an alternative form. This alternative expression of the cost functional allows us, among many other things, to solve the aforementioned minimizing problem in a rather straightforward manner, with the aid of SDREs (3.5). 7

Lemma 3.1. For any initial condition (τ, ξ, i), all v ∈ U m (τ, T ) and every P = (P1 , P2 , · · · , Pd ) : [τ, T ] 7→ Snd being continuously differentiable, the cost functional defined as (3.3) can be expressed as J(v; τ, ξ, i) = hξ, Pi (τ )ξi − E{hx(T ), PrT (T )x(T )i | rτ = i} Z T        x(t) x(t) ˙ +E hx(t), Prt (t)x(t)i + , Mrt (t, P (t)) dt|rτ = i , v(t) v(t) τ where we define M (t, P ) = (M1 (t, P ), M2 (t, P ), · · · , Md (t, P )) by means of the following matrices:   Γi (t, P ) − Ci0 (t)Ci (t) Φi (t, P ) Mi (t, P ) := , Φ0i (t, P ) Λγi (t, P ) with Γi (t, P ) :=

A0i (t)Pi

+ Pi Ai (t) +

(A0i (t))0 Pi A0i (t)

Z + E

Ei0 (t, θ)Pi Ei (t, θ)ν(dθ) +

d X

qij Pj ,

j=1

and Φi (t, P ) and Λγi (t, P ) defined by (3.6) and (3.7), respectively. Proof. Applying Itˆ o’s formula (2.2) to hx(t), Prt (t)x(t)i and noting J(v; τ, ξ, i) = J(v; τ, ξ, i) Z T  0 0 0 +E d(x(t) Prt (t)x(t)) − x (T )PrT (T )x(T ) + x(τ ) Prτ (τ )x(τ ) | rτ = i , τ

we can easily obtain the desired result. The proof is complete. Now we are in the position to give the main result of this section. Theorem 3.2. (Stochastic jump bounded real lemma) Given γ > 0, kLk < γ if and only if the corresponding system of Riccati type differential equations (SRDEs) (3.5) parameterized by γ has a unique negative semidefinite solution P = (P1 , P2 , · · · , Pd ) ∈ C([0, T ]; Snd− ). The next proposition proves the above theorem in one direction, i.e., establishing a relation between kLk < γ and the existence of some P ∈ Snd− such that the SRDEs (3.5) has a solution. Proposition 3.3. If there exists γ > 0 and P ∈ Snd− such that SRDEs (3.5) has a solution, then kLk < γ. Proof. Suppose that the SRDEs (3.5) is solvable for some pair (γ, P ) with γ > 0 and P 6 0, and let x be the solution of (3.1) with xτ = ξ. Applying Itˆ o’s formula (2.2) to hPrt (t)x(t), x(t)i, together with considering (3.5) and using the completion of squares, we have for all i ∈ D: Z T  0 0 0 J(v; τ, ξ, i) = J(v; τ, ξ, i) + E d(x(t) Prt (t)x(t)) − x (T )PrT (T )x(T ) + x(τ ) Prτ (τ )x(τ ) | rτ = i τ Z T 

γ  0 = ξ Pi (τ )ξ + E Λrt (t, P (t))(v(t) − Ψrt (t, P (t))x(t)), v(t) − Ψrt (t, P (t))x(t) dt | rτ = i τ

> ξ 0 Pi (τ )ξ,

(3.8)

where Ψi (t, P ) = −Λγi (t, P )−1 Φi (t, P ), for all (t, i) ∈ [0, T ]×D. It follows immediately that the optimal feedback control would be v(t) = Ψrt (t, P (t))x(t−) if the corresponding solution to the system equation exists. In this case, the optimal cost reads min J(v; τ, ξ, i) = ξ 0 Pi (τ )ξ, for all i ∈ D. In fact, when v(t) = Ψrt (t, P (t))x(t−), v∈U m [τ,T ]

the system (3.1) reduces to stochastic differential equations (SDEs): dx(t) = [Art (t) + Brt (t)Ψrt (t, P (t))]x(t)dt + [A0rt (t) + Br0t (t)Ψrt (t, P (t))]x(t)dW (t) 8

Z + E

([Ert (t, θ) + Frt (t, θ)Ψrt (t, P (t))]x(t−)˜ µ(dθ, dt),

x(τ ) = ξ.

(3.9)

In view of the third positive definiteness constraint in (3.5), there exists a sufficient small  > 0 such that Λγrt (t, P (t)) > I, for all t ∈ [τ, T ]. Moreover, since P is negative definite and uniformly bounded, all the coefficients of (3.9) are continuous and uniformly bounded. Therefore, the linear SDE (3.9) indeed has a unique solution x ∈ SF2 (τ, T ; Rn ), thus, v = Ψx ∈ U m [τ, T ]. Here and in the following, v = Ψx denotes {v(t) = Ψrt (t, P (t))x(t−)}t∈(τ,T ] , where Ψi (t, P (t)) ∈ C(τ, T ; Rm×n ). From (3.8), we derive that J(v; 0, x(0), i) > x(0)0 Pi (0)x(0). In particular, if x(0) = 0, then J(v; 0, 0, i) > 0. Hence J(v; 0, 0) > 0, which is equivalent to kLk 6 γ. To show kLk < γ, we define an operator T : U m [0, T ] 7−→ U m [0, T ],

T v(t) = ve(t)

with its realization dx(t) = (Art (t)x(t) + Brt (t)v(t))dt + (A0rt (t)x(t) + Br0t (t)v(t))dW (t) Z µ(dθ, dt), x(0) = 0, + (Ert (t, θ)x(t−) + Frt (t, θ)v(t))˜ E

and ve(t) = v(t) + Λγrt (t, P (t))−1 Φrt (t, P (t))x(t−). Based on the estimate for SDE (2.1), we conclude that the operator T is well defined. Moreover, it is a bounded linear operator. Then T −1 exists, which is determined by dx(t) = {[Art (t) − Brt (t)Λγrt (t, P (t))−1 Φrt (t, P (t))]x(t) + Brt (t)e v (t)}dt + {[A0rt (t) − Br0t (t)Λγrt (t, P (t))−1 Φrt (t, P (t))]x(t) + Br0t (t)e v (t)}dW (t) Z + {[Ert (t, θ) − Frt (t, θ)Λγrt (t, P (t))−1 Φrt (t, P (t))]x(t−) + Frt (t, θ)e v (t)}˜ µ(dθ, dt),

x(0) = 0,

E

and vˆ(t) = −Λγrt (t, P (t))−1 Φrt (t, P (t))x(t−) + ve(t). It follows from the inverse operator theorem in functional analysis that kT −1 k is bounded. So, there exists a  positive constant c = kT −1 such that k2 J(v; 0, 0) =

=

d X i=1 d X

Z

T

0

(v(t) − Ψrt (t, P (t))x(t))

E 0

Z E

i=1

0

T

Λγrt (t, P (t))(v(t)

(T v(t))0 Λγrt (t, P (t))(T v(t))dt | r0 = i

 − Ψrt (t, P (t))x(t))dt | r0 = i



2

>kT v(t)k > ckv(t)k2 > 0, which is equivalent to kLk < γ. The proof is complete. In order to prove the second part of the SJBRL, Theorem 3.2, we proceed in several steps. To this end, we first establish some intermediate results. The following proposition is necessary for subsequent analysis. It provides existence and uniqueness results for stochastic differential equations to be studied in the sequel. Proposition 3.4. There exist X = (X1 , X2 , · · · , Xd ) and Y = (Y1 , Y2 , · · · , Yd ) mapping [0, T ] into Snd− such that, for all i ∈ D, Z Xi˙(t) + A0i (t)Xi (t) + Xi (t)Ai (t) + (A0i (t))0 Xi (t)A0i (t) + Ei0 (t, θ)Xi (t)Ei (t, θ)ν(dθ) E

+

d X

qij Xj (t) − Ci0 (t)Ci (t) = 0,

Xi (T ) = 0,

j=1

9

(3.10)

and Yi ˙(t) + A0i (t)Yi (t) + Yi (t)Ai (t) + (A0i (t))0 Yi (t)A0i (t) + +

d X

Z E

Ei0 (t, θ)Yi (t)Ei (t, θ)ν(dθ)

qij Yj (t) − (Yi (t)Bi (t))0 Ri−1 (t)Yi (t)Bi (t) − Ci0 (t)Ci (t) = 0,

Yi (T ) = 0,

(3.11)

j=1

where R = (R1 , · · · , Rd ) ∈ Snd and Ri > 0, for all i ∈ D. Moreover, X and Y are unique, continuous, and continuously differentiable. Proof. Eq. (3.10) is a Lyapunov type equation on Snd and has a unique bounded and negative semi-definite solution (refer to Proposition 4.8 in [8]). As for Eq. (3.11), the proof of existence and uniqueness of Y is completed by utilizing the solvability of (3.10), together with the technique of quasi-linearization and successive approximation. The proof can be adapted from that of Theorem 4.1 of [12]. Lemma 3.5. Suppose that ϕ = (ϕ1 , ϕ2 , · · · , ϕd ) ∈ C(τ, T ; Mdm×n ) and P γ,ϕ = (P1γ,ϕ , P2γ,ϕ , · · · , Pdγ,ϕ ) ∈ C(τ, T ; Snd ), where P γ,ϕ satisfies the linear differential matrix-valued equations X˙ i (t) + A0i (t)Xi (t) + Xi (t)Ai (t) + (A0i (t))0 Xi (t)A0i (t) +

Z E

Ei0 (t, θ)Xi (t)Ei (t, θ)ν(dθ)

+

d X

qij Xj (t)

j=1

− Ci0 (t)Ci (t) + ϕ0i (t)Φi (t, X(t))0 + ϕ0i (t)Λγi (t, X(t))ϕi (t) + Φi (t, X(t))ϕi (t) = 0,

Xi (T ) = 0,

i ∈ D. (3.12)

Then if v ∈ U m (τ, T ), we have ϕ

J(v + ϕx ; τ, ξ, i) =

hξ, Piγ,ϕ (τ )ξi + hNrt (t, P

Z +E τ

γ,ϕ

T

[hv(t), N (t, Prγ,ϕ (t))xϕ (t)i t

ϕ

(t))x (t), v(t)i +

hv(t), Λγrt (t, P γ,ϕ )(t)v(t)i]dt

 | rτ = i ,

(3.13)

where xϕ = xϕ (·, v(·); τ, ξ) = x(·, v(·) + ϕ(·)xϕ (·); τ, ξ) is the solution of dx(t) = [(Art (t) + Brt (t)ϕ(t))x(t) + Brt (t)v(t)]dt + [(A0rt (t) + Br0t (t)ϕ(t))x(t) + Br0t (t)v(t)]dW (t) Z + [(Ert (t, θ) + Frt (t, θ)ϕ(t))x(t−) + Frt (t, θ)v(t)]˜ µ(dθ, dt), x(τ ) = ξ,

(3.14)

E

and we define N (t, P ) = (N1 (t, P ), N2 (t, P ), . . . , Nd (t, P )) by means of the following way: Ni (t, P γ,ϕ (t)) = Φ0i (t, P γ,ϕ (t)) + Λγi (t, P γ,ϕ (t))ϕi (t). In particular, if v = 0, then J(ϕxϕ ; τ, ξ, i) = hξ, Piγ,ϕ (τ )ξi.

(3.15)

Proof. As P γ,ϕ satisfies X˙ rt (t) +



I ϕ(t)

0

 Mrt (t, X(t))

10

I ϕ(t)

 = 0,

XrT (T ) = 0,

applying Lemma 3.1 with P = P γ,ϕ and v + ϕxϕ for v (and recalling the definition of M in Lemma 3.1), we obtain Z T D E γ,ϕ ϕ ϕ J(v + ϕx ; τ, ξ, i) = hξ, Pi (τ )ξi + E xϕ (t), P˙rγ,ϕ (t)x (t) t τ       xϕ (t) xϕ (t) γ,ϕ + , Mrt (t, P (t)) dt | rτ = i , v(t) + ϕ(t)xϕ (t) v(t) + ϕ(t)xϕ (t) Z T  = hξ, Piγ,ϕ (τ )ξi + E hv(t), Nrt (t, P γ,ϕ (t))xϕ (t)i τ  

 γ,ϕ ϕ γ γ,ϕ + hNrt (t, P (t))x (t), v(t)i + v(t), Λrt (t, P (t))v(t) dt | rτ = i . Hence, Eq. (3.13) holds. By setting v = 0 in (3.13), we immediately obtain (3.15). In the above proof, we should notice that as all the coefficients of equation (3.14) are continuous and uniformly bounded. From Lemma 2.1, there exists a unique solution xϕ ∈ SF2 (τ, T ; Rm ) to that equation. Therefore, ϕxϕ ∈ U m [τ, T ] and v + ϕxϕ ∈ U m [τ, T ] hold true. The following lemma shows that the cost associated to any admissible disturbance is uniformly bounded from below. Lemma 3.6. If kLk < γ, then there exists µ > 0, such that for all (τ, ξ, i) ∈ [0, T ] × Rn × D and v ∈ U m [τ, T ], we have J(v; τ, ξ, i) > −µ|ξ|2 . Proof. The inequality kLk < γ implies that there exists some constant  > 0 such that 2 6 γ 2 − kLk2 . So, we can derive that for any v ∈ U m [0, T ], Z T  d X 2 0 0 E [h(γ I − Drt (t)Drt (t))v(t), v(t)i − hCrt (t)Crt (t)x(t), x(t)i]dt | r0 = i 0

i=1

Z d X > E

T

2

 hv(t), v(t)idt | r0 = i .

0

i=1

Then it follows immediately Z T  d X d X 2 0 0 pij (τ )E [h(γ I − Drt (t)Drt (t))v(t), v(t)i − hCrt (t)Crt (t)x(t), x(t)i]dt | rτ = j τ

i=1 j=1 2

>

d X d X

T

Z pij (τ )E

 hv(t), v(t)idt | rτ = j .

(3.16)

τ

i=1 j=1

We conclude that for any j ∈ D, Z T  d X 2 0 0 pij (τ )E [h(γ I − Drt (t)Drt (t))v(t), v(t)i − hCrt (t)Crt (t)x(t), x(t)i]dt | rτ = j τ

i=1 2

>

d X

Z

T

 hv(t), v(t)idt | rτ = j .

pij (τ )E

(3.17)

τ

i=1

Otherwise, if there exists some u ˜ ∈ U m [τ, T ] and some k ∈ D such that Z T  d X 2 0 0 pik (τ )E [h(γ I − Drt (t)Drt (t))˜ u(t), u ˜(t)i − hCrt (t)Crt (t)x(t), x(t)i]dt | rτ = k τ

i=1 2

6

d X i=1

Z

T

 h˜ u(t), u ˜(t)idt | rτ = k .

pik (τ )E τ

11

(3.18)

Taking v(t) = u ˜(t)χrτ =k , ∀t ∈ [τ, T ) in the left hand side of (3.16), and combining with (3.18), we obtain d X d X

=

[h(γ I − τ

Z

6

= 2

T 2

[h(γ I −

pik (τ )E τ

i=1 2

2

pij (τ )E

i=1 j=1 d X

T

Z

d X

−

−

hCr0 t (t)Crt (t)x(t), x(t)i]dt

hCr0 t (t)Crt (t)x(t), x(t)i]dt

 | rτ = j

 | rτ = k

 h˜ u(t), u ˜(t)idt | rτ = k

pik (τ )E i=1 d X d X

τ T

Z

 h˜ u(t)χrτ =k , u ˜(t)χrτ =k idt | rτ = j

pij (τ )E

d X d X

Dr0 t (t)Drt (t))˜ u(t), u ˜(t)i

T

Z

τ

i=1 j=1

= 2

Dr0 t (t)Drt (t))˜ u(t)χrτ =k , u ˜(t)χrτ =k i

T

Z pij (τ )E

 hv(t), v(t)idt | rτ = j ,

τ

i=1 j=1

which contradicts with (3.16). Thus, (3.17) holds and moreover, Z

T 2

Dr0 t (t)Drt (t))v(t), v(t)i

[h(γ I − Z T  2 > E hv(t), v(t)idt | rτ = j .

E

τ

−

hCr0 t (t)Crt (t)x(t), x(t)i]dt

 | rτ = j (3.19)

τ

By linearity, the solution x(t, v; τ, ξ) of system equation (3.1) satisfies x(t, v; τ, ξ) = x(t, v; τ, 0) + x(t, 0; τ, ξ). Denoting by X(t) = (X1 (t), X2 (t), · · · , Xd (t)) the solution of (3.10) and applying (3.13) with ϕ = 0, we get J(v; τ, ξ, i) − J(v; τ, 0, i) Z T  = ξ 0 Xi (τ )ξ + E [hv(t), Nrt (t, X(t))x(t, 0; τ, ξ)i + hx(t, 0; τ, ξ), Nrt (t, X(t))v(t)i] dt | rτ = i , τ

where Ni (t, X(t)) = Φ0i (t, X(t)). Combining with (3.19) gives J(v;τ, ξ, i) > ξ 0 Xi (τ )ξ Z T   2  +E  hv(t), v(t)i + hv(t), Nrt (t, X(t))x(t, 0; τ, ξ)i + hNrt (t, X(t))x(t, 0; τ, ξ), v(t)i dt | rτ = i τ Z T    0 −1 2 −1 2 = ξ Xi (τ )ξ + E |v(t) +  Nrt (t, X(t))x(t, 0; τ, ξ)| − | Nrt (t, X(t))x(t, 0; τ, ξ)| dt | rτ = i τ Z T  0 −1 2 > ξ Xi (τ )ξ − E | Nrt (t, X(t))x(t, 0; τ, ξ)| dt | rτ = i . τ

(3.20) According to the estimate for SDE (2.1), there exists c0 > 0 such that Z E

T

 |x(t, 0; τ, ξ)|2 dt | rτ = i 6 c0 |ξ|2 .

τ

12

Hence, by (3.15) there exist constants c1 , c2 > 0 such that 0 > hξ, Xi (τ )ξi = J(0; τ, ξ, i) Z T 

0  =−E Crt (t)Crt (t)x(t, 0; τ, ξ), x(t, 0; τ, ξ) dt | rτ = i > −c1 |ξ|2 , τ

that is, for all i ∈ D, τ ∈ [0, T ], −c1 I 6 Xi (τ ) 6 0. Moreover, for any t ∈ [0, T ], Z 0 0 0 Er0 t (t, θ)Xrt (t)Frt (t, θ)ν(dθ)k 6 c2 . kNrt (t, X(t))k = kXrt (t)Brt (t) + (Art (t)) Xrt (t)Brt (t) + E

Thus, by (3.20) J(v; τ, ξ, i) > −c1 |ξ|2 − c22 −2 c0 |ξ|2 := −µ|ξ|2 . The proof is complete. Lemma 3.7. Suppose that kLk < γ, ϕ ∈ C(τ, T ; Mdm×n ), and P γ,ϕ satisfies (3.12) with Piγ,ϕ (T ) = 0, for any i ∈ D. Then, Λγi (t, P γ,ϕ (t)) > (γ 2 − kLk2 )I, (3.21) for any (t, i) ∈ [τ, T ] × D. Proof. We will first prove that Λγi (t, P γ,ϕ (t)) > 0, for any (t, i) ∈ [τ, T ] × D. Suppose that this is not true and  there exist tˆ ∈ [τ, T ], i0 ∈ D and v˜ ∈ Rm with k˜ v k = 1 such that v˜, Λγi0 (t, P γ,ϕ (t))˜ v 6 −η for some η > 0. Suppose tˆ < T . Since all the coefficient matrices are continuous functions, there exists δ > 0 sufficiently small such that

 1 v˜, Λγi0 (t, P γ,ϕ (t))˜ v 6 − η, t ∈ [tˆ, tˆ + δ] ⊂ [τ, T ]. 2 Define  0, t ∈ [τ, tˆ) ∪ (tˆ + δ, T ], vδ (t) = v˜χrt =i0 , t ∈ [tˆ, tˆ + δ]. Let xδ be the solution of the system (3.1) corresponding to this input vδ . Then, xδ (t) = 0 for t ∈ [0, tˆ] . Now applying Lemma 3.5 gives J(v + ϕxϕ ; 0, 0, i) Z T  

 γ,ϕ = E vδ (t), Nrt (t, P γ,ϕ (t))xϕ (t)), vδ (t)i δ (t) + hNrt (t, P 0



 γ γ,ϕ + vδ (t), Λrt (t, P (t))vδ (t) dt | r0 = i Z

tˆ+δ

=E tˆ

Z

tˆ+δ

=E tˆ

Z 6E

tˆ

 ϕ  2(xδ (t))0 Nrt (t, P γ,ϕ (t))˜ v − v˜0 Λγrt (t, P γ,ϕ (t))˜ v χrt =i0 dt | r0 = i



 d X ϕ 0 γ,ϕ 0 γ γ,ϕ [2(xδ (t)) Nj (t, P (t))˜ v − v˜ Λj (t, P (t))˜ v ]χrt =j χrt =i0 dt | r0 = i j=1

tˆ+δ

0 γ,ϕ 2(xϕ (t))˜ v χrt =i0 dt δ (t)) Ni0 (t, P



1 | r0 = i − ηE 2

Z tˆ

tˆ+δ

 χrt =i0 dt | r0 = i .

(3.22)

On the other hand, using estimate (2.1), we deduce that there exists c1 > 0 not depending upon δ such that sup 06t6T

2 E{|xϕ δ (t)|

Z | r0 = i} 6 c1 E 0

13

T

 |vδ (t)| dt | r0 = i 6 c1 δ. 2

Thus we conclude that there exists c2 > 0 not depending upon δ such that   Z tˆ+δ ϕ γ,ϕ 0 v χrt =i0 dt | r0 = i E (xδ (t)) Ni0 (t, P (t))˜ tˆ

   1   Z 2 ϕ 2 6 E sup |xδ (t)| | r0 = i E

tˆ+δ

|Ni0 (t, P

tˆ

06t6T

Since

 pii0 (t) =

γ,ϕ

 1 p 2 (t))˜ v | | r0 = i 6 c1 δc2 δ. 2

i 6= i0 , qii0 t, 1 + qi0 i0 t, i = i0 ,

where qi0 i0 < 0, and qii0 is bounded, then d Z X i=1

tˆ

d

tˆ+δ

pii0 (t)dt > δ +

1X 1 qii0 δ 2 > δ + qi0 i0 δ 2 . 2 2 i=1

Thus, for c > 0 and a sufficiently small δ > 0, we have d d Z ˆ X √ √ 1 X t+δ 1 1 ϕ J(v + ϕx ; 0, 0, i) 6 c δδ − η pii0 (t)dt 6 c δδ − η(δ + qi0 i0 δ 2 ) < 0. 2 2 2 tˆ i=1

i=1

Pd

However, according to Lemma 3.6, we know i=1 J(v + ϕxϕ ; 0, 0, i) > 0. This is a contradiction and thereby Λγi (t, P γ,ϕ (t)) > 0. If tˆ = T , a similar proof applies, only with the interval [tˆ, tˆ + δ] replaced by [T − δ, T ]. 1 Now we consider 0 <  < γ and γ˜ = (γ 2 − 2 ) 2 such that kLk < γ˜ < γ. Repeating the previous steps with γ˜ in place of γ, we conclude that, for the corresponding solution P γ˜,ϕ (t) of (3.12), Λγi˜ (t, P γ˜,ϕ (t)) > 0, ∀i ∈ D. By (3.15), we obtain, for any (τ, ξ, i) ∈ [0, T ) × Rn × D, hξ, Piγ,ϕ (τ )ξi = J(ϕxϕ ; τ, ξ, i) Z T   2  0  0 ϕ ϕ ϕ ϕ =E (γ I − Drt (t)Drt (t))ϕ(t)x (t), ϕ(t)x (t) − Crt (t)Crt (t)x (t), x (t) dt | rτ = i τ Z T   2  0  0 ϕ ϕ ϕ ϕ >E (˜ γ I − Drt (t)Drt (t))ϕ(t)x (t), ϕ(t)x (t) − Crt (t)Crt (t)x (t), x (t) dt | rτ = i τ D E = ξ, Piγ˜,ϕ (τ )ξ . It follows that for any i ∈ D, if γ > γ˜ , then Piγ,ϕi (τ ) > Piγ˜,ϕi (τ ). Therefore, Z γ γ,ϕ 2 0 0 0 γ,ϕ 0 Λi (t, P (t)) = γ I − (Di (t)) Di (t) + (Bi (t)) Pi (t)Bi (t) + Fi0 (t, θ)Piγ,ϕ (t)Fi (t, θ)ν(dθ) ZE > γ˜ 2 I − (Di (t))0 Di (t) + (Bi0 (t))0 Piγ˜,ϕ (t)Bi0 (t) + Fi0 (t, θ)Piγ˜,ϕ (t)Fi (t, θ)ν(dθ) E

=

Λγi˜ (t, P γ˜,ϕ (t))

> 0.

That is, Λγi (t, P γ,ϕ (t)) > 2 I, for any t ∈ [0, T ] and i ∈ D. Since this holds for arbitrary 2 < γ 2 − k L k2 , (3.21) follows. The proof is complete. Now, we are ready to prove Theorem 3.2. It remains to show the converse of Proposition 3.3. To this end, we will now study SRDEs(3.5). The function Z f (Prt (t)) = A0rt (t)Prt (t) + P (t)Art (t) + (A0rt (t))0 Prt (t)A0rt (t) + Er0 t (t, θ)Prt (t)Ert (t, θ)ν(dθ) E

+

d X

qrt ,j Pj (t) − Cr0 t (t)Crt (t) − Φ0rt (t, P (t))Λγrt (t, P (t))−1 Φrt (t, P (t))

j=1

14

is continuously differentiable on its domain defined by Df = {P : det(Λγrt (t, P (t))) 6= 0}. According to Proposition 3.4, there exists a unique solution to (3.5) backwards in time on a maximal interval (σ, T ]. The following theorem shows that (3.5) has a global solution on [0, T ]. Lemma 3.8. Suppose kLk < γ. Then SRDEs (3.5) has a unique solution P on [0, T ]. Moreover, the optimal feedback control is v(t) = Ψrt (t, P (t))xΨ (t−),

Ψrt (t, P (t)) = −Λγrt (t, P (t))−1 Φ0rt (t, P (t)),

t ∈ [0, T ],

where xΨ satisfies dx(t) = (Art (t) + Brt (t)Ψrt (t, P (t))x(t)dt + (A0rt (t) + Br0t (t)Ψrt (t, P (t))x(t)dW (t) Z µ(dθ, dt), x(τ ) = ξ, + (Ert (t, θ) + Frt (t, θ)Ψrt (t, P (t))x(t−)˜ E

and the optimal cost is min

v∈U m [τ,T ]

J(v; τ, ξ, i) = hξ, Pi (τ )ξi .

(3.23)

Proof. As kLk < γ, for some given sufficiently small 0 <  < γ, it holds true γ 2 I − Di0 (t)Di (t) > 2 I, for any t ∈ [0, T ] and i ∈ D. Furthermore, for every i ∈ D, there exists a maximal interval [δi , T ] such that Z γ 2 0 0 0 0 Λi (t, P (t)) = γ I − Di (t)Di (t) + (Bi (t)) Pi (t)Bi (t) + Fi0 (t, θ)Pi (t)Fi (t, θ)ν(dθ) > 2 I E

due to Pi (T ) = 0. Taking T0 = max{δ1 , · · · , δd }, then P is continuously differentiable on [T0 , T ]. We should note that T0 depends on the given . According to Proposition 3.4, the SRDEs (3.5) has a unique solution P on [T0 , T ]. Setting ϕ replaced by Ψ = −Λγ (t, P )−1 Φ0 (t, P ) ∈ C(T0 , T ; Mdn×m ) on the left hand side of (3.12), we obtain  0   I I P˙rt (t) + Mrt (t, P (t)) = P˙rt (t) + A0rt (t)Prt (t) + Prt (t)Art (t) Ψrt (t, P (t)) Ψrt (t, P (t)) Z 0 0 0 0 − Crt (t)Crt (t) + (Art (t)) Prt (t)Art (t) + Er0 t (t, θ)Prt (t)Ert (t, θ)ν(dθ) E

+

d X

qrt ,j Pj (t) − Φrt (t, P (t))0 Λγrt (t, P (t))−1 Φrt (t, P (t)) = 0,

t ∈ [T0 , T ].

j=1

The last equality holds because of the local solvability of SRDEs (3.5). Hence, P satisfies (3.12) on [T0 , T ] with Ψ = ϕ, i.e., P γ,Ψ (t) = P (t). Moreover, with this choice of Ψ, we have Nrt (t, P (t)) = Φ0rt (t, P (t)) + Λγrt (t, P (t))Ψrt (t, P (t)) = 0,

t ∈ [T0 , T ].

So, it follows from Lemma 3.5 that Z J(v + Ψx; τ, ξ, i) = hξ, Pi (τ )ξi + E τ

T



v(t), Λγrt (t, P (t))v(t)



 dt | rτ = i ,

τ ∈ [T0 , T ].

But by Lemma 3.7, Λγi (t, P (t)) = Λγi (t, P γ,Ψ (t)) > (γ 2 − k L k2 )I > 0,

∀t ∈ [T0 , T ], ∀i ∈ D.

Hence, the control v(t) = Ψrt (t, P (t))xΨ (t−) minimizes J(v; τ, ξ, i) and the optimal cost is given by (3.23), for any t ∈ [T0 , T ]. As a consequence, we obtain that for any τ ∈ [T0 , T ], Z T 

 0 hξ, Pi (τ )ξi = J(Ψx; τ, ξ, i) 6 J(0; τ, ξ, i) = E −Crt (t)Crt (t)x(t), x(t) dt | rτ = i 6 0. τ

15

On the other hand, by Lemma 3.6, hξ, Pi (τ )ξi = J(Ψx; τ, ξ, i) > −µ|ξ|2 ,

∀τ ∈ [T0 , T ].

Thus, −µI 6 Pi (τ ) 6 0,

∀τ ∈ [T0 , T ],

∀i ∈ D.

Now, suppose that there exists a solution of (3.5) backwards in time on a maximal interval (σ, T ] ⊂ [0, T ]. As t ↓ σ, Pi (t) becomes unbounded, i.e., (3.5) exhibits the phenomenon of a finite escape time. It suffices to show that the existence of a finite escape time will lead to a contraction. In fact, by the above discussion, the following estimates hold in the interval [σ + σ , T ] with σ > 0 sufficiently small, − µI 6 Pi (τ ) 6 0,

Λγi (t, P (t)) > (γ 2 − kLk2 )I,

∀i ∈ D.

(3.24)

As the constant µ is independent of the left interval endpoint σ + σ , letting σ ↓ 0 yields −µI 6 Pi (τ ) 6 0 on (σ, T ]. Thus, the solution P of (3.5) cannot escape to ∞ as t ↓ σ. It follows that there exists a boundary point P 0 , det(Λγ (P 0 )) = 0 of the domain Df which is a limit point of Pi (t) as t ↓ σ. But this contradicts the fact of (3.24), i.e., Λγi (t, P (t)) > (γ 2 − kLk2 )I as t ↓ σ. Therefore, the maximal solution interval is [0, T ]. The uniqueness of the solution follows from (3.23). This concludes the proof. So far, we have shown Theorem 3.2, i.e., the stochastic jump bounded real lemma for the Poisson jumpdiffusion system with Markovian switching parameters. Theoretically, by virtue of this theorem, the infimum of all these given disturbance attenuation levels γ > 0 such that the corresponding SRDEs (3.5) have a unique solution, can be used as an estimate of kLk.

4

Solution to stochastic H2 /H∞ control problem

In this section, the H2 /H∞ control problem is investigated for the Poisson jump-diffusion system with Markovian switching parameters (2.3). The finite horizon stochastic H2 /H∞ control problem can be stated as follows. Definition 4.1. Given a disturbance attenuation level γ > 0, the objective of the stochastic H2 /H∞ control problem is to find a state feedback control u∗ ∈ U s [0, T ], such that (1) kLu∗ k < γ with kLu∗ k :=

sup v∈U m [0,T ]

kzk kvk

v6=0,x(0)=0

RT 0 0 ∗ 0 ∗ 0 1 0 1 i=1 E{ 0 (x (t)Crt (t)Crt (t)x(t) + (u (t)) u (t) + v(t) (Drt (t)) Drt (t)v(t))dt
1 RT Pd 0 2 i=1 E{ 0 v(t) v(t)dt | r0 = i}

Pd =

sup v∈U m [0,T ] v6=0,x(0)=0


1 | r0 = i} 2

where kLu∗ k is a perturbation operator Lu∗ : U m [0, T ] → U q [0, T ] associated with system (2.3) defined as Lu∗ (v(t)) = z(t) |x(0)=0 ,

t ∈ [0, T ];

(2) when the worst case disturbance v ∗ ∈ U m [0, T ] is applied to system (2.3), u∗ minimizes the output energy ∗

J2 (u, v , i) =

d X

Z E

(x(t) 0

i=1

T

0

Cr0 t (t)Crt (t)x(t)

0

∗

0

+ u(t) u(t) + (v (t))

where v ∗ is defined as v ∗ := argminJ1 (u∗ , v) v

16

(Dr1t (t))0 Dr1t (t)v ∗ (t))dt

 | r0 = i ,

=

Z d X argmin E v

T

 (γ v(t) v(t) − z(t) z(t))dt | r0 = i . 0

2

0

0

i=1

If the aforementioned (u∗ , v ∗ ) exists, we say that the finite horizon H2 /H∞ control is solvable. To give our main results, we need the following lemma, which can be obtained by a direct application of Theorem 4.18 in [10]. Lemma 4.1. The system of differential Riccati type equations  d X   0 0 0 0 0  ˙ Xi (t) + Ai (t)Xi (t) + Xi (t)Ai (t) + (Ai (t)) Xi (t)Ai (t) + Ci (t)Ci (t) + qij Xj (t)     j=1 Z   + Ei0 (t, θ)Xi (t)Ei (t, θ)ν(dθ) − ∆i (X(t))0 Θi (X(t))−1 ∆i (X(t)) = 0, Xi (T ) = 0,    E   Θi (X(t)) > 0, t ∈ [0, T ], i ∈ D,

(4.1)

where ∆i (X(t)) := Xi (t)Bi (t) +

(A0i (t))0 Xi (t)Bi02 (t)

Z + E

Ei0 (t, θ)Xi (t)Fi2 (t, θ)ν(dθ),

and Θi (X(t)) := I + (Bi02 (t))0 Xi (t)Bi02 (t) +

Z E

(Fi2 (t, θ)0 Xi (t)Fi2 (t, θ)ν(dθ),

admits a unique bounded and semi-positive solution X = (X1 , X2 , · · · , Xd ) ∈ Snd+ . Remark 4.1. Note the relationship between the Riccati equations (4.1) and the linear quadratic (LQ) optimal control problem with the dynamic system: dx(t) = [Art (t)x(t) + Br2t (t)u(t)]dt + [A0rt (t)x(t) + Br02t (t)u(t)]dW (t) Z + [Ert (t, θ)x(t−) + Fr2t (t, θ)u(t)]˜ µ(dθ, dt), x(τ ) = ξ, E

and the cost functional: Z J2 (u; τ, ξ, i) = E

T 2

2

(|Cx| + |u| )dt | r0 = i



Z

T

=E

[x(t) τ

τ

0

Cr0 t (t)Crt (t)x(t)

 + u(t) u(t)]dt | r0 = i . 0

Then, we immediately obtain that the optimal feedback control is u(t) = Σrt (X(t))x(t−),

t ∈ [τ, T ],

where x(t) satisfies dx(t) = [Art (t) + Br2t (t)Σ(Xrt (t))]x(t)dt + [A0rt (t) + Br02t (t)Σ(Xrt (t))]x(t)dW (t) Z + [Ert (t, θ) + Fr2t (t, θ)Σ(Xrt (t))]x(t−)˜ µ(dθ, dt), x(τ ) = ξ, E

and Σrt (X(t)) = −Θrt (X(t))−1 ∆rt (X(t)), and the optimal cost is

min J2 (u; τ, ξ, i) = ξ 0 Xi (τ )ξ, where X is the solution to equations(4.1).

u∈U s [τ,T ]

17

In the following, we shall give sufficient and necessary conditions for the existence of the linear state feedback pair (u∗ , v ∗ ), which generalizes the result of Chen and Zhang [2] to the case of Poisson jump-diffusion systems with Markovian switching. Here and in what follows, we use the following notations for convenience: Z 2 1 0 1 01 0 01 Υ1,i (P1 ) := γ I − (Di ) Di + (Bi ) P1,i Bi + (Fi1 (θ))0 P1,i Fi1 (θ)ν(dθ), E Z Υ2,i (P2 ) := I + (Bi02 )0 P2,i Bi02 + (Fi2 (θ))0 P2,i Fi2 (θ)ν(dθ), E Z 1 0 02 0 01 Π1,i (P1 , K2 ) := P1,i Bi + (Ai + Bi K2,i ) P1,i Bi + (Ei (θ) + Fi2 (θ)K2,i )0 P1,i Fi1 (θ)ν(dθ), ZE 2 0 01 0 02 Π2,i (P2 , K1 ) := P2,i Bi + (Ai + Bi K1,i ) P2,i Bi + (Ei (θ) + Fi1 (θ)K1,i )0 P2,i Fi2 (θ)ν(dθ). E

Theorem 4.2. For the stochastic system (2.3), if for any i ∈ D, the following two sets of coupled systems of differential Riccati type equations (with the time argument t suppressed)  P˙1,i + (Ai + Bi2 K2,i )0 P1,i + P1,i (Ai + Bi2 K2,i ) + (A0i + Bi02 K2,i )0 P1,i (A0i + Bi02 K2,i ) − Ci0 Ci     Z  d  X  0  − K2,i K2,i + (Ei (θ) + Fi2 (θ)K2,i )0 P1,i (Ei (θ) + Fi2 (θ)K2,i )ν(dθ) + qi,j P1,j (4.2) E j=1     − Π01,i (P1 , K2 )Υ1,i (P1 )−1 Π1,i (P1 , K2 ) = 0, P1,i (T ) = 0,     Υ1,i (P1 ) > 0, K1,i = −Υ1,i (P1 )−1 Π1,i (P1 , K2 ),

(4.3)

and  P˙2,i + (Ai + Bi1 K1,i )0 P2,i + P2,i (Ai + Bi1 K1,i ) + (A0i + Bi01 K1,i )0 P2,i (A0i + Bi01 K1,i ) + Ci0 Ci     Z  d  X  0 1 0 1 1 0 1  + K1,i (Di ) Di K1,i + (Ei (θ) + Fi (θ)K1,i ) P2,i (Ei (θ) + Fi (θ)K1,i )ν(dθ) + qi,j P2,j E j=1     − Π02,i (P2 , K1 )Υ2,i (P2 )−1 Π2,i (P2 , K1 ) = 0, P2,i (T ) = 0,     Υ2,i (P2 ) > 0, K2,i = −Υ2,i (P2 )−1 Π2,i (P2 , K1 ).

(4.4)

(4.5)

admit a set of solutions (P1 (t), P2 (t), K1 (t), K2 (t)) ∈ Snd ×Snd ×Mdn×m ×Mdn×s such that P1,i (t) 6 0 and P2,i (t) > 0 on [0, T ] × D, where P1 = (P1,1 , P1,2 , · · · , P1,d ), P2 = (P2,1 , P2,2 , · · · , P2,d ), K1 = (K1,1 , K1,2 , · · · , K1,d ), and K2 = (K2,1 , K2,2 , · · · , K2,d ). Then the H2 /H∞ control problem admits a pair of solutions (u∗ , v ∗ ) satisfying u∗ (t) = K2,rt (t)x(t−),

v ∗ (t) = K1,rt (t)x(t−),

t ∈ (0, T ].

(4.6)

Moreover, setting x0 = 0 and u = u∗ yields kLu∗ k < γ, for any v ∈ U m [0, T ]. Proof. Suppose that the two sets of coupled systems of differential Riccati type equations (4.2)-(4.5) are solvable.

18

Applying Itˆo’s formula (2.2) and the completion of squares, we have for any i ∈ D, J1 (u, v, i)  Z T   2 0 0 0 0 0 0 0 =E (γ v(t) v(t) − z(t) z(t)dt + dx(t) P1,rt (t)x(t) + (x ) P1,rt (0)x − x(T ) P1,rT (T )x(T ) | r0 = i 0 Z T  0 0 0 = (x ) P1,i (0)x + E hΥ1,rt (P1 )(v − v ∗ ), v − v ∗ i − u0 u + (u∗ )0 u∗ + x0 P1 Br2t (u − K2,rt x) (A0rt

0 0 02 Brt K2,rt ) P1,rt (A0rt

u∗

v∗

+ Br02t K2,rt )x + (u − K2,rt x)0 Br2t P1,rt x + (A0rt x + Br02t u)0 P1,rt (A0rt x + Br02t u) −x + Z + (Ert (θ)x + Fr2t (θ)u)0 P1,rt (Ert (θ)x + Fr2t (θ)u)ν(dθ) − x0 (A0rt + Br02t K2,rt )0 P1,rt (A0rt + Br02t K2,rt )x EZ Z 02 0 01 0 2 0 2 0 Fr1t (θ)P1 Fr2t (θ)ν(dθ)] (Ert (θ) + Frt (θ)K2,rt ) P1,rt (Ert (θ) + Frt (θ)K2,rt )ν(dθ)x + v [(Brt ) P1,rt Brt + −x E E   Z 1 2 0 01 0 02 Frt (θ)P1,rt Frt (θ)ν(dθ)]v dt | r0 = i , × (u − K2,rt x) + (u − K2,rt x) [(Brt ) P1,rt Brt + 0

E

where

and

are given by (4.6). Letting u(t) = u∗ (t) = K2,rt (t)x(t−), we obtain Z T  J1 (u∗ , v, i) = (x0 )0 P1,i (0)x0 + E hΥ1,rt (P1 )(v − v ∗ ), v − v ∗ i dt | r0 = i . 0

So, for any i ∈ D, J1 is minimized by i.e., v ∗ is the worst case disturbance corresponding to u∗ and the optimal value is (x0 )0 P1,i (0)x0 . A similar argument as in Proposition 3.3 yields kLu∗ k < γ. Similarly, (u∗ , v, i)

v∗,

J2 (u, v, i) = (x0 )0 P2,i (0)x0 + E

Z

T

0



0 hΥ2,rt (P2 )(u − u∗ ), u − u∗ i − x0 K1,r (Dr1t )0 Dr1t K1,rt x + v 0 (Dr1t )0 Dr1t v t

+ x0 P2 Br1t (v − K1,rt x) + (v − K1,rt x)0 (Br1t )0 P2,rt x + (A0rt x + Br01t v)0 P2,rt (A0rt x + Br01t v) Z + (Ert (θ)x + Fr1t (θ)v)0 P2,rt (Ert (θ)x + Fr1t (θ)v)ν(dθ) − x0 (A0rt + Br01t K1,rt )P2,rt (A0rt + Br01t K1,rt )x EZ Z − x0 (Ert (θ) + Fr1t (θ)K1,rt )0 P2,rt (Ert (θ) + Fr1t (θ)K1,rt )ν(dθ)x + u0 [(Br01t )0 P2,rt Br02t + Fr1t (θ)P2,rt Fr2t (θ)ν(dθ)] E  E Z × (v − K1,rt x) + (v − K1,rt x)0 [(Br01t )0 P2,rt Br02t + Fr1t (θ)P2,rt Fr2t (θ)ν(dθ)]u dt | r0 = i , E

(4.7) Setting v = v ∗ leads to J2 (u∗ , v ∗ , i) 6 J2 (u, v ∗ , i) and J2 (u∗ , v ∗ , i) = (x0 )0 P2,i (0)x0 , for any i ∈ D. The above derivation shows that the finite horizon H2 /H∞ control has a pair of solutions (u∗ , v ∗ ) with u∗ and v ∗ defined in (4.6). Next, we deduce the signs of P1,i (t) and P2,i (t). By completing squares similar to that which led to (4.7), together with setting u = u∗ and v = v ∗ , we finally obtain that for every i ∈ D and for arbitrary x(t)  Z T 0 0 0 0 ∗ 0 ∗ ∗ 0 1 0 1 ∗ x (t)P2,i (t)x(t) = E [x(s) Crs (s) Crs (s)x(s) + (u (s)) u (s) + (v (s)) (Drs ) Drs v (s)]ds | rt = i > 0. t

Consequently, P2,i (t) > 0, for any t ∈ [0, T ] and i ∈ D. A similar calculation using J1 (·, ·, i) with u = u∗ and v = 0 gives Z T  x0 (t)P1,i (t)x(t) = −E [x(s)0 Crs (s)0 Crs (s)x(s) + (u∗ (s))0 u∗ (s) + hΥ1,rs (P1 )v ∗ (s), v ∗ (s)i]ds | rt = i 6 0. t

Thus, P1,i (t) 6 0, for any t ∈ [0, T ] and i ∈ D. The proof is complete. 19

Theorem 4.3. Assume that the finite horizon H2 /H∞ control problem admits a pair of linear state feedback solutions (u∗ (t), v ∗ (t)) defined by v ∗ (t) = G1rt x(t−) and u∗ (t) = G2rt x(t−), where G1 = (G11 , G12 , · · · , G1d ) and G2 = (G21 , G22 , · · · , G2d ) are continuous matrix-valued functions on [0, T ]. Then the coupled matrix-valued Riccati equations (4.2)-(4.5) have a unique solution (P1 , P2 , G1 , G2 ) such that P1,i 6 0 and P2,i > 0 for any i ∈ D. Proof. If the H2 /H∞ control problem has the solution (u∗ , v ∗ ), then we will prove that the two sets of systems of differential Riccati type equations (4.2)-(4.5) are solvable. (I) Implementing u∗ (t) = G2rt (t)x(t−) in system (2.3) with G2 to be determined, we obtain  dx(t) = [(Art (t) + Br2t (t)G2 (t))x(t) + Br1t (t)v(t)]dt + [(A0rt (t) + Br02t (t)G2rt (t))x(t) + Br01t (t)v(t)]dW (t)    Z  µ(dθ, dt), x(τ ) = ξ + [(Ert (t, θ) + Fr2t (t, θ)G2rt (t))x(t−) + Fr1t (t, θ)v(t)]˜  E    z(t) = (Crt (t) + Dr2t (t)G2rt (t))x(t) + Dr1t (t)v(t). Since the finite horizon H2 /H∞ control is solvable, by Definition 4.1, we can obtain kLu∗ k < γ. Then by applying Lemma 3.8, we conclude that Riccati type equations  ˙ t + (Art + Br2 G2r )0 P1,rt + P1,rt (Art + Br2 G2r ) + (A0r + Br02 G2r )0 P1,rt (A0r + Br02 G2r )  P1,r  t t t t t t t t t t  Z     − Cr0 t Crt − (G2rt )0 G2rt + (Ert + Fr2t G2rt )0 P1,rt (Ert + Fr2t G2rt )ν(dθ)    E (4.8) d X  2 0 −1 2   − Π (P , G ) Υ (P ) Π (P , G ) + q P = 0, P (T ) = 0 1,rt 1 1,rt 1 1,rt 1 rt ,j 1,j 1,rT     j=1    Υ (P ) > 0, 1,rt 1 have a unique solution P1 = (P1,1 , · · · , P1,d ) with P1,i 6 0, for any i ∈ D. Moreover, the optimal control problem min

v∈U m [0,T ]

J1 (u∗ , v)

has a unique solution v ∗ (t) = −Υ1,rt (P1 )−1 Π1,rt (P1 , G2 )x(t−),

t ∈ (0, T ].

Therefore, G1 = (G11 , G12 , · · · , G1d ) is determined by G1i = −Υ1,i (P1 )−1 Π1,i (P1 , G2 ),

i ∈ D.

(4.9)

(II) Substituting v(t) = v ∗ (t) = G1rt (t)x(t−) into system (2.3), we derive  dx(t) = [(Art (t) + Br1t (t)G1rt (t))x(t) + Br2t (t)u(t)]dt + [(A0rt (t) + Br01t (t)G1rt (t))x(t) + Br02t (t)u(t)]dW (t)    Z  + [(Ert (t, θ) + Fr1t (t, θ)G1rt (t))x(t−) + Fr2t (t, θ)u(t)]˜ µ(dθ, dt), x(τ ) = ξ, (4.10)  E    z(t) = (Crt (t) + Dr1t (t)G1rt (t))x(t) + Dr2t (t)u(t), As min J2 (u, v ∗ , i)

u∈U s [0,T ]

is a standard stochastic linear quadratic optimal control problem, it follows from Remark 4.1 that there exists a unique optimal control u∗ (t) = −Υ2,rt (P2 )−1 Π2,rt (P2 , G1 )x(t−),

20

t ∈ (0, T ],

where P2 = (P2,1 , · · · , P2,d ) with P2,i > 0, for any i ∈ D solves the following equation   P˙2,rt + (Art + Br1t G1rt )0 P2,rt + P2,rt (Art + Br1t G1rt ) + (A0rt + Br01t G1rt )0 P2,rt (A0rt + Br01t G1rt )   Z    1 0 1 0 1 1 0  + (Grt ) (Drt ) Drt Grt + Crt Crt + (Ert + Fr1t G1rt )0 P2,rt (Ert + Fr1t G1rt )ν(dθ)    E d X  1 −1 1 0   (P , G ) + (P ) Π qrt ,j P2,j = 0, P2,rT (T ) = 0, (P , G ) Υ − Π 2 2 2,rt 2 2,rt 2,rt     j=1    Υ (P ) > 0, 2 2,rt

(4.11)

Hence, G2rt = −Υ2,rt (P2 )−1 Π2,rt (P2 , G1 ).

(4.12)

Therefore, the coupled Riccati type equations (4.2)-(4.5) have a unique solution (P1 , P2 , G1 , G2 ) with P1,i 6 0 and P2,i > 0 for all i ∈ D. The proof is complete. Remark 4.2. (i) When D = 1 and q11 = 0, the system (2.3) reduces to a Poisson random jump-diffusion system and the corresponding Riccati equation can be further simplified. The H2 /H∞ control problem for such a system has been discussed in Wang [36]. (ii) When Ert (t, θ) = Fr1t (t, θ) = Fr2t (t, θ) = 0, the system (2.3) reduces to the Ito-diffusion system with Markovian switching parameters. Accordingly, we can obtain the Markovian switching bounded real lemma and the corresponding H2 /H∞ control. Gao [14] has considered H2 /H∞ control for such a system, which can be seen as a special case of our results.

5

Concluding remarks

This paper discussed the H2 /H∞ control problem for Poisson jump-diffusion systems with Markovian switching. Necessary and sufficient conditions for the existence of a state feedback H2 /H∞ control were respectively given in terms of the solutions to two sets of coupled systems of differential Riccati type equations. It is noteworthy that the stochastic jump bounded real lemma plays a central role in the analysis of the H∞ control problem and stands out on its own as a very interesting theoretical result. There also remain many interesting topics deserving further study. For example, how to solve the two sets of systems of differential Riccati type equations (4.2)-(4.5) is a challenging problem and worths further study.

References [1] Arapostathis, A., Ghosh, M. K. and Marcus, S. I., Optimal control of switching diffusions with application to flexible manufacturing systems, SIAM J. Control Optim., 31, 1993, 1183-1204. [2] Chen, B. and Zhang, W., Stochastic H2 /H∞ control with state-dependent noise, IEEE Trans. Automat. Control, 49(1), 2004, 45-57. [3] Costa, O. L. V. and Marques, R. P., Mixed H2 /H∞ control of discrete-time Markovian jump linear systems, IEEE Trans. Automat. Control, 43, 1998, 95-100. [4] Faiz, R. and Sing, K. N., Robust H∞ State Feedback Control of NCSs with Poisson Noise and Successive Packet Dropouts. INT J Control Autom, 13(1), 2015, 45-57. [5] Chen, W. and Chen, B., Robust stabilization design for nonlinear stochastic system with Poisson noise via fuzzy interpolation method, Fuzzy Sets Syst., 217(217), 2013, 41-61. [6] Doyle, J. C., Glover, K., Khargonekar P. P. and Francis B., State-space solutions to standard H2 and H∞ problems, IEEE Trans. Automat. Control., 34, 1989, 831-847. [7] Dragan, V., Morozan, T. and Stoica, A., Mathematical methods in robust control of discrete-time linear stochatic systems, New York: Springer-Verlag, 2010.

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[8] Dragan, V. and Morozan, T., Stability and robust stabilization to linear stochastic systems described by differential equations with markovian jumping and multiplicative white noise, Stoch. Anal. Appl., 20(1), 2002, 33-92. [9] Dragan, V. and Morozan, T., The linear quadratic optimization problems for a class of linear stochastic systems with multiplicative white noise and Markovian jumping, IEEE Trans. Automat. Control, 49(5), 2004, 665-675. [10] Dragan, V. and Morozan, T., Systems of matrix rational differential equations arising in connnection with linear stochastic systems with Markovian jumping, J. Differ. Equations, 194, 2003, 1-38. [11] Fragoso, M.D. and Rocha, N.C.S., Stationary filter for continuous-time Markovian jump linear systems, SIAM J. Control Optim., 44(3), 2006, 801-815. [12] Fragoso, M.D. and Baczynski, J., on a infinite dimensional perturbed Riccati differential equation arising in stochastic control, Linear Algebra Appl., 406, 2005, 165-176. [13] Framstad, N., Oksendal, B. and Sulem, A., Optimal consumption and portfolio in a jump diffusion market with proportional transaction costs, J. Math. Econ., 35(2), 2001, 233-257. [14] Gao, M. , Sheng, L. and Zhang, W., Finite horizon H2 /H∞ control of time-varying stochastic systems with Markov jumps and (x, u, v)-dependent noise, IET Control Theory A., 8(14), 2014, 1354-1363. [15] Gershon, E., Shaked, U. and Yaesh, I. H∞ control and estimation of state-multiplicative linear systems, London: Springer-Verlag, 2005. [16] Gershon, E. and Shaked, U. H∞ output-feedback control of discrete-time systems with state-multiplicative noise, Automatica 44(2), 2008, 574-579. [17] Ghosh, K. and Bagchi, A., Modeling stochastic hybrid systems, IFIP International Federation for Information Processing, 66, 2005, 269-280. [18] Hinrichsen, D. and Pritchard, A. J., Stochastic H∞ , SIAM J. Control Optim., 36(5), 1998, 1504-1538. [19] Hou, T., Zhang, W. and Ma, H, Infinite horizon H2 /H∞ optimal control for discrete-time Markov jump systems with (x, u, v)-dependent noise, J. Glob. Optim., 57, 2013, 1245-1262. [20] Hu, S. and Zhu, A., Risk-minimizing pricing and hedging foreign currency options under regime-switching jumpdiffusion models, Commun Stat-Theor M, 46(4), 2017, 1821-1842. [21] Ikeda, N. and Watanabe, S., Stochastic Differential Equations and Diffusion Processes, North-Holland Publishing Company, Amsterdam, 1981. [22] Khargonekar, P. P., and Rotea, M. A., Mixed H2 /H∞ control: A convex optimization approach, IEEE Trans. Automat. Control, 36(7), 1991, 824-837. [23] Li, D., Rong, X. and Zhao, H., Optimal reinsurance and investment problem for an insurer and a reinsurer with jump-diffusion risk process under the Heston model, Comp. Appl. Math., 35, 2016, 533-557. [24] Limebeer, D. J. N., Anderson, B. D. O. and Hendel, B., A Nash game approach to mixed H2 /H∞ control, IEEE Trans. Automat. Control, 39(1), 1994, 69-82. [25] Lin, Z., Lin, Y. and Zhang, W., A unified design for state and output feedback H∞ control of nonlinear stochastic Markov jump systems with state and disturbance-dependent noise, Automatica, 45(12), 2009, 2955-2962. [26] Lin, X. and Zhang, R., H∞ control for stochastic systems with Poisson jumps, J. Syst. Sci. Complex, 24(4), 2011, 683-700. [27] Ma, L., Zhang, T. and Zhang, W., H∞ control for continuous time mean-field stochastic systems, Asian J. Control, 18(6), 2016, 1-11. [28] Ma, L., Zhang, T., Zhang W. and Chen B., Finite horizon mead-field stochastic H2 /H∞ control for continuous-time systems with (x, v)-dependent noise, J. Franklin I., 352, 2015, 5393-5414. [29] Shi, P. and Boukas, E.K., H∞ control for Markovian jumping linear systems with parametric uncertainty, J. Optimiz. Theory Appl, 95(1), 1997, 75-99. [30] Situ, R., Option pricing in mathematical financial market with jumps and related problems, Vietnam J. Math., 30(2), 2002, 103-122. [31] Siqueria, A.A.G. and Terra, M.H., A fault-tolerant manipulator robot base on H2 , H∞ and H2 /H∞ Markov controls, IEEE Trans. Automat. Control, 14(2), 2009, 257-263. [32] Sweriduk, G. D. and Calise, A. J., Differential game approach to the mixed H2 /H∞ problem, J. Guid. Control Dyna., 20, 1997, 1229-1234. [33] Todorov, M.G. and Fragoso, M.D., Infinite Markov jump-bounded real lemma, Systems Control Lett., 57, 2008, 64-70. [34] Todorov, M.G. and Fragoso, M.D., A new perspective on the robustness of Markov jump linear systems, Automatica, 49, 2013, 735-747.

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[35] Wang, M., Stochastic H2 /H∞ control with random coefficients, Chinese Ann. Math. Ser. B, 34, 2013, 733-752. [36] Wang, M., Stochastic H2 /H∞ control for Poisson jump-diffusion systems, Chinese Ann. Math. Ser. B, 37(5), 2016, 643-664. [37] Wu, S. and Zeng, Y., The term structure of interest rates under regime shifts and jumps, Econom. Lett., 93, 2006, 215-221. [38] Yan, H. and Liu, S., Pricing options on stocks driven by Poisson jump-diffusion process, J. Eng Math., 20, 2003, 35-40. [39] Yin, G and Xi, F., Stability of regime-switching jump-diffusions, SIAM J. Control Optim., 48(7), 2010, 4525-4549. [40] Zhang, W. and Chen, B., State feedback H∞ control for a class of nonlinear stochastic systems, SIAM J. Control Optim., 44(6), 2006, 1973-1991. [41] Zhang, W., Zhang, H. and Chen B., Stochastic H2 /H∞ control with (x, u, v)-dependent noise: Finite horizon case, Automatica, 42, 2006, 1891-1898. [42] Zhang, C. and Liang, Z., Portfolio optimization for jump-diffusion risky assets with common shock dependence and state dependent risk aversion, Optim. Control Appl. Meth., 38, 2017, 229-246. [43] Zhang, W., Huang, Y. & Xie, L., Infinite horizon stochastic H2 /H∞ control for discrete-time systems with state and disturbance dependent noise, Automatica, 44, 2008, 2306-2316. [44] Zhang, C., Zhu, H., Zhou, H. and Ning, B., Game Theory Approach to Stochastic H2 /H∞ Control of Markov Jump Singular Systems, Springer International Publishing, 2017. [45] Zhang, Q., H2 /H∞ control for stochastic systems with delay, Int J. Control, 88(12), 2015, 2554-2562.

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