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Controllability of Semilinear Stochastic System with Multiple Delays in Control
Third International Conference on Advances in Control and Optimization of Dynamical Systems March 13-15, 2014. Kanpur, India
Controllability of Semilinear Stochastic System with Multiple Delays in Control Anurag Shukla, N Sukavanam, D.N. Pandey Department of Mathematics, Indian Institute of Technology Roorkee Roorkee-247667, India
Abstract Finite dimensional stationary dynamic system described by semilinear stochastic ordinary differential equations with multiple delays in control is considered, under the basic and readily verified conditions to guarantee the existence of the solutions to a system. In this paper, we prove the complete controllability of the stochastic differential system with multiple delays in control. Keywords: Controllability, Semilinear Systems, Stochastic Control System, Delayed Control, Multiple Delays. 1. Introduction Controllability is one of the fundamental concepts in mathematical control theory, and plays an important role in both deterministic and stochastic control theories. Conceived by Kalman, controllability study was started systematically at the beginning of the sixties. Since then various researches have been carried out extensively in the context of finite dimensional linear systems, nonlinear systems and infinite dimensional systems using different kind of approaches([1-5]). Stochastic differential equations (SDEs) are used to model diverse phenomenon such as fluctuating stock prices or physical system subjected to thermal fluctuations. In the literature, there are different definitions of controllability for SDEs both for linear and nonlinear dynamical systems . For linear systems, N.I.Mahmudov [8] got results for controllability of linear stochastic systems. Klamka [2-3] generalized the result with single and multiple delays in control in linear stochastic system. In [9] N.I.Mahmudov obtained results for controllability on semilinear stochastic system. Lijuan Shen, Jitao Sun [7] generalized these results and showed the complete controllability of semilinear stochastic system with single point delay in control. However, to best of our knowledge, there are no result on the complete controllability of semilinear SDEs with multiple delays in control as treated in the current paper. Definition 1.1. In this paper we adopt the following notation: (i) (Ω, z, P):Let z be the σ algebra generated by Ω and P : z → [0, 1] be the probability measure on z. Then the triple (Ω, z, P) is called a probability space. (ii) {zt |t ∈ [0, T ]} is the filtration generated by {ω(s) : 0 ≤ s ≤ t}. (iii) L2 (Ω, zT , Rn )=the Hilbert space of all zT -measurable square integrable variables with values in Rn . n n (iv) Lz P ([0, T ], R ) is the Banach space of all p-integrable and zt -measurable processes with values in R , p ≥ 2.
for
(v) H2 =the Banach space of all square integrable and zt -adapted processes ϕ(t) with norm ||ϕ||2 = sup E||ϕ(t)||2 t∈[0,T ]
(vi) L(X, Y) is the space of all linear bounded operators from a Banach space X in to a Banach space Y. (vii) φ(t) = exp(At) and Uad = L2z ([0, T ], Rm ). Preprint submitted to Elsevier 978-3-902823-60-1 © 2014 IFAC
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February 25, 2014 10.3182/20140313-3-IN-3024.00107
2014 ACODS March 13-15, 2014. Kanpur, India
The problem of controllability of linear stochastic system with multiple delays in control dx(t) = [Ax(t) +
i=M X
Bi u(t − hi )]dt + σ(t, x(t))dω(t)
(1.1)
i=0
x(0) = x0 ∈ L2 (Ω, FT , Rn ) and u(t) = 0 for t ∈ [−h M , 0] has been studied by various authors (see J.Klamka (2009) for references). The problem of controllability of semi-linear stochastic system dx(t) = [Ax(t) + Bu(t) + f (t, x(t))]dt + σ(t, x(t))dω(t) x(0) = x0 ∈ Rn has been studied by various authors (see N.I.Mahmudov and S.Zorlu 2003 and N.Sukavanam and Mohit Kumar 2011 for references). In this paper we examine the controllability of the following semi-linear stochastic system with discrete multiple delays in control term of following problem: dx(t) = [Ax(t) +
i=M X
Bi u(t − hi ) + f (t, x(t))]dt + σ(t, x(t))dω(t) for t ∈ [0, T ]
(1.2)
i=0
(1.3) with initial conditions: x(0) = x0 ∈ L2 (Ω, FT , Rn ) and u(t) = 0 for t ∈ [−h M , 0]
(1.4)
where the state x(t) ∈ L2 (Ω, zt , R ) = X and the control u(t) ∈ R = U, A is an n × n constant matrix,Bi , i = 0, 1, 2, 3, ..., M ,are n × m constant matrices respectively. σ : [0, T ] × Rn → Rn×n , f : [0, T ] × Rn → Rn and( f, σ) are continuous nonlinear functions on [0, T ] × Rn , ω is a n-dimensional Wiener process and 0 = h0 < h1 < h2 < ..... < h M are constant point delays. n
m
2. Preliminaries It is well known([1], [2], [7], [8]) that for given initial conditions, any admissible control u ∈ Uad and suitable nonlinear function f (t, x(t) for t ∈ [0, T ] there exists unique solution x(t; x0 , u) ∈ L2 (Ω, zt , Rn ) of the linear stochastic differential state equation (1.2) which can be represented in the following integral form Z t Z t i=M X x(t; x0 , u) = exp(At)x0 + exp(A(t − s))( Bi u(s − hi ) + f (s, x(s)))ds + exp(A(t − s))σ(s)dω(s) 0
0
i=0
Thus,without loss of generality, taking into account the zero initial control for t ∈ [−h M , 0) and changing the order of integration, the solution x(t; x0 , u) for hk < t ≤ hk+1 , k = 0, 1, 2, ..., M − 1, t ∈ [0, h M ] has the following form, which is more convenient for further deliberations j=i i=k−1 X Z t−hi X x(t; x0 , u) = exp(At)x0 + ( exp(A(t − s − h j ))B j )u(s)ds i=0
+
Z
j=k t−hk X
(
0
+
Z
t−hi+1 j=0
exp(A(t − s − h j ))B j )u(s)ds
j=0 t
exp(A(t − s)) f (s, x(s))ds +
0
Z
t
exp(A(t − s))σ(s)dω(s)
(2.1)
0
Similarly for t > h M x(t; x0 , u) =
exp(At)x0 +
i=M−1 X Z t−hi i=0
+
Z
j=M t−h M X
(
0
+
Z
(
j=i X
exp(A(t − s − h j ))B j )u(s)ds
t−hi+1 j=0
exp(A(t − s − h j ))B j )u(s)ds
j=0 t
exp(A(t − s)) f (s, x(s))ds +
0
Z
exp(A(t − s))σ(s)dω(s) 0
2
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t
(2.2)
2014 ACODS March 13-15, 2014. Kanpur, India
Now, for a given final time T, using the form of the integral solution x(t : x0 , u), let us introduce operators and sets which will be used in next sections of the paper. First of all, for hk < T < hk+1 and for k = 0, 1, 2, ..., M −1, we define the following linear and bounded control operator LT ∈ L2 ([0, T ], Rm ) → L2 (Ω, zT , Rn ):
LT u
i=k−1 X Z T −hi
=
T −hi+1 j=0
i=0
j=k T −hk X
Z
+
j=i X ( exp(A(t − s − h j ))B j )u(s)ds
(
0
exp(A(T − s − h j ))B j )u(s)ds
j=0
Moreover, for T > h M we have i=M−1 X Z T −hi
=
LT u
i=0
+
Z
j=i X ( exp(A(t − s − h j ))B j )u(s)ds
T −hi+1 j=0 j=M T −h M X
(
0
exp(A(T − s − h j ))B j )u(s)ds
j=0
and its adjoint LT∗ : L2 (Ω, zT , Rn ) → LT ∈ L2 ([0, T ], Rm ) is a linear and bounded operator given by ∗ B0 exp(A∗ (T − t))E{z|zT } f or t ∈ [0, T − h M ] j=i X LT∗ z = B∗ exp(A∗ (T − t − h j )))E{z|zT } f or t ∈ (T − hi+1 , T − hi ], ( j=1 j
i = 0, 1, 2, ...M − 1
Let RT (Uad ) denotes the set of all states reachable from the initial state x(0) = x0 ∈ L2 (Ω, zT , Rn ) in time T > 0 using admissible controls Hence RT (Uad ) =
{x(T ; x0 , u) ∈ L2 (Ω, zT , Rn ) : u ∈ Uad } Z T exp(At)x0 + ImLT + exp(A(T − s))σ(s)dω(s)
=
0
Now, let us define the linear controllability operator ΠT0 ∈ L(L2 (Ω, zT , Rn ), L2 (Ω, zT , Rn ), which is strongly associated with the control operator LT as ΠT0 {.} =
LT LT∗ {.} i=k−1 XZ = i=0
+
T −hi+1 j=0
j=k T −hk X
Z
(
0
if hi+1 < T < hi , ΠT0 {.}
j=i j=i X X ( exp(A(t − s − h j ))B j )( B∗j exp(A∗ (T − t − h j )))E{.|zT }dt
T −hi
j=0 j=k X
exp(A(T − s − h j ))B j )(
j=0
B∗j exp(A∗ (T − s − h j )))E{.|zT }dt
j=0
i = 0, 1, 2, ..., M − 1 =
LT LT∗ {.} =
i=M−1 X Z T −hi i=0
+
Z
T −hi+1 j=0
j=M T −h M X
(
0
j=i j=i X X ( exp(A(t − s − h j ))B j )( B∗j exp(A∗ (T − t − h j )))E{.|zT }dt j=0
exp(A(T − s − h j ))B j )(
j=0
j=M X j=0
3
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B∗j exp(A∗ (T − s − h j )))E{.|zT }dt
2014 ACODS March 13-15, 2014. Kanpur, India
if T > h M Now let us define n × n-dimensional deterministic controllability matrix for hi+1 < T < hi , 0, 1, 2, ..., M − 1 ΓTs
=
i =
LT (s)LT∗ (s) j=i j=i i=k−1 X Z T −hi X X = ( exp(A(t − s − h j ))B j )( B∗j exp(A∗ (T − t − h j )))dt i=0
+
T −hi+1 j=0
j=k T −hk X
Z
(
s
j=0
j=k X exp(A(T − s − h j ))B j )( B∗j exp(A∗ (T − s − h j )))dt
j=0
j=0
and for T > h M ΓTs
=
LT (s)LT∗ (s) j=i j=i i=M−1 X Z T −hi X X = ( exp(A(t − s − h j ))B j )( B∗j exp(A∗ (T − t − h j )))dt T −hi+1 j=0
i=0
+
Z
j=M T −h M X
(
0
j=0
exp(A(T − s − h j ))B j )(
j=0
j=M X
B∗j exp(A∗ (T − s − h j )))dt
j=0
Definition 2.1. The system (1.2) is completely controllable on [0, T ] if RT (Uad ) = L2 (Ω, zT , Rn ) 3. Main Result Lemma 1. Assume that the operator (ΠT0 ) is invertible. Then for arbitrary xT ∈ L2 (Ω, zT , Rn ), f (.) ∈ L2 ([0, T ], Rn ),σ(.) ∈ L2 ([0, T ], Rn×n , the control defined as: u(t) =
j=i X B∗j exp(A∗ (T − t − h j )))E{(ΠT0 )−1 p(x)|zT }, (
t ∈ (T − hi+1 , T − hi ]
j=1
i = 0, 1, 2, ..., M − 1 B∗0 exp(A∗ (T − t))E{(ΠT0 )−1 p(x)|zT }, t ∈ [0, T − h M ] RT where p(x) = xT − φ(T )x0 − 0 φ(T − s)( f (s, x(s))ds + σ(s)dω(s)). transfers the system (2.3),(2.4) from x0 ∈ Rn to xT at time T and x(t; x0 , u) =
φ(t)x0 + Πt0 [φ∗ (T − t)((ΠT0 )−1 × (xT − φ(T )x0 − Z T − φ(T − r)σ(r)dω(r))] 0 Z t Z t + φ(t − s) f (s)ds + φ(t − s)σ(s)dω(s) 0
Z
T
φ(T − r) f (r)dr
0
(3.1) (3.2)
0
Proof: By substituting u(t) in (2.3) and (2.4) and taking into account of the form of the operator ΠT0 we can easily obtain the following x(t; x0 , u) =
φ(t)x0 + Πt0 [φ∗ (T − t)((ΠT0 )−1 × (xT − φ(T )x0 − Z T − φ(T − r)σ(r)dω(r))] 0 Z t Z t + φ(t − s) f (s)ds + φ(t − s)σ(s)dω(s) 0
0
4
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Z
T
φ(T − r) f (r)dr
0
(3.3) (3.4)
2014 ACODS March 13-15, 2014. Kanpur, India
Put t = T in above equation we get x(T ; x0 , u)
x(T ; x0 , u) =
0
T
Z
= φ(T )x0 + ΠT0 [φ∗ (T − T )((ΠT0 )−1 × (xT − φ(T )x0 − Z T − φ(T − s)σ(s)dω(s))] 0 Z T Z T + φ(T − s) f (s)ds + φ(T − s)σ(s)dω(s)
φ(T − s) f (s)ds
0
0
xT
We impose the following conditions (A1) ( f, σ) satisfies the Lipschitz condition with respect to x || f (t, x1 ) − f (t, x2 )||2 + ||σ(t, x1 ) − σ(t, x2 )||2 ≤ L||x1 − x2 ||2 (A2) ( f, σ) is continuous on [0, T ] × Rn and satisfies || f (t, x)||2 + ||σ(t, x)||2 ≤ L(||x||2 + 1) (A3) The linear system (1.1) is completely controllable. From lemma(1), the control u(t) transfers the system (2.3) and (2.4) from the initial state x0 to the final state xT provided that the operator S has a fixed point. So, if the operator S has a fixed point then the system (1.2) is completely controllable. To apply the Banach fixed point theorem, define the operator S for (3.1) for t ∈ [0, T ] as follows
S(x)(t) =
φ(t)x0 + Πt0 [φ∗ (T − t)((ΠT0 )−1 × (xT − φ(T )x0 − Z T − φ(T − r)σ(r)dω(r))] 0 Z t Z t + φ(t − s) f (s)ds + φ(t − s)σ(s)dω(s) 0
Z
T
φ(T − r) f (r)dr
0
0
Now for convenience, let us introduce the notation l1 = max||φ(t)||2 : t ∈ [0, T ],
l2 = max{||Bi ||2 , i = 0, 1, 2, ...M − 1}
l3 = E||xT ||2 ,
M1 = max||ΓTs ||2 : s ∈ [0, T ]
Lemma 2. For every z ∈ L2 (Ω, zT , Rn ), there exists a process ϕ(.) ∈ L2 ([0, T ], Rn×n ) such that Z T z = Ez + ϕ(s)dω(s) 0 Z T ΠT0 z = GT (0)Ez + ΓTs ϕ(s)dω(s) 0
Moreover E||ΠT0 z||2
≤ M1 E||E{z|zT }||2 ≤ M1 E||z||2 ,
z ∈ L2 (Ω, zT , Rn )
Note that if the assumption (A3) holds, then for some γ > 0 EhΠT0 z, zi ≥ γE||z||2 ,
f or
all
z ∈ L2 (Ω, zT , Rn )
see Mahmudov (2001a) and consequently E||(ΠT0 )−1 ||2 ≤
1 = l4 γ
Theorem 3.1. Assume that the conditions (A1),(A2) and (A3) hold. If the inequality 4l1 L(M1 l1 l4 + 1)(T + 1)T < 1
(3.5)
holds, then the system (1.2) is completely controllable. Proof: As mentioned above, to prove the complete controllability it is enough to show that S has a fixed point in H2 . 5
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2014 ACODS March 13-15, 2014. Kanpur, India
To do this, we use the contraction mapping principle. To apply the contraction mapping principle, first we show that S maps H2 into itself. Now by Lemma 1 we have E||(Sx)(t)||2
E||φ(t)x0 + Πt0 [φ∗ (T − t) × (ΠT0 )−1 (xT − φ(T )x0 Z T Z T − φ(T − r) f (r, x(r))dr − φ(T − r)σ(r, x(r))dω(r))] 0 0 Z t Z t + φ(t − s) f (s, x(s))ds + φ(t − s)σ(s, x(s))dω(s)||2
=
0
0
≤ 4||φ(t)x0 ||2 + 4E||Πt0 [φ∗ (T − t) × (ΠT0 )−1 (xT − φ(T )x0 Z T Z T − φ(T − r) f (r, x(r))dr − φ(T − r)σ(r, x(r))dω(r))]||2 0 0 Z t Z t 2 2 +4t ||φ(t − r)|| E|| f (r, x(r)|| dr + 4 ||φ(t − r)||2 E||σ(r, x(r))||2 dr 0
0
≤ 4l1 ||x0 ||2 + 16M1 l1 l4 (E||xT ||2 + l1 ||x0 ||2 Z T Z T +T ||φ(T − r)||2 E|| f (r, x(r))||2 dr + ||φ(T − r)||2 E||σ(r, x(r))||2 dr) 0 0 Z t +4l1 (T E|| f (r, x(r))||2 + E||σ(r, x(r))||2 )dr 0 Z t ≤ B1 + B2 ( (T E|| f (r, x(r))||2 + E||σ(r, x(r))||2 )dr) 0
where B1 > 0 and B2 > 0 are suitable constants. It follows from above and the condition (A2) that there exists C1 > 0 such that Z T E||(S x)(t)||2 ≤ C1 (1 + E||x(r)||2 dr) 0
≤ C1 (1 + T sup E||x(r)||2 ) 0≤r≤T
for all t ∈ [0, T ]. Therefore S maps H2 into itself. Secondly, we show that S is a contraction mapping. indeed 2
E||(Sx1 )(t) − (Sx2 )(t)||
=
T
Z E||Πt0 [φ∗ (T +
Z
+
Z
+
Z
T
−
t)(ΠT0 )−1
×(
φ(T − s)( f (s, x2 (s)) − f (s, x1 (s)))ds
0
φ(T − s)(σ(s, x2 (s)) − σ(s, x1 (s)))dω(s))]
0 t
φ(t − s)( f (s, x1 (s)) − f (s, x2 (s)))ds
0 t
φ(t − s)(σ(s, x2 (s)) − σ(s, x1 (s)))dω(s)||2 0 Z T ≤ 4M1 l12 l4 (T E|| f (s, x1 (s)) − f (s, x2 (s))||2 ds 0 Z T + E||σ(s, x1 (s)) − σ(s, x2 (s))||2 ds) 0 Z t +4l1 (T E|| f (s, x1 (s)) − f (s, x2 (s))||2 ds 0 Z t + E||σ(s, x1 (s)) − σ(s, x2 (s))||2 ds) 0 Z T = 4M1 l12 l4 L(T + 1) E||x1 (s) − x2 (s)||2 ds 0 Z t +4l1 L(T + 1) E||x1 (s) − x2 (s)||2 ds 0 Z T ≤ 4l1 L(M1 l1 l4 + 1)(T + 1) E||x1 (s) − x2 (s)||2 ds 0
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2014 ACODS March 13-15, 2014. Kanpur, India
It results that sup E||(Sx1 )(t) − (Sx2 )(t)||2 ≤ 4l1 L(M1 l1 l4 + 1)(T + 1)T sup E||x1 (t) − x2 (t)||2 t∈[0,T ]
t∈[0,T ]
Therefore S is a contraction mapping if the inequality (3.2) holds. Then the mapping S has a unique fixed point x(.) in H2 which is the solution of the equation (1.2). Thus the system (1.2) is completely controllable. The theorem is proved. Remark 3.1. hypothesis (3.2) is fulfilled if L is sufficiently small. Remark 3.2. If the semilinear system has no delay (as described by N.I.Mahmudov) then Z T M1 = max||ΓTs ||2 : s ∈ [0, T ] = max|| φ(T − t)BB∗ φ∗ (T − t)dt||2 : s ∈ [0, T ] s
≤ l12 l22 T If the semilinear system is in the form of (1.1) then hi+1 < T < hi ,
f or
i = 0, 1, 2, ..., M − 1
M1 = max||ΓTs ||2 : s ∈ [0, T ] = max||
M1
i=k−1 X Z T −hi
+
j=k T −hk X
(
s
j=i X
T −hi+1 j=0
i=0
Z
(
j=i X exp(A(t − s − h j ))B j )( B∗j exp(A∗ (T − t − h j )))dt j=0 j=k X
exp(A(T − s − h j ))B j )(
j=0
j=0
≤ [l12 l22 M(p(
B∗j exp(A∗ (T − s − h j )))dt||2
(M − 1)(2M − 1) + 1) + M(T − h M−1 ))] where 6 f or
p = max|hk+1 − hk |
T > hM
M1 = max||ΓTs ||2 : s ∈ [0, T ] M1
=
max||
i=M−1 X Z T −hi i=0
+
(
s
≤
T −hi+1 j=0
j=M T −h M X
Z
[l12 l22 (M
(
j=i X
j=0
j=i X B∗j exp(A∗ (T − t − h j )))dt exp(A(t − s − h j ))B j )( j=0
exp(A(T − s − h j ))B j )(
j=M X
B∗j exp(A∗ (T − s − h j )))dt||2
j=0
M(2M + 1) + 1)(p( + 1) + (M + 1)(T − h M ))] where 6
p = max|hk+1 − hk |
References [1] A.Pazy, Semigroup of linear operators and application to partial differential equations, Springer Verlag, New York,(1983). [2] J. Klamka, Stochastic controllability of linear systems with delay in control, Bulletin of the polish academy of sciences. 55(2007),pp.23-29. [3] J. Klamka, Stochastic controllability of systems with multiple delays in control, Int. J. Appl. Math. Comput. Sci. 19 (2009), no. 1, 39–47. MR2515022 [4] J.P. Dauer and N.I. Mahmudov, Approximate Controllability of semilinear function equations in Hilbert spaces, Journal of Math.Analysis and Applications. 273(2002),pp.310-327. [5] K. Balachandran and J. P. Dauer, Controllability of nonlinear systems in Banach spaces: a survey, J. Optim. Theory Appl. 115 (2002), no. 1, 7–28. [6] K. Naito, Controllability of semilinear control systems dominated by the linear part, SIAM J. Control Optim. 25 (1987), no. 3, 715–722. [7] L. Shen and J. Sun, Relative controllability of stochastic nonlinear systems with delay in control, Nonlinear Anal. Real World Appl. 13 (2012), no. 6, 2880–2887. MR2927233 [8] N. I. Mahmudov, Controllability of linear stochastic systems, IEEE Trans. Automat. Control 46 (2001), no. 5, 724–731. MR1833025 (2002c:93020) [9] N.I. Mahmudov and S.Zorlu, Controllability of nonlinear stochastic systems, Journal of Control vol 76(2003),pp.95-104. [10] N. Sukavanam and M. Kumar, S -controllability of an abstract first order semilinear control system, Numer. Funct. Anal. Optim. 31 (2010), no. 7-9, 1023–1034. [11] R.E. Kalman, Controllability of linear systems,Contributions to Differential Equations.1(1963),pp.190-213. [12] R. Triggiani, Controllability and Observability in Banach spaces with bounded operators, SIAM Journal of Control and Optimization 10(1975),pp.462-291. [13] Stephen Barnett, Introduction to mathematical Control Theory, Clarendon press, Oxford 1975.
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Controllability of Semilinear Stochastic System with Multiple Delays in Control Anurag Shukla, N Sukavanam, D.N. Pandey Department of Mathematics, Indian Institute of Technology Roorkee Roorkee-247667, India
Abstract Finite dimensional stationary dynamic system described by semilinear stochastic ordinary differential equations with multiple delays in control is considered, under the basic and readily verified conditions to guarantee the existence of the solutions to a system. In this paper, we prove the complete controllability of the stochastic differential system with multiple delays in control. Keywords: Controllability, Semilinear Systems, Stochastic Control System, Delayed Control, Multiple Delays. 1. Introduction Controllability is one of the fundamental concepts in mathematical control theory, and plays an important role in both deterministic and stochastic control theories. Conceived by Kalman, controllability study was started systematically at the beginning of the sixties. Since then various researches have been carried out extensively in the context of finite dimensional linear systems, nonlinear systems and infinite dimensional systems using different kind of approaches([1-5]). Stochastic differential equations (SDEs) are used to model diverse phenomenon such as fluctuating stock prices or physical system subjected to thermal fluctuations. In the literature, there are different definitions of controllability for SDEs both for linear and nonlinear dynamical systems . For linear systems, N.I.Mahmudov [8] got results for controllability of linear stochastic systems. Klamka [2-3] generalized the result with single and multiple delays in control in linear stochastic system. In [9] N.I.Mahmudov obtained results for controllability on semilinear stochastic system. Lijuan Shen, Jitao Sun [7] generalized these results and showed the complete controllability of semilinear stochastic system with single point delay in control. However, to best of our knowledge, there are no result on the complete controllability of semilinear SDEs with multiple delays in control as treated in the current paper. Definition 1.1. In this paper we adopt the following notation: (i) (Ω, z, P):Let z be the σ algebra generated by Ω and P : z → [0, 1] be the probability measure on z. Then the triple (Ω, z, P) is called a probability space. (ii) {zt |t ∈ [0, T ]} is the filtration generated by {ω(s) : 0 ≤ s ≤ t}. (iii) L2 (Ω, zT , Rn )=the Hilbert space of all zT -measurable square integrable variables with values in Rn . n n (iv) Lz P ([0, T ], R ) is the Banach space of all p-integrable and zt -measurable processes with values in R , p ≥ 2.
for
(v) H2 =the Banach space of all square integrable and zt -adapted processes ϕ(t) with norm ||ϕ||2 = sup E||ϕ(t)||2 t∈[0,T ]
(vi) L(X, Y) is the space of all linear bounded operators from a Banach space X in to a Banach space Y. (vii) φ(t) = exp(At) and Uad = L2z ([0, T ], Rm ). Preprint submitted to Elsevier 978-3-902823-60-1 © 2014 IFAC
306
February 25, 2014 10.3182/20140313-3-IN-3024.00107
2014 ACODS March 13-15, 2014. Kanpur, India
The problem of controllability of linear stochastic system with multiple delays in control dx(t) = [Ax(t) +
i=M X
Bi u(t − hi )]dt + σ(t, x(t))dω(t)
(1.1)
i=0
x(0) = x0 ∈ L2 (Ω, FT , Rn ) and u(t) = 0 for t ∈ [−h M , 0] has been studied by various authors (see J.Klamka (2009) for references). The problem of controllability of semi-linear stochastic system dx(t) = [Ax(t) + Bu(t) + f (t, x(t))]dt + σ(t, x(t))dω(t) x(0) = x0 ∈ Rn has been studied by various authors (see N.I.Mahmudov and S.Zorlu 2003 and N.Sukavanam and Mohit Kumar 2011 for references). In this paper we examine the controllability of the following semi-linear stochastic system with discrete multiple delays in control term of following problem: dx(t) = [Ax(t) +
i=M X
Bi u(t − hi ) + f (t, x(t))]dt + σ(t, x(t))dω(t) for t ∈ [0, T ]
(1.2)
i=0
(1.3) with initial conditions: x(0) = x0 ∈ L2 (Ω, FT , Rn ) and u(t) = 0 for t ∈ [−h M , 0]
(1.4)
where the state x(t) ∈ L2 (Ω, zt , R ) = X and the control u(t) ∈ R = U, A is an n × n constant matrix,Bi , i = 0, 1, 2, 3, ..., M ,are n × m constant matrices respectively. σ : [0, T ] × Rn → Rn×n , f : [0, T ] × Rn → Rn and( f, σ) are continuous nonlinear functions on [0, T ] × Rn , ω is a n-dimensional Wiener process and 0 = h0 < h1 < h2 < ..... < h M are constant point delays. n
m
2. Preliminaries It is well known([1], [2], [7], [8]) that for given initial conditions, any admissible control u ∈ Uad and suitable nonlinear function f (t, x(t) for t ∈ [0, T ] there exists unique solution x(t; x0 , u) ∈ L2 (Ω, zt , Rn ) of the linear stochastic differential state equation (1.2) which can be represented in the following integral form Z t Z t i=M X x(t; x0 , u) = exp(At)x0 + exp(A(t − s))( Bi u(s − hi ) + f (s, x(s)))ds + exp(A(t − s))σ(s)dω(s) 0
0
i=0
Thus,without loss of generality, taking into account the zero initial control for t ∈ [−h M , 0) and changing the order of integration, the solution x(t; x0 , u) for hk < t ≤ hk+1 , k = 0, 1, 2, ..., M − 1, t ∈ [0, h M ] has the following form, which is more convenient for further deliberations j=i i=k−1 X Z t−hi X x(t; x0 , u) = exp(At)x0 + ( exp(A(t − s − h j ))B j )u(s)ds i=0
+
Z
j=k t−hk X
(
0
+
Z
t−hi+1 j=0
exp(A(t − s − h j ))B j )u(s)ds
j=0 t
exp(A(t − s)) f (s, x(s))ds +
0
Z
t
exp(A(t − s))σ(s)dω(s)
(2.1)
0
Similarly for t > h M x(t; x0 , u) =
exp(At)x0 +
i=M−1 X Z t−hi i=0
+
Z
j=M t−h M X
(
0
+
Z
(
j=i X
exp(A(t − s − h j ))B j )u(s)ds
t−hi+1 j=0
exp(A(t − s − h j ))B j )u(s)ds
j=0 t
exp(A(t − s)) f (s, x(s))ds +
0
Z
exp(A(t − s))σ(s)dω(s) 0
2
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t
(2.2)
2014 ACODS March 13-15, 2014. Kanpur, India
Now, for a given final time T, using the form of the integral solution x(t : x0 , u), let us introduce operators and sets which will be used in next sections of the paper. First of all, for hk < T < hk+1 and for k = 0, 1, 2, ..., M −1, we define the following linear and bounded control operator LT ∈ L2 ([0, T ], Rm ) → L2 (Ω, zT , Rn ):
LT u
i=k−1 X Z T −hi
=
T −hi+1 j=0
i=0
j=k T −hk X
Z
+
j=i X ( exp(A(t − s − h j ))B j )u(s)ds
(
0
exp(A(T − s − h j ))B j )u(s)ds
j=0
Moreover, for T > h M we have i=M−1 X Z T −hi
=
LT u
i=0
+
Z
j=i X ( exp(A(t − s − h j ))B j )u(s)ds
T −hi+1 j=0 j=M T −h M X
(
0
exp(A(T − s − h j ))B j )u(s)ds
j=0
and its adjoint LT∗ : L2 (Ω, zT , Rn ) → LT ∈ L2 ([0, T ], Rm ) is a linear and bounded operator given by ∗ B0 exp(A∗ (T − t))E{z|zT } f or t ∈ [0, T − h M ] j=i X LT∗ z = B∗ exp(A∗ (T − t − h j )))E{z|zT } f or t ∈ (T − hi+1 , T − hi ], ( j=1 j
i = 0, 1, 2, ...M − 1
Let RT (Uad ) denotes the set of all states reachable from the initial state x(0) = x0 ∈ L2 (Ω, zT , Rn ) in time T > 0 using admissible controls Hence RT (Uad ) =
{x(T ; x0 , u) ∈ L2 (Ω, zT , Rn ) : u ∈ Uad } Z T exp(At)x0 + ImLT + exp(A(T − s))σ(s)dω(s)
=
0
Now, let us define the linear controllability operator ΠT0 ∈ L(L2 (Ω, zT , Rn ), L2 (Ω, zT , Rn ), which is strongly associated with the control operator LT as ΠT0 {.} =
LT LT∗ {.} i=k−1 XZ = i=0
+
T −hi+1 j=0
j=k T −hk X
Z
(
0
if hi+1 < T < hi , ΠT0 {.}
j=i j=i X X ( exp(A(t − s − h j ))B j )( B∗j exp(A∗ (T − t − h j )))E{.|zT }dt
T −hi
j=0 j=k X
exp(A(T − s − h j ))B j )(
j=0
B∗j exp(A∗ (T − s − h j )))E{.|zT }dt
j=0
i = 0, 1, 2, ..., M − 1 =
LT LT∗ {.} =
i=M−1 X Z T −hi i=0
+
Z
T −hi+1 j=0
j=M T −h M X
(
0
j=i j=i X X ( exp(A(t − s − h j ))B j )( B∗j exp(A∗ (T − t − h j )))E{.|zT }dt j=0
exp(A(T − s − h j ))B j )(
j=0
j=M X j=0
3
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B∗j exp(A∗ (T − s − h j )))E{.|zT }dt
2014 ACODS March 13-15, 2014. Kanpur, India
if T > h M Now let us define n × n-dimensional deterministic controllability matrix for hi+1 < T < hi , 0, 1, 2, ..., M − 1 ΓTs
=
i =
LT (s)LT∗ (s) j=i j=i i=k−1 X Z T −hi X X = ( exp(A(t − s − h j ))B j )( B∗j exp(A∗ (T − t − h j )))dt i=0
+
T −hi+1 j=0
j=k T −hk X
Z
(
s
j=0
j=k X exp(A(T − s − h j ))B j )( B∗j exp(A∗ (T − s − h j )))dt
j=0
j=0
and for T > h M ΓTs
=
LT (s)LT∗ (s) j=i j=i i=M−1 X Z T −hi X X = ( exp(A(t − s − h j ))B j )( B∗j exp(A∗ (T − t − h j )))dt T −hi+1 j=0
i=0
+
Z
j=M T −h M X
(
0
j=0
exp(A(T − s − h j ))B j )(
j=0
j=M X
B∗j exp(A∗ (T − s − h j )))dt
j=0
Definition 2.1. The system (1.2) is completely controllable on [0, T ] if RT (Uad ) = L2 (Ω, zT , Rn ) 3. Main Result Lemma 1. Assume that the operator (ΠT0 ) is invertible. Then for arbitrary xT ∈ L2 (Ω, zT , Rn ), f (.) ∈ L2 ([0, T ], Rn ),σ(.) ∈ L2 ([0, T ], Rn×n , the control defined as: u(t) =
j=i X B∗j exp(A∗ (T − t − h j )))E{(ΠT0 )−1 p(x)|zT }, (
t ∈ (T − hi+1 , T − hi ]
j=1
i = 0, 1, 2, ..., M − 1 B∗0 exp(A∗ (T − t))E{(ΠT0 )−1 p(x)|zT }, t ∈ [0, T − h M ] RT where p(x) = xT − φ(T )x0 − 0 φ(T − s)( f (s, x(s))ds + σ(s)dω(s)). transfers the system (2.3),(2.4) from x0 ∈ Rn to xT at time T and x(t; x0 , u) =
φ(t)x0 + Πt0 [φ∗ (T − t)((ΠT0 )−1 × (xT − φ(T )x0 − Z T − φ(T − r)σ(r)dω(r))] 0 Z t Z t + φ(t − s) f (s)ds + φ(t − s)σ(s)dω(s) 0
Z
T
φ(T − r) f (r)dr
0
(3.1) (3.2)
0
Proof: By substituting u(t) in (2.3) and (2.4) and taking into account of the form of the operator ΠT0 we can easily obtain the following x(t; x0 , u) =
φ(t)x0 + Πt0 [φ∗ (T − t)((ΠT0 )−1 × (xT − φ(T )x0 − Z T − φ(T − r)σ(r)dω(r))] 0 Z t Z t + φ(t − s) f (s)ds + φ(t − s)σ(s)dω(s) 0
0
4
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Z
T
φ(T − r) f (r)dr
0
(3.3) (3.4)
2014 ACODS March 13-15, 2014. Kanpur, India
Put t = T in above equation we get x(T ; x0 , u)
x(T ; x0 , u) =
0
T
Z
= φ(T )x0 + ΠT0 [φ∗ (T − T )((ΠT0 )−1 × (xT − φ(T )x0 − Z T − φ(T − s)σ(s)dω(s))] 0 Z T Z T + φ(T − s) f (s)ds + φ(T − s)σ(s)dω(s)
φ(T − s) f (s)ds
0
0
xT
We impose the following conditions (A1) ( f, σ) satisfies the Lipschitz condition with respect to x || f (t, x1 ) − f (t, x2 )||2 + ||σ(t, x1 ) − σ(t, x2 )||2 ≤ L||x1 − x2 ||2 (A2) ( f, σ) is continuous on [0, T ] × Rn and satisfies || f (t, x)||2 + ||σ(t, x)||2 ≤ L(||x||2 + 1) (A3) The linear system (1.1) is completely controllable. From lemma(1), the control u(t) transfers the system (2.3) and (2.4) from the initial state x0 to the final state xT provided that the operator S has a fixed point. So, if the operator S has a fixed point then the system (1.2) is completely controllable. To apply the Banach fixed point theorem, define the operator S for (3.1) for t ∈ [0, T ] as follows
S(x)(t) =
φ(t)x0 + Πt0 [φ∗ (T − t)((ΠT0 )−1 × (xT − φ(T )x0 − Z T − φ(T − r)σ(r)dω(r))] 0 Z t Z t + φ(t − s) f (s)ds + φ(t − s)σ(s)dω(s) 0
Z
T
φ(T − r) f (r)dr
0
0
Now for convenience, let us introduce the notation l1 = max||φ(t)||2 : t ∈ [0, T ],
l2 = max{||Bi ||2 , i = 0, 1, 2, ...M − 1}
l3 = E||xT ||2 ,
M1 = max||ΓTs ||2 : s ∈ [0, T ]
Lemma 2. For every z ∈ L2 (Ω, zT , Rn ), there exists a process ϕ(.) ∈ L2 ([0, T ], Rn×n ) such that Z T z = Ez + ϕ(s)dω(s) 0 Z T ΠT0 z = GT (0)Ez + ΓTs ϕ(s)dω(s) 0
Moreover E||ΠT0 z||2
≤ M1 E||E{z|zT }||2 ≤ M1 E||z||2 ,
z ∈ L2 (Ω, zT , Rn )
Note that if the assumption (A3) holds, then for some γ > 0 EhΠT0 z, zi ≥ γE||z||2 ,
f or
all
z ∈ L2 (Ω, zT , Rn )
see Mahmudov (2001a) and consequently E||(ΠT0 )−1 ||2 ≤
1 = l4 γ
Theorem 3.1. Assume that the conditions (A1),(A2) and (A3) hold. If the inequality 4l1 L(M1 l1 l4 + 1)(T + 1)T < 1
(3.5)
holds, then the system (1.2) is completely controllable. Proof: As mentioned above, to prove the complete controllability it is enough to show that S has a fixed point in H2 . 5
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2014 ACODS March 13-15, 2014. Kanpur, India
To do this, we use the contraction mapping principle. To apply the contraction mapping principle, first we show that S maps H2 into itself. Now by Lemma 1 we have E||(Sx)(t)||2
E||φ(t)x0 + Πt0 [φ∗ (T − t) × (ΠT0 )−1 (xT − φ(T )x0 Z T Z T − φ(T − r) f (r, x(r))dr − φ(T − r)σ(r, x(r))dω(r))] 0 0 Z t Z t + φ(t − s) f (s, x(s))ds + φ(t − s)σ(s, x(s))dω(s)||2
=
0
0
≤ 4||φ(t)x0 ||2 + 4E||Πt0 [φ∗ (T − t) × (ΠT0 )−1 (xT − φ(T )x0 Z T Z T − φ(T − r) f (r, x(r))dr − φ(T − r)σ(r, x(r))dω(r))]||2 0 0 Z t Z t 2 2 +4t ||φ(t − r)|| E|| f (r, x(r)|| dr + 4 ||φ(t − r)||2 E||σ(r, x(r))||2 dr 0
0
≤ 4l1 ||x0 ||2 + 16M1 l1 l4 (E||xT ||2 + l1 ||x0 ||2 Z T Z T +T ||φ(T − r)||2 E|| f (r, x(r))||2 dr + ||φ(T − r)||2 E||σ(r, x(r))||2 dr) 0 0 Z t +4l1 (T E|| f (r, x(r))||2 + E||σ(r, x(r))||2 )dr 0 Z t ≤ B1 + B2 ( (T E|| f (r, x(r))||2 + E||σ(r, x(r))||2 )dr) 0
where B1 > 0 and B2 > 0 are suitable constants. It follows from above and the condition (A2) that there exists C1 > 0 such that Z T E||(S x)(t)||2 ≤ C1 (1 + E||x(r)||2 dr) 0
≤ C1 (1 + T sup E||x(r)||2 ) 0≤r≤T
for all t ∈ [0, T ]. Therefore S maps H2 into itself. Secondly, we show that S is a contraction mapping. indeed 2
E||(Sx1 )(t) − (Sx2 )(t)||
=
T
Z E||Πt0 [φ∗ (T +
Z
+
Z
+
Z
T
−
t)(ΠT0 )−1
×(
φ(T − s)( f (s, x2 (s)) − f (s, x1 (s)))ds
0
φ(T − s)(σ(s, x2 (s)) − σ(s, x1 (s)))dω(s))]
0 t
φ(t − s)( f (s, x1 (s)) − f (s, x2 (s)))ds
0 t
φ(t − s)(σ(s, x2 (s)) − σ(s, x1 (s)))dω(s)||2 0 Z T ≤ 4M1 l12 l4 (T E|| f (s, x1 (s)) − f (s, x2 (s))||2 ds 0 Z T + E||σ(s, x1 (s)) − σ(s, x2 (s))||2 ds) 0 Z t +4l1 (T E|| f (s, x1 (s)) − f (s, x2 (s))||2 ds 0 Z t + E||σ(s, x1 (s)) − σ(s, x2 (s))||2 ds) 0 Z T = 4M1 l12 l4 L(T + 1) E||x1 (s) − x2 (s)||2 ds 0 Z t +4l1 L(T + 1) E||x1 (s) − x2 (s)||2 ds 0 Z T ≤ 4l1 L(M1 l1 l4 + 1)(T + 1) E||x1 (s) − x2 (s)||2 ds 0
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2014 ACODS March 13-15, 2014. Kanpur, India
It results that sup E||(Sx1 )(t) − (Sx2 )(t)||2 ≤ 4l1 L(M1 l1 l4 + 1)(T + 1)T sup E||x1 (t) − x2 (t)||2 t∈[0,T ]
t∈[0,T ]
Therefore S is a contraction mapping if the inequality (3.2) holds. Then the mapping S has a unique fixed point x(.) in H2 which is the solution of the equation (1.2). Thus the system (1.2) is completely controllable. The theorem is proved. Remark 3.1. hypothesis (3.2) is fulfilled if L is sufficiently small. Remark 3.2. If the semilinear system has no delay (as described by N.I.Mahmudov) then Z T M1 = max||ΓTs ||2 : s ∈ [0, T ] = max|| φ(T − t)BB∗ φ∗ (T − t)dt||2 : s ∈ [0, T ] s
≤ l12 l22 T If the semilinear system is in the form of (1.1) then hi+1 < T < hi ,
f or
i = 0, 1, 2, ..., M − 1
M1 = max||ΓTs ||2 : s ∈ [0, T ] = max||
M1
i=k−1 X Z T −hi
+
j=k T −hk X
(
s
j=i X
T −hi+1 j=0
i=0
Z
(
j=i X exp(A(t − s − h j ))B j )( B∗j exp(A∗ (T − t − h j )))dt j=0 j=k X
exp(A(T − s − h j ))B j )(
j=0
j=0
≤ [l12 l22 M(p(
B∗j exp(A∗ (T − s − h j )))dt||2
(M − 1)(2M − 1) + 1) + M(T − h M−1 ))] where 6 f or
p = max|hk+1 − hk |
T > hM
M1 = max||ΓTs ||2 : s ∈ [0, T ] M1
=
max||
i=M−1 X Z T −hi i=0
+
(
s
≤
T −hi+1 j=0
j=M T −h M X
Z
[l12 l22 (M
(
j=i X
j=0
j=i X B∗j exp(A∗ (T − t − h j )))dt exp(A(t − s − h j ))B j )( j=0
exp(A(T − s − h j ))B j )(
j=M X
B∗j exp(A∗ (T − s − h j )))dt||2
j=0
M(2M + 1) + 1)(p( + 1) + (M + 1)(T − h M ))] where 6
p = max|hk+1 − hk |
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