Controllability of Semilinear Stochastic Systems N. I. Mahmudov and S. Zorlu Eastern Mediterranean University Gazimagusa, Mersin 10 Turkey Phone : (0090392) 630 1002/630 1421 e-mail:
[email protected] [email protected] April 21, 2004 Abstract The contraction mapping principle is used to derive sufficient conditions for approximate and complete controllability of semilinear stochastic system under the assumption that the corresponding linear system is appropriately controllable.
Keywords: Approximate controllability, complete controllability, semilinear stochastic systems, contraction mapping principle.
1
Introduction
The concepts of controllability play important roles in analysis and the design of control systems. Any control system is said to be controllable if every state corresponding to this process can be affected or controlled in respective time by some control signals. In many dynamical systems, it is possible to steer the dynamical system from an arbitrary initial state to an arbitrary final state using the set of admissible controls; that is there are systems which are completely controllable. If the system cannot be controlled completely then different types of controllability can be defined such as approximate, null, local null, local approximate null controllability, etc. 1
For deterministic systems, the basic controllability concepts have been well investigated by Mirza and Womack 1971, Balachandran and Dauer 1987, Do 1990, Zabczyk 1992, Klamka 2000. The classical theory of controllability for deterministic systems is extended to linear stochastic systems. There are limited number of papers dealing with the stochastic controllability problems. For fixed ε, p Sunahara et. al. 1974 and Klamka and Socha 1977, Klamka and Socha 1980 gave conditions for 0 ∈ Apε (T, x0 ) via the Lyapunov approach for several type of nonlinear stochastic systems, where Apε (T, x0 ) is the set of nonrandom (ε, p)- attainable points from x0 in time T defined by Apε (T, x0 ) = h ∈ Rn : ∃u ∈ Uad , P kx (T ) − hk2 ≤ ε ≥ p . Dauer and Balachandran 1997 studied the sample controllability for nonlinear random differential equations. Mahmudov and Zorlu 2003 studied complete controllability and local null controllability of non-linear systems. The problem of controllability of a linear stochastic system
dx(t) = [Ax(t) + Bu(t)]dt + σ1 (t)dw(t)
(1)
x(0) = x0 , t ∈ [0, T ] has been studied by various authors. Dubov and Mordukhovich 1978 studied the null approximate controllability of the linear stochastic systems. Zabczyk 1981 studied the stochastic controllability of linear stochastic systems. Mahmudov 2000, Mahmudov 2001 showed that complete and approximate controllability notions for linear stochastic systems are equivalent. In this paper, the approximate and complete controllability of the following semilinear
2
stochastic system is studied using the contraction mapping principle, dx(t) = [Ax(t) + Bu(t) + f (t, x(t))]dt + σ(t, x(t))dw(t)
(2)
x(0) = x0 ∈ Rn where A and B are matrices of dimensions n×n, n×m respectively, , f : [0, T ]×Rn → Rn , σ : [0, T ] × Rn → Rn×n , σ1 : [0, T ] → Rn×n (see (1)) and w is n-dimensional Wiener process. The paper is organized as follows: Section 2 contains definitions and some preliminary results. In Section 3, we obtain some controllability conditions via one of the fixed point methods, namely the contraction mapping principle. Assuming controllability of the associated linear system under some natural conditions, we prove the approximate controllability of the system (2). Additionally, the complete controllability of the same system is studied under conditions that the matrix A is nonnegative and self-adjoint and the matrix BB ∗ is positive.
2
Definitions
In this paper, the following notations are adopted :
• (Ω, F, P) := The probability space with probability measure P on Ω. • {Ft | t ∈ [0, T ]} := The filtration generated by {w (s) : 0 ≤ s ≤ t}. • L2 (Ω, FT ,Rn ) := The Hilbert space of all FT -measurable square integrable variables with values in Rn . n • LF 2 ([0, T ] , R ) := The Hilbert space of all square integrable and Ft -measurable
processes with values in Rn . 3
• C ([0, T ] , L2 (Ω, F, P, X)) := The Banach space of continuous maps from [0, T ] into L2 (Ω, F, P, X) satisfying the condition sup E kx(t)k2 < ∞. t∈[0,T ]
• H2 := The Banach space with norm topology given by kxk2H2 = sup E kx(t)k2 t∈[0,T ]
which is a closed subspace of C ([0, T ] , L2 (Ω, F, P, X)) consisting of measurable and Ft - adapted processes x (t) . • L(X, Y ) := The space of all linear bounded operators from a Banach space X to a Banach space Y . • S(t) = exp(At). Now let us introduce the following operators and sets. m n 1. The operator LT0 ∈ L LF 2 ([0, T ] , R ) , L2 (Ω, FT , R ) is defined by LT0 u
Z
T
S (T − s) Bu(s)ds
= 0
Clearly the adjoint LT0
∗
m : L2 (Ω, FT , Rn ) → LF 2 ([0, T ] , R ) is defined by
(LT0 )∗ z = B ∗ S ∗ (T − t)E {z | Ft } . 2. The controllability matrix ΓTs ∈ L (Rn , Rn ) ΓTs
Z
T
=
S (T − t) BB ∗ S ∗ (T − t) dt, 0 ≤ s < t
s
and the resolvent operator −1 R α, ΓTs = αI + ΓTs , 0 ≤ s ≤ T.
3. Set of all states attainable from x0 in time t > 0 m Rt (x0 ) = {x(t; x0 , u) : u (·) ∈ LF 2 ([0, T ] , R )},
4
m where x (t, x0 , u) is the solution of (2) corresponding to x0 ∈ Rn , u (·) ∈ LF 2 ([0, T ] , R ) .
Now for convenience, let us introduce the following notations :
M = kBk2 , M1 = max ||S(t)||2 : t ∈ [0, T ] , n o 2 M2 = max Γts : s, t ∈ [0, T ] . Definition 1 The linear stochastic system (2) is approximately controllable on [0, T ] if RT (x0 ) = L2 (Ω, FT , Rn ), that is if it is possible to steer the system from the initial point x0 to within a distance ε > 0 from all the final points in the state space L2 (Ω, FT , Rn ) at time T.
Definition 2 The linear stochastic system (2) is completely controllable on [0, T ] if RT (x0 ) = L2 (Ω, FT , Rn ), that is if all the points in L2 (Ω, FT , Rn ) can be reached from the point x0 at time T.
The following conditions on data of the problem are imposed : (H1) (f, σ) satisfies the Lipschitz condition with respect to x kf (t, x1 ) − f (t, x2 )k2 + kσ(t, x1 ) − σ(t, x2 )k2 ≤ L kx1 − x2 k2 . (H2) (f, σ) is continuous on [0, T ] × Rn and satisfies kf (t, x)k2 + kσ(t, x)k2 ≤ L.
(H3) The linear system (1) is approximately controllable. 5
(H4) The linear system (1) is completely controllable. (H5) A is nonnegative and self-adjoint. (H6) BB ∗ is positive, that is there exists γ1 > 0 such that hBB ∗ x, xi ≥ γ1 kxk2 .
(AC) αR α, ΓT0 → 0 as α → 0+ . Note that the assumptions (AC) , (H3) and (H4) are equivalent, see Mahmudov 2001. The following lemma gives a formula for a control steering the state x0 to some neighborhood of an arbitrary point h. n F n×n Lemma 3 For arbitrary f (·) ∈ LF ), h ∈ L2 (Ω, F, Rn ) 2 ([0, T ] , R ), σ (·) ∈ L2 ([0, T ] , R
the control −1 uα (t) = B ∗ S ∗ (T − t) αI + ΓT0 (Eh − S(T )x0 ) Z t −1 − B ∗ S ∗ (T − t) αI + ΓTr S(T − r)f (r)dr 0 Z t −1 ∗ ∗ − B S (T − t) αI + ΓTr (S(T − r)σ(r) − ϕ (r)) dw(r)
(3)
0
transfers the system Z
t
S (t − s) Bu(s)ds x(t) = S(t)x0 + 0 Z t Z t + S (t − s) f (s)ds + S (t − s) σ(s)dw(s) 0
(4)
0
from x0 ∈ Rn to some neighborhood of h at time T and −1 (Eh − S(T )x0 ) xα (T ) = h − α αI + ΓT0 Z T −1 α αI + ΓTr S(T − r)f (r)dr + 0 Z T −1 + α αI + ΓTr (S(T − r)σ (r) − ϕ (r)) dw(r), 0
where h has the following representation h = Eh + Shiryaev 1977. 6
RT 0
ϕ (r) dw (r) , see Lipster and
Proof. By substituting (3) in (4), one can easily obtain that Z
t
xα (t) = S(t)x0 + S (t − r) f (r)dr 0 Z t + S(t − r)σ(r)dw(r) 0
−1 + Γt0 S ∗ (T − t) αI + ΓT0 (Eh − S (T ) x0 ) Z t −1 − Γtr S ∗ (T − r) αI + ΓTr S(T − r)f (r) dr 0 Z t −1 (S(T − r)σ (r) − ϕ (r)) dw(r). − Γtr S ∗ (T − r) αI + ΓTr
(5)
0
The equation (5) can be rewritten at t = T. Hence, −1 (Eh − S(t)x0 ) xα (T ) = h − α αI + ΓT0 Z T −1 α αI + ΓTr + S(T − r)f (r)dr 0 Z T −1 α αI + ΓTr + (S(T − r)σ (r) − ϕ (r)) dw(r). 0
Lemma 4 Let Assumption (H4), (H5) and (H6) holds. Then there exists C > 0 such n that for all g (·) ∈ LF 2 (0, T ; R ) the following inequality holds
Z t
2 −1
t ∗
S (t − r) g (r) dr lim− E
Γr S (T − t) ΓTr t→T 0 Z T ≤C E kg (r)k2 dr.
(6)
0
Proof. By assumption (H4) , the linear stochastic system is completely controllable on [0, T ] . Thus, the system (1) is completely controllable on every subinterval [r, t] , 0 ≤ r < t ≤ T , that is one can find γt,r > 0 such that
Γtr x, x ≥ γt,r kxk2 ,
7
t −1 consequently (Γr ) ≤ 1/γt,r . To estimate the integral on the left hand side of (6), we need to find an explicit relation between γt,r and (t − r) . More precisely, we have to find a constant C1 > 0 satisfying 1 1 ≤ . γt,r C1 (t − r)
(7)
To prove (7), the nonnegativeness and self-adjointness of A and positiveness of BB ∗ will be used.
Γtr x, x
Z
t
hS (t − s) BB ∗ S ∗ (t − s) x, xi ds
= Zr t =
hBB ∗ S ∗ (t − s) x, S ∗ (t − s) xi ds
Zr t
kB ∗ S ∗ (t − s) xk2 ds r Z t ≥ γ1 hS (t − s) S ∗ (t − s) x, xi ds Zr t
2A(t−s) = γ1 e x, x ds Zr t
= γ1 I + 2A (t − s) + 2A2 (t − s)2 ... x, x ds r Z t
2 ≥ γ1 kxk (t − r) + γ1 2A (t − s) + 2A2 (t − s) + ... x, x ds =
r
≥ γ1 kxk2 (t − r) . Hence, γt,r is a constant multiple of (t − r) . That is, γt,r = C1 (t − r) for C1 ≤ γ1 . It yields that Z
t
2 −1
E Γtr S ∗ (T − t) ΓTr S (T − r) g (r) dr 0 Z t 1 4 2 ≤ M1 M (t − r)2 2 E kg (r)k2 dr γT,r 0 Z M 4 M 2 t (t − r)2 2 = 12 2 E kg (r)k dr C1 0 (T − r) 4 2 Z T M M ≤ 12 E kg (r)k2 dr C1 0
Taking limit on (8) as t → T − , the inequality (6) with C = 8
M14 M 2 C12
is obtained.
(8)
3
Controllability via Contraction Mapping Principle
In this section, using the contraction mapping principle.some controllability conditions for the semilinear stochastic system (2) are derived.
Remark 5 In Mahmudov 2000 and Mahmudov 2001 it is shown that complete controllability and approximate controllability of the linear system (1) coincide. That is why, we study the approximate and complete controllability of the semilinear stochastic system (2) separately. This may not always be true for semilinear stochastic systems.
The solution of the system (2) is referred to the solution of the following nonlinear integral equation Z
t
x(t) = S(t)x0 + S(t − s)Buα (s)ds 0 Z t Z t S (t − s) σ(s, x (s))dw(s) S (t − s) f (s, x (s))ds + +
(9)
0
0
where −1 uα (t) = B ∗ S ∗ (T − t) αI + ΓT0 (E {h|Ft } − S(T )x0 ) Z t −1 ∗ ∗ − B S (T − t) αI + ΓTr S(T − r)f (r, x (r))dr 0 Z t −1 ∗ ∗ − B S (T − t) αI + ΓTr S(T − r)σ(r, x (r))dw(r) 0
9
(10)
To apply the contraction mapping principle, it is essential to introduce the nonlinear operator Fα , α > 0 from H2 to H2 which is defined as follows Z
t
(Fα x) (t) = S(t)x0 + S (t − r) f (r, x (r))dr 0 Z t + S (t − r) σ (r, x (r)) dw (r) 0
−1 + Γt0 S ∗ (T − t) αI + ΓT0 (Eh − S (T ) x0 ) Z t −1 (S(T − r)σ (r, x (r)) − ϕ (r)) dw (r) − Γtr S ∗ (T − t) αI + ΓTr 0 Z t −1 S (T − r) f (r, x (r)) dr − Γtr S ∗ (T − t) αI + ΓTr
(11)
0
If α = 0 the nonlinear operator F0 is defined by Z
t
(F0 x) (t) = S(t)x0 + S (t − r) f (r, x (r))dr 0 Z t S (t − r) σ (r, x (r)) dw (r) + 0
−1 + Γt0 S ∗ (T − t) ΓT0 (Eh − S (T ) x0 ) Z t −1 Γtr S ∗ (T − t) ΓTr S (T − r) f (r, x (r)) dr − 0 Z t −1 − Γtr S ∗ (T − t) ΓTr (S(T − r)σ (r, x (r)) − ϕ (r)) dw (r)
(12)
0
Theorem 6 Assume that the conditions (H1) and (H2) hold, then for each α > 0 the operator Fα has a fixed point. Proof. As mentioned above, showing the fixed point of the nonlinear operator Fα in H2 will confirm the approximate controllability. To do this, the contraction mapping principle is used. This is achieved by firstly showing that Fα maps H2 into itself. E k(Fα x) (t)k2 ≤ 6M1 kx0 k2 + 6M1 L (t + 1) 6 2 2 M M 2 kEhk + 2M kx k 1 2 1 0 α2 Z T 6 2 + 2 M1 M2 LM1 (t + 2) + 2 E kϕ (r)k dr α 0 +
10
for all t ∈ [0, T ] . Therefore Fα maps H2 into itself. Secondly, Fnα is a contraction mapping. That is, for each α > 0 there exists C (α) > 0 such that E k(Fα x1 )(t) − (Fα x2 )(t)k2 Z t = M1 t E kf (s, x1 (s)) − f (s, x2 (s))k2 ds Z t0 + M1 E kσ(s, x1 (s)) − σ(s, x2 (s))k2 ds 0 Z t 1 + 2 M1 M2 t E kf (s, x1 (s)) − f (s, x2 (s))k2 ds α 0 Z t 2 + E kσ(s, x1 (s)) − σ(s, x2 (s))k ds 0 Z t 1 ≤ M1 (t + 1) L + 2 M1 M2 (t + 1) L E kx1 (s) − x2 (s)k2 ds α 0 Z t ≤ C (α) E kx1 (s) − x2 (s)k2 ds 0
≤ C (α) t kx1 − x2 k2H2 . and
2 E (F2α x1 )(t) − (F2α x2 )(t) Z t ≤ C (α) E k(Fα x1 )(s) − (Fα x2 )(s)k2 ds Z0 t Z s ≤ C 2 (α) E kx1 (r) − x2 (r)k2 drds 0 0 Z t 2 2 ≤ C (α) sup E kx1 (s) − x2 (s)k s ds t∈[0,T ]
0
2
= C 2 (α)
t kx1 − x2 k2H2 2!
11
Evidently,
2 E (F3α x1 )(t) − (F3α x2 )(t) Z t
2
≤ C (α) E (F2α x1 )(s) − (F2α x2 )(s) ds Z0 t Z s ≤ C 2 (α) E k(Fα x1 ) (r) − (Fα x2 ) (r)k2 drds Z0 t Z0 s Z r ≤ C 3 (α) E kx1 (τ ) − x2 (τ )k2 dτ drds 0
0
0
t3 ≤ C 3 (α) kx1 − x2 k2H2 3! In a similar way, it can be shown that
sup
E k(Fnα x1 )(t)
−
(Fnα x2 )(t)k2
t∈[0,T ]
For sufficiently large n, C
n (α)T n
n!
Tn ≤ C (α) kx1 − x2 k2H2 . n! n
< 1. This results that Fnα is a contraction mapping for
sufficiently large n. Then the mapping Fα has a unique fixed point x(·) in H2 which is the solution of the equation (9). The theorem is proved.
Theorem 7 Assume hypotheses (H1) , (H2) , (H4) , (H5) and (H6) hold. Then the operator F0 has a fixed point.
Proof. The proof is similar to that of Theorem 6. Using Lemma 4 and some basic assumptions, it is shown that F0 maps H2 into itself, that is E k(F0 x) (t)k2 ≤ 6M1 kx0 k2 + 6M1 L (t + 1) +
6 2 γT,0
M1 M2 2 kEhk2 + 2M1 kx0 k2
Z
T 2
E kϕ (r)k dr .
+ 6C M1 L (t + 2) + 2 0
12
Fn0 is a contraction mapping. By Lemma 4, E k(F0 x1 )(t) − (F0 x2 )(t)k2 Z t = M1 t E kf (s, x1 (s)) − f (s, x2 (s))k2 ds Z t0 + M1 E kσ(s, x1 (s)) − σ(s, x2 (s))k2 ds 0Z t +C t E kf (s, x1 (s)) − f (s, x2 (s))k2 ds 0 Z t 2 + E kσ(s, x1 (s)) − σ(s, x2 (s))k ds 0 Z t ≤ {M1 (t + 1) L + C} E kx1 (s) − x2 (s)k2 ds 0 Z t ≤ C2 E kx1 (s) − x2 (s)k2 ds 0
≤ C2 kx1 − x2 k2H2 . and
2
E (F20 x1 )(t) − (F20 x2 )(t) Z t ≤ C2 E k(F0 x1 )(s) − (F0 x2 )(s)k2 ds Z0 t Z s ≤ C22 E kx1 (r) − x2 (r)k2 drds 0 0 Z t 2 2 ≤ C2 sup E kx1 (s) − x2 (s)k s ds t∈[0,T ]
0
2
= C22
t kx1 − x2 k2H2 2!
Furthermore,
2 E (F30 x1 )(t) − (F30 x2 )(t) Z t
2
≤ C2 E (F20 x1 )(s) − (F20 x2 )(s) ds Z0 t Z s ≤ C22 E k(F0 x1 ) (r) − (F0 x2 ) (r)k2 drds Z0 t Z0 s Z r ≤ C23 E kx1 (τ ) − x2 (τ )k2 dτ drds 0
0
0
t3 ≤ C23 kx1 − x2 k2H2 . 3! 13
Similarly, sup E k(Fn0 x1 )(t) − (Fn0 x2 )(t)k2 ≤ C2n t∈[0,T ]
For sufficiently large n,
C2n T n n!
Tn kx1 − x2 k2H2 . n!
< 1 and by the above inequality Fn0 is a contraction mapping.
Thus thanks to the Contraction Mapping Principle that F0 has a fixed point. The theorem is proved.
Theorem 8 Assume hypotheses (H1) , (H2) and (H3) are satisfied. Then the system (9) is approximately controllable. Proof. Let xα (·) be a fixed point of Fα in H2 . By Lemma 3, any fixed point of Fα is a solution of (9) and satisfies −1 (Eh − S(T )x0 ) xα (T ) = h − α αI + ΓT0 Z T −1 + α αI + ΓTr S(T − r)f (r, xα (r))dr 0 Z T −1 + α αI + ΓTr (S(T − r)σ(r, xα (r)) − ϕ (r)) dw(r) 0
14
(13)
By (13) and the assumption (H2) ,
2 E kxα (T ) − hk2 ≤ 3 αR α, ΓT0 (Eh − S (T ) x0 ) Z T
2 + 3T E αR α, ΓTr S (T − r) f (r, xα (r)) dr 0 Z T
2 +6 E αR α, ΓTr S (T − r) σ (r, xα (r)) dr 0 Z T
2 +6 E αR α, ΓTr ϕ (r) dr 0
2 ≤ 3 αR α, ΓT0 kEh − S (T ) x0 k2 Z T
αR α, ΓTr 2 E kS (T − r) f (r, xα (r))k2 dr + 3T 0 Z T
αR α, ΓTr 2 E kS (T − r) σ (r, xα (r))k2 dr +6 0 Z T
αR α, ΓTr 2 E kϕ (r)k2 dr +6 0
2 ≤ 3 αR α, ΓT0 kEh − S (T ) x0 k2 Z T
αR α, ΓTr 2 dr + 3M1 L (T + 2) 0 Z T
αR α, ΓTr 2 E kϕ (r)k2 dr +6 0
2 2 Since αR α, ΓTr ≤ 1, αR α, ΓTr → 0 as α → 0+ for all 0 ≤ r < T, by the Lebesque dominated convergence theorem E kxα (T ) − hk2 → 0 as α → 0+ . This gives the approximate controllability. Theorem 9 Assume hypotheses (H1) , (H2) , (H4), (H5) and (H6) are satisfied. Then the system (2) is completely controllable. Proof. By Theorem 7, the operator F0 has a fixed point. So, the control −1 u0 (t) = B ∗ S ∗ (T − t) ΓT0 (Eh − S(T )x0 ) Z t −1 − B ∗ S ∗ (T − t) ΓTr S(T − r)f (r, x (r))dr 0 Z t −1 ∗ ∗ − B S (T − t) ΓTr (S(T − r)σ(r, x (r)) − ϕ (r)) dw(r) 0
15
transfers the system (2) from x0 to h. The theorem is proved.
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