1. Introduction
Stochastic differential equations have been successfully used in recent years in many applied problems in physics, economics, electricity, mechanics, etc. Many real systems and biological procedures exhibit some form of dynamic action under random perturbation, with continuous and discrete properties. In the last few decades, controllability concepts (approximate/exact approximate/finite-approximate controllability and so on) for different types of stochastic semilinear evolutionary systems have been studied in many articles using various methods. We divide scientific articles devoted to stochastic controllability concepts into groups as follows.
Linear stochastic systems: Approximate controllability notions for stochastic linear systems were studied in [
1,
2,
3,
4,
5,
6,
7,
8,
9,
10]. In [
1,
2], stochastic Ljapunov methods are used to give sufficient conditions for these types of stochastic observability and controllability. In [
3,
4], the authors study the controllability of linear dynamical systems in the presence of random perturbations. In [
7], with the help of dual equations the duality between approximate controllability and observability is deduced. In [
8,
9], necessary and sufficient conditions, in terms of uniform and strong convergence of a certain sequence of operators involving the resolvent of the negative of the controllability operator, are formulated.
Semilinear stochastic systems: Studies on the approximate controllability concepts of semilinear/nonlinear stochastic systems have progressed slowly as compared to linear stochastic systems, see [
11,
12,
13,
14,
15,
16,
17,
18,
19,
20]. There are several approaches: a resolvent approach applied together with fixed point methods, integral contractor, sequencing method and the monotone technique. Several researchers—Sunahara et al. [
11,
12], Mahmudov [
9], George [
13], Sakthivel and Kim [
14], Tand and Zhang [
15], Mokkedem and Fu [
21], Ain et al. [
22], Anguraj and Ramkumar [
23]—have used different methods to study approximate controllability for several stochastic evolution systems.
Non-Lipschitz stochastic systems: Approximate controllability of non-Lipschitz stochastic systems was considered in Sing et al. [
24], Ren et al. [
25], Mahmudov et al. [
26].
Finite-approximate controllability: Simultaneous mean square approximate and finite-dimensional mean exact controllability, referred to as the finite-approximate mean square controllability of linear/semilinear stochastic systems in infinite-dimensional spaces, is studied in [
10,
27].
As far as we know, no attempts have been made to study the analogue of mean square finite-approximate controllability for linear stochastic evolution systems as well as for semilinear stochastic evolution systems with non-Lipschitz coefficients. In contrast, approximate controllability problems for the mean square finite-approximate controllability for linear/semilinear stochastic systems investigated in this manuscript have not been tackled in the existing literature. This study explores the mean square approximate controllability for linear/semilinear stochastic systems with non-Lipschitz drift and diffusion coefficients and fills this gap in the literature.
Therefore, motivated by the above discussions, we study the mean square finite-approximate controllability of the following stochastic differential equation:
Here, is a Hilbert space, is the state process, is the control process, is a Hilbert space, is an infinitesimal generator of -semigroup, is a linear continuous operator, are functions to be defined later.
We introduce mean square finite-approximate controllability for Equation (
1).
Definition 1. Let M be a subspace of with finite-dimension. is the orthogonal projection operator. System (1) is said to be mean square approximately controllable if for a given and , there exists a control process such that the solution to (1) satisfies Definition 2. Let M be a subspace of with finite-dimension. is the orthogonal projection operator. System (1) is said to be exact mean finite-dimensional controllable if for a given , there exists a control process such that the solution to (1) satisfies Definition 3. ([
10]).
Let M be a subspace of with finite-dimension. is the orthogonal projection operator. System (1) is said to be mean square finite-approximately controllable if for a given and , there exists a control process such that the solution to (1) satisfies Simultaneous exact mean finite-dimensional and approximate mean square controllability is referred to as mean square finite-approximate controllability. A control process
can be selected such that
satisfies (
2) and a finite number of constrains (
3). It is clear that mean square finite-approximate controllability implies both exact mean finite-dimensional and approximate mean square controllability. However, the converse is not obvious.
The following are the main contributions of the paper.
- (i)
We introduce and study the simultaneous mean exact finite-dimensional and approximate mean square controllability (mean square finite-approximate controllability) concept for the linear/semilinear infinite-dimensional stochastic systems.
- (ii)
We prove that the finite-approximate mean square controllability of the stochastic linear system (
4) is equivalent to the mean square approximate controllability of the system (
4). We give an explicit analytical form of the control that provides finite-dimensional mean square controllability of the linear stochastic system (
1) in terms of stochastic resolvent-like operators.
- (iii)
We present sufficient conditions for the mean square finite-dimensional controllability semilinear stochastic differential systems in infinite dimensional Hilbert spaces. We prove that mean square approximate controllability of the linear part of the stochastic system implies the mean square finite-approximate controllability of the semilinear stochastic differential equation with non-Lipschitz coefficients. Our results are new even for the semilinear stochastic differential equation with Lipschitz coefficients.
The following is how the rest of this paper is structured: In
Section 2, we provide some fundamental notations and definitions, as well as some relevant assumptions. In
Section 3, we show that for a linear stochastic evolution system (
5) approximate mean square controllability on
is equivalent to finite-approximate controllability in the mean square sense on
. Necessary and sufficient conditions are given for a finite-approximate mean square controllability concept of linear stochastic evolutionary systems in Hilbert spaces in terms of stochastic resolvent-like operators. In addition, we find an explicit form of the finitely approximating control in terms of the stochastic resolvent-like operator
. In
Section 4, by applying the Picard approximation method, we establish sufficient conditions for the mean square finite-dimensional controllability of (
1). Finally, to illustrate the theoretical findings, we provide numerical examples.
2. Preliminaries
We give notations and some preliminary results needed to present our principal results.
For any pair and of separable real Hilbert spaces, we denote by the space of bounded (continuous) linear operators from to .
is a normal filtration, is a probability space.
is a Wiener process on
. The covariance operator
with tr
satisfies the following assumption: there exists a basis
in
K, a bounded sequence of positive real numbers
such that
and a sequence of independent Brownian motions
such that
and
, where
is the sigma algebra generated by
and are separable Hilbert spaces.
is the space of all Hilbert–Schmidt operators with the inner product tr.
is the (Hilbert) space of all -measurable square integrable functions .
is the Hilbert space of all square integrable and -adapted processes .
is the Banach space of continuous maps from into satisfying the condition .
is a closed subspace consisting of measurable and -adapted -valued processes endowed with the norm .
is a
-semigroup generated by
and
such that
To formulate and prove our main results, we require the following assumptions.
(H1). is a function that satisfies the following conditions:
- (a)
The function is measurable strongly for all and the function is continuous in for each
- (b)
The function is measurable strongly for all and the function is continuous in for each
- (c)
For any
and
, there exist non-decreasing functions
such that
(H2). The functions
and
are continuous in
p for each fixed
and locally integrable in
for each fixed
. Moreover, the integral equation
admits a solution for all
and
(H3). There exist non-decreasing functions
such that for all
and
(H4). The functions
are continuous in
p for each fixed
and locally integrable with
. Moreover, if the inequality
is satisfied by a nonnegative continuous function
for
subject to
for some
, then
for all
- (AC)
The stochastic linear system
is mean square approximately controllable on
Here
Remark 1. (i) If , then the functions in the assumption (H) become the Lipschitz functions.
- (ii)
If are concave and for all then the Jensen inequality implies (H). - (iii)
For some concrete examples, see [25].
We present the following definition of mild solutions to (
1).
Definition 4. ([
28]).
Stochastic process is said to be a mild solution of (1) if for any it satisfies the following stochastic integral equation 3. Linear Systems: Finite-Approximate Controllability
In this section, we study the mean square finite-approximate controllability of the stochastic linear evolution system:
The continuous linear operator
defined by
is called a controllability operator. Its adjoint is defined by
The controllability Gramian operator is defined by
The resolvent operator
is known to be useful in studying the approximate/exact controllability properties of linear and semilinear deterministic/stochastic evolution systems, see [
1,
6]. In this regard, a new criterion for finite-approximation controllability of a linear stochastic evolutionary system (
5) is formulated in terms of a resolvent-like operator
. We show that for a linear stochastic evolution system (
5) approximate mean square controllability on
is equivalent to finite-approximate controllability in the mean square sense on
. Necessary and sufficient conditions are given for a finite-approximate mean square controllability concept of linear stochastic evolutionary systems in Hilbert spaces in terms of stochastic resolvent-like operators. In addition, we find an explicit form of the finitely approximating control in terms of the stochastic resolvent-like operator
.
The following two types of operators:
Operator is said to be nonnegative if for all
Operator is said to be positive if for all with
Firstly, we present two properties on the resolvent operator .
Lemma 1. Assume that is a linear positive operator. Then
- (a)
For any , we have
- (b)
is continuous in ε and
Proof. It is clear that
maps
into a finite-dimensional subspace of
and
To show that
, in contrast, suppose that there exists a sequence
such that
It follows from Equation (
6) that
is a sequence of finite-dimensional vectors and
Taking the limit as
we obtain
Now, from Equation (
7), it follows that
as
, which is a contradiction. The lemma is proved. □
The next lemma establishes a connection between the stochastic resolvent operator and the stochastic resolvent-like operator
Lemma 2. If is a non-negative linear operator, then the operator is invertible andwhere . Moreover, if is a linear positive operator then Proof. We write
as follows.
It is clear that
It follows that
is invertible and inequality (
8) is satisfied. If
is a positive linear operator then by Lemma 1,
exists. On the other hand, since
is invertible and
the operator
is boundedly invertible and (
9) is satisfied. □
Next, we present new criteria for the mean square finite-approximate controllability of linear stochastic system (
5).
Theorem 1. The following assertions are equivalent:
- (i)
Linear stochastic system (5) is mean square approximately controllable on - (ii)
is positive;
- (iii)
For any , we have as ;
- (iv)
For any , we have as
- (v)
Linear stochastic system (5) is mean square finite-approximately controllable on .
Proof. We show that (i)⟺(ii). By definition, system (
5) is approximately controllable if
is dense in
. Then, we know that
Moreover
It follows that
We show that (ii)⟺(iii).
Suppose (iii) fails. Then, for some
, we have
Set
. Then,
and taking the limit of both sides, we obtain
for nonzero
z, which contradicts the positivity of
Now, assuming that (ii) fails, for some nonzero
, we have
It follows that
which leads to a contradiction.
To prove the implication (iii)⟹(iv), suppose that
It follows from (
9) that for any
On the other hand, from
it follows that
is continuous in
Indeed,
By (
10), the continuity of
and Lemma 1, we have
Thus,
converges to zero as
in the strong operator topology.
To prove the equivalence (iii)⟺(v), we take any
, and consider the functional
defined as follows:
Suppose that (iii) (⇔(ii)) is satisfied. It is obvious that
is Gateaux differentiable and
is strictly monotonic. The positivity of
implies that the functional
is strictly convex. Thus,
has a unique minimum and can be calculated as follows:
For the control
Since (iii)⇒(iv), we have
That is, system (
5) is finite-approximately mean square controllable. Thus, (iii) implies (v). The implication (v) ⇒ (iii) is obvious, since mean square finite-approximate controllability implies the mean square approximate controllability. (iv)⇒(v) follows from equality (
11). Thus, we have
□
Theorem 2. The (deterministic) systemis approximately controllable on every if and only if the linear stochastic system (5) is (mean square) approximate controllable on Proof. Suppose that the deterministic system (
12) is approximately controllable on every
Then, it is known that
is positive. On the other hand, by the martingale representation theorem, for any
, there exists a stochastic process
such that
see, for example, [
9]. Using the above representation, we can write
in terms of matrix
Therefore, for any nonzero
Thus, the positivity of the operator
is equivalent to the positivity of
,
Therefore, by Theorem 1, the stochastic linear system (
5) is approximately mean square controllable on
if and only if the deterministic counterpart (
12) is approximately controllable on any
. □
4. Semilinear Systems: Mean Square Finite-Approximate Controllability
The proof of the main result of this section is based on the Picard approximation method. To apply the Picard method, for any
we introduce the non-linear operator
→
which is defined as follows
where
and
comes from the representation
of
Lemma 3. Under assumptions (H)–(H), the operator is well defined and there exist positive numbers such that for then Proof. Firstly, we estimate
as follows.
Next, we estimate
Inserting inequality (
14) into Equation (
13), we obtain
where
□
Lemma 4. Under assumptions (H)–(H), the operator is well defined and there exist positive numbers such that for , then Proof. It is clear that
Firstly, we estimate
as follows.
Using assumption (H
), we obtain
where
Next, we estimate
Combining inequalities (
15) and (
16), we obtain
□
Lemma 5. Under assumptions (H)–(H), the sequence is bounded in
Proof. By Lemma 3, for any
, we have
where
,
are constants independent of
n. Let
be a global solution of the equation
with an initial condition
We will prove by mathematical induction that
For
inequality (
17) holds by definition of
p. Suppose that
Then, by inequality (
18) we obtain that
It follows that
is bounded in
□
Lemma 6. Under assumptions (H)–(H), the sequence is a Cauchy sequence in .
Proof. Define
The functions
,
,
are well defined, uniformly bounded and evidently nondecreasing. Then, there exist nondecreasing functions
,
,
such that
By Lemma 4, we obtain that
from which in turn it follows that
By the Lebesgue dominated convergence theorem, we obtain
If follows that
By the Bihari inequality, it follows that
. However,
Therefore
□
Theorem 3. Under assumptions (H)–(H), the operator has a unique fixed point in .
Proof. By Lemma 6,
is a Cauchy sequence in
. The completeness of
implies the existence of a process
such that
Taking the limit
we see that
is a fixed point of
.
Further, if
are two fixed points of
, then Lemma 4 would imply that
So, as in the proof of Lemma 6, we obtain that
Therefore,
and
has a unique fixed point in
. □
Theorem 4. Let assumptions (H)–(H) and (AC) hold. Assume that the operator , is compact and analytic. Moreover, suppose the functions and are uniformly bounded. Then, system (1) is mean square finite-approximately controllable on . Proof. Let
be a fixed point of
in
. Then
Since the functions
and
are uniformly bounded, there exists a constant
such that
Then, there exists a subsequence still denoted by
which converges weakly to say
. Now, due to compactness,
it follows that
in
From equation (
20), we have
On the other hand,
strongly as
and
. Therefore, by the Lebesgue dominated convergence theorem, we can easily obtain that
as
This implies the approximate controllability in the mean square of system (
1). Mean exact finite-dimensional controllability follows from Equation (
20):
□
5. Applications
Example 1. We consider a system governed by the semilinear heat equation with lumped control
where
is the characteristic function of
. Let
,
and
with
We define the bounded linear operator
by
and the nonlinear operator
f is assumed to be bounded.
Set
and denote by
the operator of the orthogonal projection
onto
.
generates a compact analytic semigroup
which is defined as follows.
where
,
is a complete orthonormal set of eigen vectors of
. Subsequently, we attain
It is clear that
as
if
which holds whenever
is an irrational number.
If
is an irrational number, then the linear determinisitic system (
21) is finite-approximately controllable on every
By Theorem 2, the following linear stochastic system is mean square finite-approximately controllable on
where
denotes a standard real valued Wiener process,
.
Example 2. Consider the following stochastic partial differential equation:
where
denotes a standard real valued Wiener process on
and
;
is continuous in
;
are continuous. Let
and define the operator
with
Then,
generates a compact analytic semigroup
which is defined as follows
where
,
is a complete orthonormal set of eigen vectors of
. From these expressions, it follows that
is a uniformly bounded compact analytic semigroup.
Define an infinite-dimensional control space
by
such that
endowed with the norm
. Next, define a continuous linear mapping
from
into
as follows
Let and define the bounded linear operator by and .
The linear deterministic system that corresponds to (
22) is approximately controllable on every
and all conditions of Theorem 4 are satisfied. Hence, by Theorem 4 the stochastic differential system (
22) is finite-approximately controllable on
.
6. Conclusions
The main aim of this work was to present:
Necessary and sufficient conditions for finite-approximate mean square controllability of linear stochastic evolution systems in infinite-dimensional separable Hilbert spaces in terms of stochastic resolvent-like operators . Moreover, we found an explicit analytical form of the contollability control which, in addition to the mean square approximate controllability property, ensures finite-dimensional mean exact controllability.
The Picard approximation method to show a mean square finite-approximate controllability of a semilinear stochastic evolution system under non-Lipschitz conditions satisfied by the nonlinear drift and diffusion coefficients depending on control.
One can assume that the results of this work apply to a class of problems determined by various types of first order and second order fractional (impulsive) stochastic evolution systems, such as Caputo SDEs, Riemann–Liouville-type SDEs, Hadamard-type SDEs, Sobolev-type fractional SDEs and so on.
On the other hand, many real-world systems can sometimes experience different types of stochastic perturbations. For example, Poisson jumps are now used to describe various types of real-world systems. In the future, the same approach could be used for different types of systems with different stochastic perturbations.