5.5.4. Observability

Consider the following time varying stochastic singular equation

$$O\_1 O\_1 v(t) = O\_2(t) v(t) dt + O\_4(t) v(t) dw(t), \\ t \ge 0, \\ v(0) = v\_0, \\ x(t) = O\_5(t) v(t), \tag{70}$$

and its dual time varying stochastic singular equation

$$O\_1^\* dv^\*(t) = O\_2^\*(t)v^\*(t)dt + O\_5^\*(t)u^\*(t)dt + O\_4^\*(t)v^\*(t)dw(t), t \ge 0,\\ v^\*(0) = v\_0^\*.\tag{71}$$

For the time varying stochastic singular Equation (70), the following concepts are defined.

The observability operator of time varying stochastic singular Equation (70) on [0, *b*] is the continuous linear operator

$$\mathcal{Q}\_{\mathcal{O}}^{T}: L^{2}(\Omega, F\_{\mathcal{b}}, P, \widetilde{D}) \to L^{2}([0, b], \Omega, \mathcal{Y}\_{3})$$

defined by *Q<sup>b</sup> <sup>O</sup>y* = *O*5(*t*)*G*(*b*, *t*)*E*(*y*|*Ft*), its dual operator

$$Q\_O^{b\*}: L^2([0, b], \Omega, Y\_3^\*) \to L^2(\Omega, F\_{b'}P\_\prime(\overline{D})^\*)$$

is defined by

$$Q\_O^{b\*} \mathfrak{x}^\* = \int\_0^b G^\*(b, t) O\_\mathbb{S}^\*(t) \mathfrak{x}^\*(t) dt.$$

**Definition 36.** *Time varying stochastic singular Equation (70) is said to be exactly observable on* [0, *b*] *if Q<sup>b</sup> <sup>O</sup> is injective and its inverse is bounded on* ran(*Q<sup>b</sup> O*)*.*

In the case of Definition 36, the state *v*<sup>0</sup> can be uniquely and continuously constructed from the knowledge of the output *x*(*t*) in *L*2([0, *b*], Ω,*Y*3).

**Definition 37.** *Time varying stochastic singular Equation (70) is said to be approximately observable on* [0, *b*] *if Q<sup>b</sup> <sup>O</sup> is injective.*

In the case of Definition 37, the state *v*<sup>0</sup> can be uniquely constructed from the knowledge of the output *x*(*t*) in *L*2([0, *b*], Ω,*Y*3).

We can obtain the following dual principle.

**Theorem 54.** *Assume that Y*<sup>1</sup> *and Y*<sup>3</sup> *are reflexive. Time varying stochastic singular Equation (70) is exactly (approximately) observable on* [0, *b*] *if, and only if, its dual time varying stochastic singular Equation (71) is exactly (approximately) controllable on* [0, *b*]*.*

**Proof.** Here, we only prove the case of exact observability. Since

$$Q\_O^{b\*} \mathfrak{x}^\* = \int\_0^b G^\*(b, t) O\_5^\*(t) \mathfrak{x}^\*(t) dt$$

happens to be the controllability operator *Q<sup>b</sup> <sup>C</sup>* of time varying stochastic singular Equation (71), so *Qb*<sup>∗</sup> *<sup>C</sup>* = *<sup>Q</sup><sup>b</sup> O*.

If the time varying stochastic singular Equation (70) is exactly observable, then there exists 1/*γ* > 0, such that

$$\| (Q\_O^b)^{-1} \mathfrak{x} \|\_{L^2(\Omega, F\_{b\ast}P, \mathfrak{D})} \le \frac{1}{\gamma} \| \mathfrak{x} \|\_{L^2([0, b]; \Omega, \mathbb{Y}^3)^{\prime\prime}}$$

for all *<sup>x</sup>* <sup>∈</sup> ran(*Q<sup>b</sup> <sup>O</sup>*). This implies that

$$\gamma \|\|\boldsymbol{y}\|\|\_{L^{2}(\Omega,\mathbb{F}\_{b},\mathcal{P},\mathcal{D})} = \gamma \|\| (\mathcal{Q}\_{O}^{b})^{-1} \mathcal{Q}\_{O}^{b} \boldsymbol{y}\|\|\_{L^{2}(\Omega,\mathbb{F}\_{b},\mathcal{P},\mathcal{D})}$$

$$\leq \|\|\mathcal{Q}\_{O}^{b} \boldsymbol{y}\|\|\_{L^{2}([0,b],\Omega,\mathcal{Y}\_{3})} = \|\|\mathcal{Q}\_{C}^{b\*} \boldsymbol{y}\|\|\_{L^{2}([0,b],\Omega,\mathcal{Y}\_{3})}$$

where

$$y = (Q\_O^b)^{-1} \mathfrak{x}, y \in L^2(\Omega, F\_{\mathfrak{b}'} P, \overline{D}).$$

According to Theorem 52 (a), we have that (71) is exactly controllable.

Assume next that the time varying stochastic singular Equation (71) is exactly controllable. From Theorem 52 (b), we have that *Q<sup>b</sup> <sup>O</sup>* is injective and has closed range. According to closed graph theorem (*Q<sup>b</sup> O*)−<sup>1</sup> is bounded on ran*Q<sup>T</sup> O*.

Theorems 52 and Definitions 36 and 37 yield the following conditions for observability of time varying stochastic singular Equation (70).

**Corollary 2.** *Time varying stochastic singular Equation (70) is exactly observable on* [0, *b*] *if, and only if, one of the following conditions holds for some <sup>γ</sup>* <sup>&</sup>gt; <sup>0</sup> *and for all y* <sup>∈</sup> *<sup>L</sup>*2(Ω, *Fb*, *<sup>P</sup>*, *<sup>D</sup>*)*:*

*(a)* -*Qb Oy*-*<sup>L</sup>*2([0,*b*],Ω,*Y*3) ≥ *γy*-*<sup>L</sup>*2(Ω,*Fb*,*P*,*D*)*; (b)* ker(*Q<sup>b</sup> <sup>O</sup>*) = {0} *and* ran(*Q<sup>b</sup> <sup>O</sup>*) *is closed.*

**Corollary 3.** *Time varying stochastic singular Equation (70) is approximately observable on* [0, *b*] *if, and only if,* ker(*Q<sup>b</sup> <sup>O</sup>*) = {0}*.*

5.5.5. An Illustrative Example

In this part, we give an example to illustrate the effectiveness of the obtained results. According to [72], in input–output economics, many models were established to describe the real economics. The economics Leontief dynamic input–output model can be extended as an ordinary differential equation of the form:

$$O\_1 \frac{dv(t)}{dt} = O\_2(t)v(t) + O\_3(t)u(t), \\ x(t) = O\_5(t)v(t) \tag{72}$$

in Banach space *Y*1, where *O*<sup>1</sup> ∈ *B*(*Y*1) and *O*2(*t*) : dom(*O*2(*t*)) ⊆ *Y*<sup>1</sup> → *Y*<sup>1</sup> is a linear and possibly unbounded operator, *O*3(*t*),*O*5(*t*) ∈ *P*([0, *b*], *B*(*Y*1)), while *v*(*t*) and *u*(*t*) are state process and control process valued in *Y*1, respectively, for *t* ≥ 0. However, in reality, there are many unpredicted parameters and different types of uncertainty that have not been implemented in the mathematical modelling process of this equation. Nonetheless, according to [85,86], we can consider a stochastic version of the singular Equation (72) with the one-dimensional standard Wiener process *w*(*t*) used to model the uncertainties of the form:

$$\mathcal{O}\_1 dv(t) = \mathcal{O}\_2(t)v(t)dt + \mathcal{O}\_3(t)u(t)dt + \mathcal{O}\_4(t)v(t)dw(t), \\ x(t) = \mathcal{O}\_5(t)v(t), \tag{73}$$

where *O*4(*t*) ∈ *P*([0, *b*], *B*(*Y*1)). This stochastic version of the input-output model is a time varying stochastic singular equation in Banach space *Y*<sup>1</sup> of the form (62).

We consider the following unforced time varying stochastic singular equation, i.e., *u*(*t*) = 0 in time varying stochastic singular Equation (73):

$$O\_1 dv(t) = O\_2(t)v(t)dt + O\_4(t)v(t)dw(t), \\ x(t) = O\_5(t)v(t). \tag{74}$$

Time varying stochastic singular Equation (74) is the form of time varying stochastic singular linear Equation (70). In what follows, we will verify the effectiveness of Corollary 3.

If for some concrete engineering practice, the following data are taken in time varying stochastic singular Equation (74):

$$O\_1 = \begin{bmatrix} \mathcal{U}\_1 & 0\\ 0 & 0 \end{bmatrix}, O\_2(t) = \begin{bmatrix} -(2t+1)\mathcal{U}\_1 & 0\\ 0 & 5(t^2+1)\mathcal{U}\_2 \end{bmatrix},$$

$$O\_4(t) = \begin{bmatrix} (2t)^{1/2}\mathcal{U}\_1 & 0\\ 0 & 3t^2\mathcal{U}\_2 \end{bmatrix}, O\_5(t) = \begin{bmatrix} \mathcal{T}(t+1)^2\mathcal{U}\_1 & 0\\ 0 & 0 \end{bmatrix}.$$

where *U*1, *U*<sup>2</sup> are identical operators in Banach spaces *Y*11,*Y*12, respectively. Time varying stochastic singular Equation (74) can be written as

$$
\begin{bmatrix}
\mathcal{U}\_1 & 0 \\
0 & 0
\end{bmatrix}
\begin{bmatrix}
dv\_1(t) \\ \upsilon\_2(t)
\end{bmatrix} = 
\begin{bmatrix}
0 & 5(t^2+1)\mathcal{U}\_2
\end{bmatrix}
\begin{bmatrix}
v\_1(t)dt \\ \upsilon\_2(t)dt
\end{bmatrix}
$$

$$
+
\begin{bmatrix}
(2t)^{1/2}\mathcal{U}\_1 & 0 \\
0 & 3t^2\mathcal{U}\_2
\end{bmatrix}
\begin{bmatrix}
v\_1(t) \\ \upsilon\_2(t)
\end{bmatrix} dw(t),
$$

$$
\mathbf{x}(t) = \begin{bmatrix}
\mathcal{T}(t+1)^2\mathcal{U}\_1 & 0 \\
0 & 0
\end{bmatrix}
\begin{bmatrix}
v\_1(t) \\ \upsilon\_2(t)
\end{bmatrix}
\tag{75}
$$

where & *<sup>v</sup>*1(*t*) *v*2(*t*) ' ∈ *Y*<sup>11</sup> ⊕ *Y*<sup>12</sup> = *Y*1. We can find that *D* = *Y*11. According to [87], we can obtain

$$G(t,s) = \begin{bmatrix} \exp[-\frac{3}{2}t^2 - t + \frac{3}{2}s^2 + s + \int\_s^t (2r)^{1/2} w(r)ds] \mathcal{U}\_1 & 0\\ 0 & 0 \end{bmatrix}$$

.

It is obvious that time varying stochastic singular Equation (75) satisfies the conditions of Lemma 3. If & *<sup>y</sup>* 0 ' <sup>∈</sup> *<sup>L</sup>*2(Ω, *Fb*, *<sup>P</sup>*, *<sup>D</sup>*), and

$$Q\_O^b \begin{bmatrix} \ y \\ 0 \end{bmatrix} = O\_5(t) G(b, t) E(\begin{bmatrix} y \\ 0 \end{bmatrix} | F\_t) = 0, t \in [0, b]\_\prime$$

then

$$(O\_5(b)G(b,b)E(\begin{bmatrix} \ y \\ 0 \end{bmatrix} | F\_b) = \mathcal{T}(b+1)^2 \begin{bmatrix} \ y \\ 0 \end{bmatrix} = 0,$$

i.e., *y* = 0. This implies that ker(*Q<sup>b</sup> <sup>O</sup>*) = {0}. Therefore time varying stochastic singular Equation (75) is approximately observable by Corollary 3.

In this section, we have discussed the controllability of some types of stochastic singular linear systems. However, the following problems still need to be studied.

**Problem 9.** *How about the controllability of the following system?*

$$L dx(t) = [A(t)x(t) + B(t)u(t)]dt + \sum\_{k=1}^{\infty} \mathcal{C}\_k(t)x(t)dw\_{1,k}(t)$$

$$+ \sum\_{j=1}^{\infty} D\_j(t)u(t)dw\_{2,j}(t), x(0) = x\_{0,i}$$

*where L* ∈ *B*(*H*) *and* ker(*L*) = {0}*, A*(*t*) : dom(*A*(*t*)) ⊆ *H* → *H is the generator of a GE-evolution operator induced by L in the Hilbert (or Banach) space H, B*(*t*) : dom(*B*(*t*)) ⊂ *U* → *H is a linear operator, U is a Hilbert (or Banach) space; Ck*(*t*) ∈ *P*([0, *b*], *B*(*H*)), *Dk*(*t*) ∈ *<sup>P</sup>*([0, *<sup>b</sup>*], *<sup>B</sup>*(*U*, *<sup>H</sup>*)), *for each i* <sup>∈</sup> <sup>N</sup>*; and in Hilbert spaces,*

$$\sum\_{k=1}^{\infty} \sup\_{0 \le t \le b} \|\mathbb{C}\_k(t)\|\_{B(H)}^2 < +\infty, \sum\_{k=1}^{\infty} \sup\_{0 \le t \le b} \|D\_k(t)\|\_{B(\mathcal{U},H)}^2 < +\infty;$$

*in Banach spaces,*

$$\sum\_{k=1}^{\infty} \sup\_{0 \le t \le b} \|C\_k(t)\|\_{B(H)} < +\infty, \sum\_{k=1}^{\infty} \sup\_{0 \le t \le b} \|D\_k(t)\|\_{B(\mathcal{U},H)} < +\infty;$$

*the countable set* {*w*1,*k*, *<sup>w</sup>*2,*j*, *<sup>k</sup>*, *<sup>j</sup>* <sup>∈</sup> <sup>N</sup>} *consists of independent standard Wiener processes defined on the stochastic basis* (Ω, *F*, {*Ft*}, *P*).
