*2.2. Exact Controllability by Feedback Controller*

In 2018, Barbu and Tubaro consider the exact controllability by feedback controller of the following stochastic linear system in [60]:

$$d\mathbf{x}(t) + A(t)\mathbf{x}(t)dt = B(t)u(t)dt + \sum\_{k=1}^{d} \mathbb{C}\_{k} \mathbf{x}(t) dw\_{k}(t), \mathbf{x}(0) = \mathbf{x}\_{0} \tag{11}$$

with the final target *<sup>x</sup>*(*τ*) = *<sup>ξ</sup>*, where *<sup>A</sup>*(·), *<sup>B</sup>*(·) <sup>∈</sup> *<sup>C</sup>*([0, <sup>∞</sup>); <sup>R</sup>*n*×*m*); for some *<sup>γ</sup>* <sup>&</sup>gt; 0, *<sup>B</sup>*(*t*)*B*(*t*)*<sup>T</sup>* <sup>≥</sup> *<sup>γ</sup>*<sup>2</sup> *In*, <sup>∀</sup>*<sup>t</sup>* <sup>∈</sup> [0, <sup>∞</sup>); *Ck* <sup>∈</sup> <sup>R</sup>*n*×*n*;

*<sup>x</sup>*(·) <sup>∈</sup> *<sup>L</sup>*2([0, *<sup>τ</sup>*], <sup>Ω</sup>, <sup>R</sup>*n*), *<sup>u</sup>*(·) <sup>∈</sup> *<sup>L</sup>*2([0, *<sup>τ</sup>*], <sup>Ω</sup>, <sup>R</sup>*m*); *<sup>x</sup>*0, *<sup>ξ</sup>* <sup>∈</sup> <sup>R</sup>*n*.

The problem we address here is the following.

**Problem 2.** *Given <sup>x</sup>*0, *<sup>ξ</sup>* <sup>∈</sup> <sup>R</sup>*<sup>n</sup> find an Ft*−*adapted feedback controller <sup>u</sup>* <sup>=</sup> *<sup>f</sup>*(*x*) *and <sup>u</sup>* <sup>∈</sup> *<sup>L</sup>*2([0, *<sup>τ</sup>*], <sup>Ω</sup>, <sup>R</sup>*m*)*, such that the solution x*(*t*) *to system (11) satisfies x*(0) = *<sup>x</sup>*0, *<sup>x</sup>*(*τ*) = *<sup>ξ</sup>*.

Let *<sup>F</sup>* <sup>∈</sup> *<sup>C</sup>*([0, *<sup>τ</sup>*]; <sup>R</sup>*n*×*n*) be the solution to equation

$$dF(t) = \sum\_{k=1}^{d} \mathbb{C}\_k F(t) dw\_k(t), t \ge 0,\\ F(0) = I\_n.$$

By the substitution *x*(*t*) = *F*(*t*)*z*(*t*) one transforms via Ito's formula equation (see [60] for details) (11) into stochastic differential equation

$$\frac{dz(t)}{dt} + F(t)^{-1}A(t)F(t)z(t) = F(t)^{-1}B(t)u(t), \\ z(0) = x\_0. \tag{12}$$

In (12), we take as *u* the feedback controller

$$u(t) = -\bar{a}\text{sign}(F(t)^{-1}B(t))^T(z(t) - z\_\pi)), t \ge 0,\tag{13}$$

where *<sup>α</sup>*˜ <sup>∈</sup> *<sup>L</sup>*2(Ω, *FT*, *<sup>P</sup>*, <sup>R</sup>), *<sup>z</sup><sup>τ</sup>* <sup>∈</sup> *<sup>L</sup>*2(Ω, *FT*, *<sup>P</sup>*, <sup>R</sup>*n*) are given and *<sup>z</sup><sup>τ</sup>* <sup>=</sup> *<sup>F</sup>*(*τ*)−1*ξ*; sign : <sup>R</sup>*<sup>n</sup>* <sup>→</sup> R*<sup>n</sup>* is the multivalued mapping sign*y* = *<sup>y</sup> y*-<sup>R</sup>*<sup>n</sup>* if *<sup>y</sup>* <sup>=</sup> 0, sign*<sup>y</sup>* <sup>=</sup> {*<sup>β</sup>* <sup>∈</sup> <sup>R</sup>*<sup>n</sup>* : *β*-<sup>R</sup>*<sup>n</sup>* ≤ 1} if *y* = 0. Arguing as in the proof of Proposition 3.1 in [60], it follows that (12) has unique absolutely continuous solution *z*(*t*). We note that if *z*(*t*) is an *Ft*−adapted solution to (12) and (13) then *x*(*t*) = *F*(*t*)*z*(*t*) is the solution to closed loop system (11) with feedback control

$$\mu(t) = -\overline{\mathfrak{a}}\text{sign}((F(t)^{-1}B(t))^T F(t)^{-1}(\mathfrak{x}(t) - F(t)F(\tau)^{-1}\mathfrak{x}(\tau))).$$

The following results were obtained in [60].

**Theorem 8** ([60])**.** *Let <sup>τ</sup>* <sup>&</sup>gt; 0, *<sup>x</sup>*<sup>0</sup> <sup>∈</sup> <sup>R</sup>*<sup>n</sup> and <sup>ξ</sup>* <sup>∈</sup> *<sup>L</sup>*2(Ω, *<sup>F</sup>τ*, *<sup>P</sup>*, <sup>R</sup>*n*) *be arbitrary but fixed. Then there is <sup>α</sup>*˜ <sup>∈</sup> *<sup>L</sup>*2(Ω, *FT*, *<sup>P</sup>*, <sup>R</sup>)*, such that the controller (13) steers <sup>x</sup>*<sup>0</sup> *in <sup>z</sup>τ, in time <sup>τ</sup>, with probability one.*

**Remark 1.** *It should be noted that, under the assumption of the Theorem 8, the solution z*(*t*) *to (12) is not adapted. Therefore, the solution x*(*t*) = *F*(*t*)*z*(*t*) *to system (11) is not Ft*−*adapted. Hence, further research is needed on Problem 2.*

**Theorem 9** ([59])**.** *Consider system (11) where <sup>A</sup>* <sup>∈</sup> <sup>R</sup>*n*×*n*, *<sup>B</sup>* <sup>∈</sup> <sup>R</sup>*n*×*m*, 1 <sup>≤</sup> *<sup>m</sup>* <sup>≤</sup> *<sup>n</sup> is time independent and satisfy the Kalman rank condition* rank[*B*, *AB*, ··· , *<sup>A</sup>n*−1*B*] = *n. Assume also that <sup>d</sup>* <sup>=</sup> 1, *<sup>C</sup>*<sup>1</sup> <sup>=</sup> *<sup>C</sup> and <sup>C</sup>*<sup>2</sup> <sup>=</sup> *aC*, *<sup>C</sup>*(R*n*) <sup>⊂</sup> *<sup>B</sup>*(R*m*) *for some <sup>a</sup>* <sup>∈</sup> <sup>R</sup>*. Let <sup>τ</sup>* <sup>&</sup>gt; <sup>0</sup> *and <sup>x</sup>*<sup>0</sup> <sup>∈</sup> <sup>R</sup>*<sup>n</sup> be arbitrary but fixed. Then there is an Ft*−*adapted controller <sup>u</sup>* <sup>∈</sup> *<sup>L</sup>*2([0, *<sup>τ</sup>*], <sup>Ω</sup>, <sup>R</sup>*m*) *which steers <sup>x</sup>*<sup>0</sup> *in origin, in time τ, with probability one.*

**Remark 2.** *One might suspect that the controller u steering x*<sup>0</sup> *in origin can be found in feedback form but the problem is open.*

See [60] (p. 22) for example of this part.
