*5.1. C*0−*Semigroup Method for a Class of Time Invariant Systems in Hilbert Spaces*

In 2015, Liaskos et al. studied the exact controllability of the following stochastic singular linear system by using the *C*0−semigroup method in the sense of strong solution in Hilbert spaces in [71].

$$dL\mathbf{x}(t) = [M\mathbf{x}(t) + \mathbb{C}u(t) + f(t)]dt + Bdw(t), \\ t \in [0, \tau], \\ \mathbf{x}(0) = \boldsymbol{\xi}. \tag{51}$$

In order to introduce the exact controllability, make the following assumptions and preparations.

Let *H*, *U*, *K* be separable and infinite dimensional Hilbert spaces, *x*(*t*) be the state process valued in *H*, *u*(*t*) be the control process valued in *U*, and *w*(*t*) be a *U*−valued standard Wiener process in (51). The closure of an operator *S* will be denoted by *S*. We use the notation *S*⊥ for the orthogonal complement of a set *S* and for the restriction of the operator *A* to a linear subset *S* the symbol *A*|*S*. For the coefficients *L*, *M*, *C*, *f* , *B*, *ξ* involved in (51), the following assumptions and definitions should be considered.

(A1) (i) *L* ∈ *B*(*H*), ker(*L*) = {0}. (ii) ker(*L*) = ker(*L*).

(A2) (i) *M* : dom(*M*) ⊆ *H* → *H* is a linear, densely defined and closed operator.

(ii) For the linear subspace *D* = {*x* ∈ dom(*M*) : *Mx* + *f*(*t*) ∈ ran(*L*)}, we assume that *D* ∩ ker(*L*) = {0} and *P*<sup>⊥</sup> <sup>1</sup> *D* is dense in *P*<sup>⊥</sup> <sup>1</sup> *H*, where *P*1, *P*<sup>⊥</sup> <sup>1</sup> are the projections onto ker(*L*) and (ker(*L*))⊥, respectively.

(A3) (i) The operator pencil *λL* − *M* : dom(*M*) → *H* is of parabolic type, i.e., the restriction of the pencil *<sup>λ</sup><sup>L</sup>* <sup>−</sup> *<sup>M</sup>* : *<sup>D</sup>* <sup>→</sup> ran(*L*) is invertible with a bounded inverse (*λ<sup>L</sup>* <sup>−</sup> *<sup>M</sup>*)−1, for all *λ* > *ω*, where *ω* is a negative real constant. This regularity on the pencil also implies that *<sup>M</sup>*(*D*) = ran(*L*) and *<sup>M</sup>*|*<sup>D</sup>* : *<sup>D</sup>* <sup>→</sup> ran(*L*) is invertible with a bounded inverse *<sup>M</sup>*−1.

(ii) The bounded pseudo-resolvent operators *<sup>R</sup>*1(*λ*)=(*λ<sup>L</sup>* <sup>−</sup> *<sup>M</sup>*)−1*<sup>L</sup>* : *<sup>H</sup>* <sup>→</sup> *<sup>D</sup>* and *<sup>R</sup>*2(*λ*) = *<sup>L</sup>*(*λ<sup>L</sup>* <sup>−</sup> *<sup>M</sup>*)−<sup>1</sup> : ran(*L*) <sup>→</sup> *<sup>L</sup>*(*D*) satisfy -*U*(*λ*)-*<sup>B</sup>*(*H*) <sup>≤</sup> *<sup>c</sup> <sup>λ</sup>*−*<sup>ω</sup>* , for all *<sup>λ</sup>* <sup>&</sup>gt; *<sup>ω</sup>*, 0 <sup>&</sup>lt; *c* < 1, where *U*(*λ*) stands for both *R*1(*λ*), *R*2(*λ*).

(A4) *<sup>f</sup>* <sup>∈</sup> *<sup>L</sup>*1([0, *<sup>τ</sup>*]; *<sup>H</sup>*) <sup>∩</sup> *<sup>L</sup>*2([0, *<sup>τ</sup>*], <sup>Ω</sup>, *Ft*, *<sup>H</sup>*), satisfying *<sup>f</sup>*(*t*) <sup>∈</sup> *<sup>L</sup>*(*D*), *<sup>P</sup>* <sup>−</sup> *<sup>a</sup>*.*s*., *<sup>a</sup>*.*e*. in [0, *<sup>τ</sup>*].

(A5) *B* : *U* → *H* is a linear operator with ran(*B*) ⊆ *L*(*D*), such that *B* ∈ *B*(*U*, *H*).

(A6) *<sup>ξ</sup>* is a *<sup>D</sup>*−valued random variable *<sup>P</sup>* <sup>−</sup> *<sup>a</sup>*,*s*., with *<sup>ξ</sup>* <sup>∈</sup> *<sup>L</sup>*2(Ω, *<sup>F</sup>*0, *<sup>P</sup>*, *<sup>H</sup>*).

(A7) *<sup>C</sup>* <sup>∈</sup> *<sup>B</sup>*(*K*, *<sup>H</sup>*), with ran(*C*) <sup>⊆</sup> *<sup>L</sup>*(*D*), such that for any *<sup>u</sup>* <sup>∈</sup> *<sup>L</sup>*2([0, *<sup>τ</sup>*], <sup>Ω</sup>, *<sup>K</sup>*), the stochastic process *Cu*(*t*), *t* ∈ [0, *τ*] satisfies

$$E[\int\_0^\pi \int\_0^t \| (L^\perp)^{-1} M\_0 S\_1(s-t) (L^\perp)^{-1} (\mathbb{C}u(s) + f(s)) \|\_{H} ds dt] < \infty,$$

where *L*<sup>⊥</sup> = *L*| *P*⊥ <sup>1</sup> *<sup>H</sup>* : *<sup>P</sup>*<sup>⊥</sup> <sup>1</sup> *H* → *Q*⊥*H*, *Q* is the projection onto ker(*L*∗); *M*<sup>0</sup> = *M*(*P*<sup>⊥</sup> <sup>1</sup> <sup>|</sup>*D*)−1, *S*1(*t*) is the *C*0-semigroup in the closed subspace *P*<sup>⊥</sup> <sup>1</sup> *<sup>H</sup>* generated by the operator (*L*⊥)−1*M*0.

**Definition 19.** *An H*−*valued stochastic process x*(*t*), *t* ∈ [0, *τ*], *is called a strong solution of the initial value problem (51), if*

*(i) x* <sup>∈</sup> *<sup>D</sup>*, *<sup>P</sup>* <sup>−</sup> *<sup>a</sup>*.*s*., *<sup>a</sup>*.*e*. *in* [0, *<sup>τ</sup>*] *and x* <sup>∈</sup> *<sup>L</sup>*1([0, *<sup>τ</sup>*]; *<sup>H</sup>*), *<sup>P</sup>* <sup>−</sup> *<sup>a</sup>*.*s*. *(ii) Lx*, *Mx* <sup>∈</sup> *<sup>L</sup>*1([0, *<sup>τ</sup>*]; *<sup>H</sup>*), *<sup>P</sup>* <sup>−</sup> *<sup>a</sup>*.*s*. *(iii) Lx*(*t*) = *Lξ* + *<sup>t</sup>* <sup>0</sup> [*Mx*(*s*) + *Cu*(*t*) + *f*(*s*)]*ds* + *Bw*(*t*), *P* − *a*.*s*., *a*.*e*. *in* [0, *τ*].

From the above, the controlled stochastic singular linear system (51) has a unique strong solution *xu*(*t*), *t* ∈ [0, *τ*], which admits the form:

$$\mathbf{x}\_{\rm u}(t) = \overline{(P\_1^\perp|\_{D})^{-1}} \mathbf{S}\_1(t) P\_1^\perp \overline{\xi} + \int\_0^t \overline{(P\_1^\perp|\_{D})^{-1}} \mathbf{S}\_1(t-s) (L^\perp)^{-1} [\mathbf{C}u(s) + f(s)] ds$$

$$+ \int\_0^t \overline{(P\_1^\perp|\_{D})^{-1}} \mathbf{S}\_1(t-s) (L^\perp)^{-1} \mathbf{B} dw(s), t \in [0, \tau]. \tag{52}$$

**Definition 20.** *Stochastic singular linear system (51) is called exactly controllable at time τ* > 0*, if for any <sup>ξ</sup> which is <sup>D</sup>*−*valued random variable <sup>P</sup>* <sup>−</sup> *<sup>a</sup>*.*s*., *with <sup>ξ</sup>* <sup>∈</sup> *<sup>L</sup>*2(Ω, *<sup>F</sup>*0, *PH*) *and for any ξτ which is also a <sup>D</sup>*−*valued random variable <sup>P</sup>* <sup>−</sup> *<sup>a</sup>*.*s*., *with <sup>ξ</sup>* <sup>∈</sup> *<sup>L</sup>*2(Ω, *<sup>F</sup>τ*, *<sup>P</sup>*, *<sup>H</sup>*)*, there exists at least one control <sup>u</sup>* <sup>∈</sup> *<sup>L</sup>*2([0, *<sup>τ</sup>*], <sup>Ω</sup>, *<sup>K</sup>*)*, such that the corresponding strong solution xu*(*t*)*, which admits the form of (52), satisfies the initial condition xu*(0) = *ξ and the terminal condition xu*(*τ*) = *ξτ.*

The following result was obtain in [71].

**Theorem 34** ([71])**.** *Suppose that L*⊥*S*1(*t*)*v*(*t*) − *f*(*t*) ∈ ran(*C*), *P* − *a*.*s*., *a*.*e*. *in* [0, *τ*]*. Then there exists at least one <sup>u</sup>* <sup>∈</sup> *<sup>L</sup>*2([0, *<sup>τ</sup>*], <sup>Ω</sup>, *<sup>K</sup>*)*, such that the corresponding strong solution xu*(*t*)*, which admits the form of (52), satisfies the initial condition xu*(0) = *ξ and the terminal condition xu*(*τ*) = *ξτ and hence stochastic singular linear system (51) is exactly controllable.*

See [71] for the details of practical example.

In 2018, Liaskos et al. studied the exact controllability of the stochastic singular linear system (51) by using the *C*0−semigroup method in the sense of strong solution in Hilbert spaces in [72].

Suppose that (A1)–(A6) hold true, and

$$E\left[\int\_0^\tau \int\_0^t \|M\_0(L^\perp)^{-1} S\_2(s-t)(L^\perp)^{-1} (\mathbb{C}u(s) + f(s))\|\_H ds dt\right] < \infty.$$

Then, the controlled stochastic singular linear system (51) has a unique strong solution *xu*(*t*), *t* ∈ [0, *τ*], which admits the form:

$$x\_{\boldsymbol{u}}(t) = \overline{(P\_1^{\perp}|\_{D})^{-1}}(L^{\perp})^{-1}S\_2(t)L\overline{\xi}$$

$$\frac{1}{t} + \int\_0^t \overline{(P\_1^{\perp}|\_{D})^{-1}} (L^{\perp})^{-1}S\_2(t-s)[\mathbb{C}u(s) + f(s)]ds$$

$$\frac{1}{t} + \int\_0^t \overline{(P\_1^{\perp}|\_{D})^{-1}} (L^{\perp})^{-1}S\_2(t-s)Bdw(s), t \in [0, \tau],\tag{53}$$

where *<sup>S</sup>*2(*t*) is the *<sup>C</sup>*0−semigroup generated by the operator *<sup>M</sup>*0(*L*⊥)−1.

The following result was obtained in [72]:

**Theorem 35** ([72])**.** *Suppose that S*2(*t*)*v*(*t*) − *f*(*t*) ∈ ran(*C*), *P* − *a*.*s*., *a*.*e*. *in* [0, *τ*]*. Then, there exists at least one <sup>u</sup>* <sup>∈</sup> *<sup>L</sup>*2([0, *<sup>τ</sup>*], <sup>Ω</sup>, *<sup>K</sup>*)*, such that the corresponding strong solution xu*(*t*)*, which* *admits the form of (53), satisfies the initial condition xu*(0) = *ξ and the terminal condition xu*(*τ*) = *ξτ and hence stochastic singular linear system (51) is exactly controllable.*
