Duality of switched DAEs

Abstract

We present and discuss the definition of the adjoint and dual of a switched differential-algebraic equation (DAE). For a proper duality definition, it is necessary to extend the class of switched DAEs to allow for additional impact terms. For this switched DAE with impacts, we derive controllability/reachability/determinability/observability characterizations for a given switching signal. Based on this characterizations, we prove duality between controllability/reachability and determinability/observability for switched DAEs.

Appendices

Appendix A: Some basics on linear algebra

Lemma 34

Let $A: \mathbb {R}^n\rightarrow \mathbb {R}^n$ be a linear mapping, and $\mathcal {S}_1$ and $\mathcal {S}_2$ be subspaces of $\mathbb {R}^n$. Then, it holds

• 1. $\left( A^{-1} {\mathcal {S}} \right) ^{\perp } = A^\top {\mathcal {S}}^{\perp }$ and $\left( A {\mathcal {S}} \right) ^{\perp } = A^{-\top } {\mathcal {S}}^{\perp }$;

• 2. $\left( \ker A\right) ^\perp = {{\mathrm{im}}}A^\top$ and $\left( {{\mathrm{im}}}A\right) ^\perp = \ker A^\top$;

• 3. $\left( \mathcal {S}_1 + \mathcal {S}_2\right) ^\perp = \mathcal {S}_1^\perp \cap \mathcal {S}_2^\perp$ and $\left( \mathcal {S}_1\cap \mathcal {S}_2\right) ^\perp = \mathcal {S}_1^\perp + \mathcal {S}_2^\perp$;

• 4. $A \left( A^{-1}\mathcal {S}_1 \cap \mathcal {S}_2 \right) = \mathcal {S}_1 \cap A\mathcal {S}_2$;

• 5. $A^{-1}\left( A \mathcal {S}_1 + \mathcal {S}_2 \right) = \mathcal {S}_1 + A^{-1}\mathcal {S}_2$.

Proof

• 1. [4, Lemma 4.1],

• 2. [3, Property A. 3.4],

• 3. The first statement is shown in [16, Lemma 4.6], the second follows by computing the orthogonal complement,

• 4. “$\subseteq$”: $A \left( A^{-1} \mathcal {S}_1 \cap \mathcal {S}_2 \right) \subseteq A A^{-1} \mathcal {S}_1 \cap A \mathcal {S}_2 = \mathcal {S}_1 \cap {{\mathrm{im}}}A \cap A \mathcal {S}_2 = \mathcal {S}_1 \cap A \mathcal {S}_2$. “$\supseteq$”: Let $x\in \mathcal {S}_1 \cap A\mathcal {S}_2$, i.e $x\in \mathcal {S}_1$ and $\exists y \in \mathcal {S}_2: \, x=Ay$. y fulfills $Ay\in \mathcal {S}_1$, and hence, $y\in A^{-1}\mathcal {S}_1$ and $y \in A^{-1}\mathcal {S}_1\cap \mathcal {S}_2$. Finally, $x=Ay\in A\left( A^{-1}\mathcal {S}_1\cap \mathcal {S}_2\right)$.

• 5. “$\supseteq$”: $\mathcal {S}_1 + A^{-1}\mathcal {S}_2 = \mathcal {S}_1 + \ker A + A^{-1}\mathcal {S}_2 = A^{-1}A\mathcal {S}_1 + A^{-1}\mathcal {S}_2 \subseteq A^{-1}\left( A\mathcal {S}_1 + \mathcal {S}_2 \right)$. “$\subseteq$”: Let $x\in A^{-1}\left( A\mathcal {S}_1 + \mathcal {S}_2 \right)$, and hence, there exist $s_1\in \mathcal {S}_1$ and $s_2\in \mathcal {S}_2$ such that $Ax=As_1 + s_2$. Therefore, $A(x-s_1) = s_2$, i.e., $x-s_1\in A^{-1}\mathcal {S}_2$. This gives $x\in \mathcal {S}_1 + A^{-1}\mathcal {S}_2$.$\square$

Lemma 35

Let $\varPi : \mathbb {R}^n\rightarrow \mathbb {R}^n$ be a projector and $\mathcal {S}$ be a subspace of $\mathbb {R}^n$. Then, it holds

• 1. $\mathcal {S}+ \ker \varPi = \varPi \mathcal {S}+ \ker \varPi$ and $\mathcal {S}\cap {{\mathrm{im}}}\varPi = {\varPi }^{-1} \mathcal {S}\cap {{\mathrm{im}}}\varPi$;

• 2. ${{\mathrm{im}}}\varPi \subseteq \mathcal {S}\, \Leftrightarrow \, {\varPi }^{-1} \mathcal {S}= \mathbb {R}^n$;

• 3. for $\ker \varPi \subseteq \mathcal {S}$: ${{\mathrm{im}}}\varPi \cap \mathcal {S}= \varPi \mathcal {S}$.

Proof

• 1. The second statement follows from the first by computing the orthogonal complement using Lemma 34.1 and renaming $\overline{\mathcal {S}}=\mathcal {S}^{\perp }$ and $\overline{\varPi } = {\varPi }^\top$. Consider the first statement: “$\subseteq$”: Let $x\in \mathcal {S}+\ker \varPi$, i.e. $\exists s\in \mathcal {S}, y\in \ker \varPi$: $x=s+y = \varPi s + \left( (I-\varPi )s+y\right) \in \varPi \mathcal {S}+ \ker \varPi$. “$\supseteq$”: Let $x\in \varPi \mathcal {S}+ \ker \varPi$ i.e. $\exists s\in \mathcal {S}, y\in \ker \varPi$: $x=\varPi s+y = s + \left( (\varPi -I)s + y\right) \in \mathcal {S}+\ker \varPi$.

• 2. Let ${{\mathrm{im}}}\varPi \subseteq \mathcal {S}$. Then, it holds $\mathbb {R}^n= {\varPi }^{-1} \left( {{\mathrm{im}}}\varPi \right) \subseteq {\varPi }^{-1} \mathcal {S}$. For the other inclusion it holds ${{\mathrm{im}}}\varPi = \varPi \mathbb {R}^n= \varPi \left( {\varPi }^{-1} \mathcal {S}\right) \subseteq \mathcal {S}$, because of $\varPi \left( {\varPi }^{-1} \mathcal {S}\right) \subseteq \mathcal {S}$.

• 3. Let $x \in {{\mathrm{im}}}\varPi \cap \mathcal {S}$. Hence, $\varPi x = x$, and thus, $x \in \varPi \mathcal {S}$. Let $x \in \varPi \mathcal {S}$, i.e., there exists $s \in \mathcal {S}$ with $x = \varPi s$. It is $s = \varPi s + \left( I - \varPi \right) s$. As $\left( I-\varPi _i\right) s \in \ker \varPi \subseteq \mathcal {S}$, it follows $x\in \mathcal {S}$, and hence, $x\in {{\mathrm{im}}}\varPi \cap \mathcal {S}$.

$\square$

Appendix B: Proofs of Section 4.2

Proof of Lemma 15

Write a and b as $a = {\alpha }_{\mathbb {D}} + \sum _{t\in \varGamma ^a} a[t]$ and $b = {\beta }_{\mathbb {D}} + \sum _{t\in \varGamma ^b} b[t]$. The product of $\sum _{t\in \varGamma ^a} a[t]$ and $\sum _{t\in \varGamma ^b} b[t]$ (or their time inversions) is zero for both causal and anticausal multiplication. The product of ${\alpha }_{\mathbb {D}}$ and ${\beta }_{\mathbb {D}}$ is the same for both kinds of multiplication, and furthermore, (25) yields $\mathscr {T}_T\left\{ {\alpha }_{\mathbb {D}} *_c {\beta }_{\mathbb {D}}\right\} = \mathscr {T}_T\left\{ {\alpha }_{\mathbb {D}}\right\} *_c \mathscr {T}_T\left\{ {\beta }_{\mathbb {D}}\right\}$.

Using the linearity of ${\mathscr {T}}_T$, it is sufficient to consider the product of a piecewise-smooth function and a Dirac impulse (or derivatives of a Dirac impulse):

$$\begin{aligned} \mathscr {T}_T\left\{ \alpha *_c \delta _t\right\}&= \mathscr {T}_T\left\{ \alpha (t^+)\delta _t\right\} = \alpha (t^+)\delta _{T-t} = \mathscr {T}_T\left\{ \alpha \right\} ((T-t)^-) \delta _{T-t} \\&= \mathscr {T}_T\left\{ \alpha \right\} *_{ac} \delta _{T-t} = \mathscr {T}_T\left\{ \alpha \right\} *_{ac} \mathscr {T}_T\left\{ \delta _t\right\} \end{aligned}$$

and analogously, $\mathscr {T}_T\left\{ \delta _t *_c a\right\} = \mathscr {T}_T\left\{ \delta _t\right\} *_{ac} \mathscr {T}_T\left\{ a\right\}$. Applying the differentiation rule of the multiplication gives inductively for $D=\delta _t,\delta _t^{(1)},\delta _t^{(2)},\ldots$:

$$\begin{aligned} \begin{aligned} \mathscr {T}_T\left\{ D' *_c \alpha \right\}&= \mathscr {T}_T\left\{ \left( D *_c \alpha \right) ' - D *_c \alpha ' \right\} \\&{\overset{(24)}{=}} - (\mathscr {T}_T\left\{ D *_c \alpha \right\} )' - \mathscr {T}_T\left\{ D *_c \alpha '\right\} \\&{\overset{\mathrm{Ind.}}{=}}- \left( \mathscr {T}_T\left\{ D\right\} *_{ac} \mathscr {T}_T\left\{ \alpha \right\} \right) ' - \mathscr {T}_T\left\{ D\right\} *_{ac} \mathscr {T}_T\left\{ \alpha '\right\} \\&{\overset{(24)}{=}}- \left( \mathscr {T}_T\left\{ D\right\} *_{ac} \mathscr {T}_T\left\{ \alpha \right\} \right) ' + \mathscr {T}_T\left\{ D\right\} *_{ac} \left( \mathscr {T}_T\left\{ \alpha \right\} \right) '\\&=- \left( \left( \mathscr {T}_T\left\{ D\right\} \right) ' *_{ac} \mathscr {T}_T\left\{ \alpha \right\} \right) \\&{\overset{(24)}{=}}\mathscr {T}_T\left\{ D'\right\} *_{ac} \mathscr {T}_T\left\{ \alpha \right\} . \end{aligned} \end{aligned}$$

and analogously for $\mathscr {T}_T\left\{ \alpha *_c D' \right\}$. Hence, the first statement is shown. The second statement follows by applying $\tilde{a}=\mathscr {T}_T\left\{ a\right\}$ and $\tilde{b}=\mathscr {T}_T\left\{ b\right\}$ to the first statement and using the involution property $D = \mathscr {T}_T\left\{ \mathscr {T}_T\left\{ D\right\} \right\}$.

Proof of Lemma 16

As in the proof of Lemma 15, we observe that it suffices to consider the product of a piecewise-smooth function $\alpha$ and a Dirac impulse:

$$\begin{aligned} \alpha *_c \delta _t =\alpha (t^+)\delta _t= \delta _t *_{ac} \alpha \quad \text { and } \alpha *_{ac} \delta _t = \alpha (t^-)\delta _t= \delta _t *_{c} \alpha . \end{aligned}$$

Hence, the entries of the matrices $\left( A*_c B\right) ^{\top }$ and $B^{\top } *_{ac} A^{\top }$ are identical.

Appendix C: Proofs of Section 5.3

While the proof for controllability can easily be deduced from the one for switched DAEs without impacts given in [15], the same is not true for observability and determinability. The proofs in [21, 25] use properties of jump and impulse of a switched DAE which do not hold true any more when impacts are added to the system. Hence, these proofs are given here together with the proof for reachability, which has not been considered before for switched DAEs.

Proof of Lemma 26

Controllability: The proof is analogous to the one for switched DAEs without impacts given in [15].

Reachability, “$\subseteq$”: Let $x_T\in \mathcal {R}_{\sigma _1}^{(0,T)}$, i.e., there exists $(u,x,y)\in \mathcal {B}_{\sigma _1}$ with $x(0^+)=0$ and $x(T^-)=x_T$. We assume u to be zero on $[t_1,t_1+\varepsilon )$ for some $\varepsilon \in (0,\tau _1)$ ([15, Lemma 3.3]). Define $\overline{u}:=u_{(-\infty ,t_1)}$, $\hat{u}=u_{[t_1,\infty )}$ and corresponding solutions $\overline{x},\hat{x}$ with zero initial condition. Clearly, $x = \overline{x}+\hat{x}$. It holds $\overline{x}(t_1^-)\in \mathcal {C}_0$, and therefore, $\overline{x}(T^-)\in \mathrm {e}^{A^\text { diff}_1\tau _1} \Pi ^\text { diff}_{1} H_{1}\mathcal {C}_0$. For $\hat{x}$, it holds $\hat{x}(t_1^-)=0$, and hence, $\hat{x}(T^-)\in \mathcal {C}_1$. This gives $x(T^-)=\overline{x}(T^-)+\hat{x}(T^-) \in \mathrm {e}^{A^\text { diff}_1\tau _1} \Pi ^\text { diff}_{1} H_{1}\mathcal {C}_0 + \mathcal {C}_1$.

Reachability, “$\supseteq$”: Let $x_T\in \mathcal {C}_1+\mathrm {e}^{A^\text { diff}_1\tau _1} \Pi ^\text { diff}_{1} H_{1}\mathcal {C}_0$. Hence, there exists $x_1\in \mathcal {C}_0$ such that $x_T-\mathrm {e}^{A^\text { diff}_1\tau _1} \Pi ^\text { diff}_{1} H_{1} x_1 \in \mathcal {C}_1$. Define $\overline{u}$ on $(0,t_1)$ such that $(\overline{u},\overline{x},\overline{y})\in \mathcal {B}_{\sigma _1}$ with zero initial condition and $\overline{x}(t_1^-)=x_1$ and define $\hat{u}$ on $(t_1+\varepsilon ,T)$ for some $\varepsilon \in (0,\tau _1)$ such that $(\hat{u},\hat{x},\hat{y})\in \mathcal {B}_{\sigma _1}$ with zero initial condition and $\hat{x}(T^-) = x_T - \mathrm {e}^{A^\text { diff}_1\tau _1} \Pi ^\text { diff}_{1} H_{1}x_1$. Note that $\overline{u}$ is zero outside $(0,t_1)$ and $\hat{u}$ is zero outside $(t_1+\varepsilon ,T)$. It holds for $(u,x,y):=(\overline{u}+\hat{u},\overline{x}+\hat{x},\overline{y}+\hat{y})\in \mathcal {B}_{\sigma _1}$: $x(0^+)=\overline{x}(0^+)+\hat{x}(0^+)=0$ and

$$\begin{aligned} x(T^-) = \overline{x}(T^-) + \hat{x}(T^-) = \mathrm {e}^{A^\text { diff}_1\tau _1} \Pi ^\text { diff}_{1} H_{1} x_1 + x_T - \mathrm {e}^{A^\text { diff}_1\tau _1} \Pi ^\text { diff}_{1} H_{1} x_1 = x_T. \end{aligned}$$

Hence, $\mathcal {C}_1 + \mathrm {e}^{A^\text { diff}_1\tau _1} \Pi ^\text { diff}_{1} H_{1}\mathcal {C}_0\subseteq \mathcal {R}_{\sigma _1}^{(0,T)}$.

Observability, “$\subseteq$”: Let $x_0 \in \mathcal {UO}_{\sigma _1}^{(0,T)}$, i.e., there exists $(0,x,y)\in \mathcal {B}_{\sigma }$ with $x(0^-)=x_0$ and $y_{(0,T)}= 0$. This gives

• 1. $y_{(0,t_1)}= 0$, which is equivalent to $x(0^-) \in \ker O_0^\text { diff}\cap \mathcal {V}^*_{0}$;

• 2. $y_{(t_1,T)}= 0$, which is equivalent to $x(t_1^+)\in \ker O_1^\text { diff}\cap \mathcal {V}^*_{1}$, and thus

  $$\begin{aligned} \begin{aligned} x(t_1^-) \overset{(14a)}{\in }\left( \varPi ^\text { diff}_{1} H_{1}\right) ^{-1}\{x(t_1^+)\}&\subseteq \left( \varPi ^\text { diff}_{1} H_{1}\right) ^{-1}\left( \ker O_1^\text { diff}\cap \mathcal {V}^*_{1} \right) \\&\subseteq \left( \varPi ^\text { diff}_{1} H_{1}\right) ^{-1} \ker O_1^\text { diff}= \ker \left( O_1^\text { diff}\varPi _1^\text { diff}H_{1} \right) ; \end{aligned} \end{aligned}$$

• 3. $y[t_1]=0$, which is by (14b) equivalent to $x(t_1^-)\in \ker \left( O_1^\text { imp}\varPi _1^\text { imp}H_{1}\right)$.

Using $x(t_1^-)=\mathrm {e}^{A_0^\text { diff}\tau _0}x(0^-)$ for the input-free solution x gives the desired inclusion

$$\begin{aligned} \mathcal {UO}_{\sigma _1}^{(0,T)} \subseteq {{\mathrm{im}}}\varPi _0 \cap \ker O_0^\text { diff}\cap \mathrm {e}^{-A_0^\text { diff}\tau _0}\left( \ker \left( O_1^\text { diff}\varPi _1^\text { diff}H_{1} \right) \cap \ker \left( O_1^\text { imp}\varPi _1^\text { imp}H_{1} \right) \right) . \end{aligned}$$

Observability, “$\supseteq$”: Let $x_0 \in {{\mathrm{im}}}\varPi _0 \cap \ker O_0^\text { diff}\cap \mathrm {e}^{-A_0^\text { diff}\tau _0}\mathcal {U}^{\text { H}}_{1}$. Then, there exists a solution $(0,x,y)\in \mathcal {B}_{\sigma _1}$ with $x(0^-)=x_0$ as $x_0\in {{\mathrm{im}}}\varPi _0$ is consistent. By the derivations above, it follows $y_{(0,T)}= 0$. Thus, $x_0\in \mathcal {UO}_{\sigma _1}^{(0,T)}.$

Determinability, “$\subseteq$”: By (39), we know that for an input-free solution $(0,x,y)\in \mathcal {B}_{\sigma _1}$ with $y_{(0,T)}= 0$, it holds $x(t_1^-)\in \widetilde{\mathcal {M}}_1 = {{\mathrm{im}}}\varPi _0 \cap \ker O_0^\text { diff}\cap \mathcal {U}^{\text { H}}_{1}$. Because of $u= 0$, this gives

$$\begin{aligned} x(T^-)=\mathrm {e}^{A_1^\text { diff}\tau _1} \Pi ^\text { diff}_{1} H_{1} x(t_1^-) \in \mathrm {e}^{A_1^\text { diff}\tau _1} \Pi ^\text { diff}_{1} H_{1} \left( {{\mathrm{im}}}\varPi _0 \cap \ker O_0^\text { diff}\cap \mathcal {U}^{\text { H}}_{1} \right) . \end{aligned}$$

Determinability, “$\supseteq$”: Let $0\ne x_T\in \mathrm {e}^{A_1^\text { diff}\tau _1} \Pi ^\text { diff}_{1} H_{1} \left( {{\mathrm{im}}}\varPi _0 \cap \ker O_0^\text { diff}\cap \mathcal {U}^{\text { H}}_1 \right)$. Using (14a), there exists an input-free solution x with $(0,x,y)\in \mathcal {B}_{\sigma _1}$ and $0\ne x(t_1^-)\in {{\mathrm{im}}}\varPi _0 \cap \ker O_0^\text { diff}\cap \mathcal {U}^{\text { H}}_1$ (as ${{\mathrm{im}}}\varPi _0 \cap \ker O_0^\text { diff}\cap \mathrm {e}^{-A_0^\text { diff}\tau _0}\mathcal {U}^{\text { H}}_1 \subseteq \mathcal {V}^*_{0}$ is a set of consistent initial values and non-empty by assumption). Using (39), this gives $y_{(0,T)}= 0$, i.e. $x_T\in \mathcal {UD}_{\sigma _1}^{(0,T)}$. $\square$

The proof of (31) gives for $\widetilde{\mathcal {M}}_1={{\mathrm{im}}}\varPi _0 \cap \ker O_0^\text { diff}\cap \mathcal {U}^{\text { H}}_1$:

$$\begin{aligned} \left\{ x(t_1^-)\ \left| \ \phantom {x(t_1^-)} (0,x,y)\in \mathcal {B}_{\sigma _1} \text { with } y_{(0,T)}=0\right. \right\} = \widetilde{\mathcal {M}}_1. \end{aligned}$$

(39)

Note that one gets the same space $\widetilde{\mathcal {M}}_1$ if one assumes only $y_{(t_1-\varepsilon _1,t_1+\varepsilon _2)}=0$ for $\varepsilon _1,\varepsilon _2>0$. The restricted switching signal $\sigma _{>t_{i-1}}$ has only one switch on the open interval $(0,t_{i+1})$. Hence, we get for $\widetilde{\mathcal {M}}_i$:

$$\begin{aligned} \left\{ x(t_{i}^-)\ \left| \ \phantom {x(t_{i}^-)} (0,x,y)\in \mathcal {B}_{\sigma _{>t_{i-1}}} \text { with } y_{(t_{i-1},t_{i+1})}\right. \right\} = \widetilde{\mathcal {M}}_{i}. \end{aligned}$$

(40)

Proof of Theorem 27

We start by proving the recursions. For controllability, the proof can be carried out analogously to [15]. The formulas are shown by induction. The induction start ($i=1$ for reachability and determinability, $i=m$ for observability) is precisely the single switch case (Lemma 26). For reachability, $i=0$ corresponds to an unswitched system.

Reachability: Analogously to (30), it holds for $i\ge 1$

$$\begin{aligned} \mathcal {R}_\sigma ^{(0,t_{i+1})} = \mathcal {C}_i + \mathrm {e}^{A^\text { diff}_i\tau _i} \Pi ^\text { diff}_{i} H_{i} \mathcal {R}_\sigma ^{(0,t_i)}. \end{aligned}$$

Hence, it holds by induction

$$\begin{aligned} \mathcal {R}_\sigma ^{(0,t_{i+1})} = \mathcal {Q}_0^{i} \; \text { for } i=0,1,\ldots ,m. \end{aligned}$$

Observability: Assume the statement holds for i: Let $x_{i-2}\in \mathcal {UO}_{\sigma _{>t_{i-2}}}^{(t_{i-2},T)}$. Hence, there exists $(0,x,y)\in \mathcal {B}_{\sigma _{>t_{i-2}}}$ with $x(t_{i-2}^+) = x_{i-2}$ and $y_{(t_{i-2},T)}= 0$. Thus, it holds

and therewith $x(t_{i-1}^-)\in \widetilde{\mathcal {M}}_{i-1} \cap \left( \varPi ^\text { diff}_{i-1} H_{i-1}\right) ^{-1}\widetilde{\mathcal {M}}_i^m$. This implies

$$\begin{aligned} x(t_{i-2}^+) \!= \!\mathrm {e}^{-A^\text { diff}_{i-2}\tau _{i-2}}x(t_{i-1}^-) \!\in \! \mathrm {e}^{-A_{i-2}\tau _{i-2}} \left( \widetilde{\mathcal {M}}_{i-1}\!\cap \! \left( \varPi ^\text { diff}_{i-1} H_{i-1}\right) ^{-1}\widetilde{\mathcal {M}}^m_i \right) \overset{\text {Def.}}{=} \widetilde{\mathcal {M}}^m_{i-1}. \end{aligned}$$

For the other direction, let $x_{i-2}\in \widetilde{\mathcal {M}}^m_{i-1}$. As $\widetilde{\mathcal {M}}^m_{i-1}$ is a subset of $\overline{ \mathcal {V}^*_{i-2} }$, there exists a solution $(0,x,y)\in \mathcal {B}_{\sigma _{>t_{i-2}}}$ with $x(t_{i-2}^+)=x_{i-2}$. It holds $x(t_{i-1}^-)\in \widetilde{\mathcal {M}}_{i-1}$, and hence, by (40) we get $y_{(t_{i-2},t_{i})}= 0$. $x(t_{i-1}^+)\in \widetilde{\mathcal {M}}^m_i$ gives by induction $y_{(t_{i-1},T)}= 0$. Hence, $y_{(t_{i-2},T)}= 0$, and thus, $x_{i-2}\in \mathcal {UO}_{\sigma _{>t_{i-2}}}^{(t_{i-2},T)}$.

Determinability: For the induction step $i-1\rightarrow i$, let $x_{i+1}\in \mathcal {UD}_\sigma ^{(0,t_{i+1})}$. Hence, there exists $(0,x,y)\in \mathcal {B}_{\sigma }$ with $x(t_{i+1}^-)=x_{i+1}$ and $y_{(0,t_{i+1})}= 0$. Thus, it holds

All in all, we obtain

$$\begin{aligned} x_{i+1}=x(t_{i+1}^-)=\mathrm {e}^{A^\text { diff}_{i}\tau _i}x(t_i^+) \in \mathrm {e}^{A^\text { diff}_i\tau _i} \Pi ^\text { diff}_{i} H_{i}\left( \widetilde{\mathcal {M}}_i \cap \widetilde{\mathcal {N}}^{i-1}_1\right) \overset{\text {Def.}}{=} \widetilde{\mathcal {N}}_1^{i}. \end{aligned}$$

For the other inclusion, let $x_{i+1}\in \widetilde{\mathcal {N}}_1^i \overset{\text {Def.}}{=} \mathrm {e}^{A^\text { diff}_i\tau _i} \Pi ^\text { diff}_{i} H_{i}\left( \widetilde{\mathcal {M}}_i\cap \widetilde{\mathcal {N}}_1^{i-1}\right)$. Thus, there exists $x_i\in \widetilde{\mathcal {M}}_i \cap \widetilde{\mathcal {N}}_1^{i-1}$ with $x_{i+1}=\mathrm {e}^{A^\text { diff}_i\tau _i} \Pi ^\text { diff}_{i} H_{i} x_i$. By the induction assumption, it holds $\widetilde{\mathcal {N}}_1^{i-1}=\mathcal {UD}_\sigma ^{(0,t_i)}$, and hence, there exists $(0,x,y)\in \mathcal {B}_{\sigma }$ with $x(t_i^-)=x_i$ and $y_{(0,t_i)}= 0$. $x(t_i^-)\in \widetilde{\mathcal {M}}_i$ gives $y_{(t_{i-1},t_{i+1})}= 0$ by (40). Hence, $y_{(0,t_{i+1})}= 0$, and thus, $x_{i+1}=x(t_{i+1}^-)\in \mathcal {UD}_\sigma ^{(0,t_{i+1})}$.

Characterization of system properties: The system is reachable on [0, T] iff it holds $\mathcal {R}_\sigma ^{[0,T]}= \overline{ \mathcal {V}^*_{m} }$. By Lemma 24, this is equivalent to $\mathcal {R}_\sigma ^{(0,T)}= \overline{ \mathcal {V}^*_{m} }$ and the claim follows from $\mathcal {R}_\sigma ^{(0,T)}=\mathcal {Q}_0^m$.

The same argument can be used for observability, determinability, and controllability. For the latter, note that $\mathcal {P}_0^m \cap \overline{ \mathcal {V}^*_{0} } = \overline{ \mathcal {V}^*_{0} }$ is equivalent to $\overline{ \mathcal {V}^*_{0} } \subseteq \mathcal {P}_0^m$.$\square$

Proof of Theorem 29

Controllability: By Theorem 27, the system is controllable on [0, T] iff $\overline{ \mathcal {V}^*_{0} } \subseteq \mathcal {P}_0^m$. As in the proof of [15, Theorem 3.6], we use ${{\mathrm{im}}}K_0^\text { imp}\subseteq \mathcal {C}_0 \subseteq \mathcal {P}_0^m$ and $\overline{ \mathcal {V}^*_{0} } = {{\mathrm{im}}}\varPi _0 \oplus {{\mathrm{im}}}K_0^\text { imp}$ to obtain as an equivalent criterion ${{\mathrm{im}}}\varPi _0 \subseteq \mathcal {P}_0^m$. By Lemma 35.2, this is equivalent to

$$\begin{aligned} \varPi _0^{-1}\mathcal {P}_0^m = \mathbb {R}^n. \end{aligned}$$

Using the recursion formula for $\mathcal {P}_i^m$, we can write $\mathcal {P}_0^m$ explicitly as

$$\begin{aligned} \mathcal {C}_0 + \mathrm {e}^{-A^\text { diff}_0\tau _0} \left( \varPi ^\text { diff}_{1} H_{1}\right) ^{-1} \left( \ldots \left( \mathcal {C}_{m-1} + \mathrm {e}^{-A^\text { diff}_{m-1}\tau _{m-1}} \left( \varPi ^\text { diff}_{m} H_{m}\right) ^{-1}\mathcal {C}_m \right) \ldots \right) . \end{aligned}$$

The statement follows then by the fact that it holds ${\varPi }^\text { diff}=\varPi$ for normalized systems.

Reachability: The recursion formula for $\mathcal {Q}_0^i$ yields:

$$\begin{aligned} \mathcal {Q}_0^m = \sum _{j=0}^i \left( \prod _{k=0}^{i-1-j} \mathrm {e}^{A^\text { diff}_{i-k}\tau _{i-k}} \Pi ^\text { diff}_{i-k} H_{i-k}\right) \mathcal {C}_j. \end{aligned}$$

By Theorem 27, this is the reachable set $\mathcal {R}_\sigma ^{(0,T)}$. It can be rewritten as

$$\begin{aligned} \left( {{\mathrm{im}}}K_m^\text { diff}+\sum _{j=0}^{m-1} \mathrm {e}^{A^\text { diff}_{m}\tau _{m}} \Pi ^\text { diff}_{m} H_{m} \left( \prod _{k=1}^{m-1-j} \mathrm {e}^{A^\text { diff}_{m-k}\tau _{m-k}} \Pi ^\text { diff}_{m-k} H_{m-k}\right) \mathcal {C}_j \right) \oplus {{\mathrm{im}}}K_m^\text { imp}. \end{aligned}$$

Normalization gives ${\varPi }^\text { diff}=\varPi$. Using the commutativity of $\mathrm {e}^{A^\text { diff}_m\tau _m}$ and $\varPi _m$ as well as ${{\mathrm{im}}}K_m^\text { diff}= \varPi \mathcal {C}_m$ gives

$$\begin{aligned} \mathcal {R}_\sigma ^{(0,T)} \!= \!\varPi _m\left( \mathcal {C}_m + \mathrm {e}^{A^\text { diff}_m \tau _m} H_{m}\sum _{j=0}^{m-1}\left( \!\prod _{k=1}^{m-1-j} \mathrm {e}^{A^\text { diff}_{m-k}\tau _{m-k}}\varPi _{m-k}H_{m-k}\right) \mathcal {C}_j \!\right) \!+\!{{\mathrm{im}}}K_m^\text { imp}. \end{aligned}$$

The system is reachable iff $\mathcal {R}_\sigma ^{(0,T)}= \overline{ \mathcal {V}^*_{m} }$, which is equivalent to

$$\begin{aligned} \varPi _m\left( \mathcal {C}_m + \mathrm {e}^{A^\text { diff}_m \tau _m} H_{m}\sum _{j=0}^{m-1}\left( \prod _{k=1}^{m-1-j} \mathrm {e}^{A^\text { diff}_{m-k}\tau _{m-k}}\varPi _{m-k}H_{m-k}\right) \mathcal {C}_j \right) +\ker \varPi _m = \mathbb {R}^n. \end{aligned}$$

By Lemma 35.1, this is equivalent to the claim.

Observability: Show by induction that it holds for the normalized system

$$\begin{aligned} \widetilde{\mathcal {M}}^m_i = {{\mathrm{im}}}\varPi _{i-1} \cap \ker O^\text { diff}_{i-1} \cap \mathrm {e}^{-A^\text { diff}_{i-1}\tau _{i-1}}H_i^{-1}\left( \bigcap \limits _{j=i}^m \left( \prod \limits _{k=i}^{j-1} \mathrm {e}^{-A_k^\text { diff}\tau _k} \varPi _k^{-1} H_{k+1}^{-1}\right) \mathcal {U}_j\!\right) \nonumber \\ \end{aligned}$$

(41)

for $i=m,\ldots ,1$.

For $i=m$, this holds true by the $\mathrm {e}^{-A_{i-1}^\text { diff}\tau _{i-1}}$-invariance of ${{\mathrm{im}}}\varPi _{i-1}$ and $\ker O^\text { diff}_{i-1}$.

By the same argument, we get for the induction step $i+1 \rightarrow i$:

$$\begin{aligned} \widetilde{\mathcal {M}}^m_i = {{\mathrm{im}}}\varPi _{i-1} \cap \ker O_{i-1}^\text { diff}\cap \mathrm {e}^{-A_{i-1}^\text { diff}\tau _{i-1}}\left( \mathcal {U}^{\text { H}}_i \cap \left( \varPi _i^\text { diff}H_i\right) ^{-1}\widetilde{\mathcal {M}}^m_{i+1}\right) . \end{aligned}$$

For normalized systems it holds $\varPi _i^\text { diff}=\varPi _i$ and $\mathcal {U}^{\text { H}}_i = H_i^{-1}\mathcal {U}_i$. This yields:

$$\begin{aligned} \widetilde{\mathcal {M}}^m_i = {{\mathrm{im}}}\varPi _{i-1} \cap \ker O_{i-1}^\text { diff}\cap \mathrm {e}^{-A_{i-1}^\text { diff}\tau _{i-1}} H_i^{-1} \left( \mathcal {U}_i \cap \varPi _i^{-1}\widetilde{\mathcal {M}}^m_{i+1}.\right) \end{aligned}$$

(42)

Observe that it holds $\varPi _i^{-1}{{\mathrm{im}}}\varPi _i = \mathbb {R}^n$ and $\varPi _i^{-1}\ker O_i^\text { diff}\supseteq \mathcal {U}_i$. Hence, we get from the induction assumption

$$\begin{aligned} \begin{aligned} \mathcal {U}_i \cap \varPi _i^{-1}\widetilde{\mathcal {M}}^m_{i+1}&= U_i \cap \mathrm {e}^{-A^\text { diff}_i\tau _i}\varPi _i^{-1}H_{i+1}^{-1}\left( \bigcap \limits _{j=i+1}^m \left( \prod \limits _{k=i+1}^{j-1} \mathrm {e}^{-A_k^\text { diff}\tau _k}\varPi _k^{-1} H_{k+1}^{-1}\right) \mathcal {U}_j\right) \\&= \bigcap \limits _{j=i}^m \left( \prod \limits _{k=i}^{j-1} \mathrm {e}^{-A_k^\text { diff}\tau _k}\varPi _k^{-1} H_{k+1}^{-1}\right) \mathcal {U}_j. \end{aligned} \end{aligned}$$

Inserting this into Eq. (42) yields the induction step. Finally, applying ${{\mathrm{im}}}\varPi _{i-1} \cap \ker O_{i-1}^\text { diff}= {{\mathrm{im}}}\varPi _{i-1} \cap \mathcal {U}_{i-1}$ to (41) for $i=1$ gives the desired result.

Determinability: We show by induction

$$\begin{aligned} \widetilde{\mathcal {N}}_1^i = \varPi _i \left( \mathcal {U}_i \cap \mathrm {e}^{A_i^\text { diff}\tau _i} H_{i} \varPi _{i-1}\left( \ldots \left( \mathcal {U}_1 \cap \mathrm {e}^{A_1^\text { diff}\tau _1} H_{1} \varPi _0 \mathcal {U}_0 \right) \ldots \right. \right) . \end{aligned}$$

For $i=1$ it holds

$$\begin{aligned} \widetilde{\mathcal {N}}_1^1&= \mathrm {e}^{A_1^\text { diff}\tau _1}\varPi _1^\text { diff}H_1 \left( {{\mathrm{im}}}\varPi _0 \cap \ker O_0^\text { diff}\cap \mathcal {U}^{\text { H}}_1\right) . \end{aligned}$$

As the system is normalized, it follows

$$\begin{aligned} \widetilde{\mathcal {N}}_1^1&= \mathrm {e}^{A_1^\text { diff}\tau _1} \varPi _1 H_1 \left( {{\mathrm{im}}}\varPi _0 \cap \ker O_0^\text { diff}\cap H_1^{-1}\mathcal {U}_1\right) . \end{aligned}$$

Applying Lemma 34.4 gives

$$\begin{aligned} \widetilde{\mathcal {N}}_1^1&= \mathrm {e}^{A_1^\text { diff}\tau _1} \varPi _1 \left( H_1 \left( {{\mathrm{im}}}\varPi _0 \cap \ker O_0^\text { diff}\right) \cap \mathcal {U}_1\right) . \end{aligned}$$

Using ${{\mathrm{im}}}\varPi _0 \cap \ker O_0^\text { diff}= \varPi _0 \ker O_0^\text { diff}$, Lemma 35.3, and the $\mathrm {e}^{A_1^\text { diff}\tau _1}$-invariance of $\mathcal {U}_1$ as well as the commutativity of $\varPi _1$ and $\mathrm {e}^{A_1^\text { diff}\tau _1}$ gives the claim for $i=1$.

By $\widetilde{\mathcal {N}}_1^i \subseteq \mathcal {V}^*_{i}$ and $\widetilde{\mathcal {N}}_1^i \subseteq \mathrm {e}^{A^\text { diff}_i \tau _i} \Pi ^\text { diff}_{i} H_{i} \mathcal {U}^{\text { H}}_{i} \subseteq \mathrm {e}^{A^\text { diff}_i \tau _i} \ker O_i^\text { diff}= \ker O_i^\text { diff}$, it follows for $\widetilde{\mathcal {M}}_{i+1}$:

$$\begin{aligned} \widetilde{\mathcal {M}}_{i+1} \cap \widetilde{\mathcal {N}}_1^{i} =\mathcal {U}^{\text { H}}_{i+1} \cap \widetilde{\mathcal {N}}_1^{i}. \end{aligned}$$

Hence, it holds for the induction step $i \rightarrow i+1$:

$$\begin{aligned} \widetilde{\mathcal {N}}_1^{i+1} = \mathrm {e}^{A_{i+1}^\text { diff}\tau _{i+1}} \Pi ^\text { diff}_{i+1} H_{i+1} \left( \mathcal {U}^{\text { H}}_{i+1} \cap \widetilde{\mathcal {N}}_1^{i} \right) . \end{aligned}$$

The same arguments as for the induction start yield

$$\begin{aligned} \widetilde{\mathcal {N}}_1^{i+1} = \varPi _{i+1} \left( \mathcal {U}_{i+1} \cap \mathrm {e}^{A_{i+1}^\text { diff}\tau _{i+1}} H_{i+1}\widetilde{\mathcal {N}}_1^{i} \right) . \end{aligned}$$

Therefore, the statement follows by Theorem 27. $\square$