A geometric approach to nonlinear dissipative balanced reduction I: exogenous signals

Abstract

This paper is divided into two parts and provides a unified geometric theory for nonlinear dissipative and Hankel balanced reduction with the help of a framework based on differential geometry, dissipativity theory, Lie-semigroups and sub manifold Hilbert theory. Part I presents a theory for the invariants of the behavior using classical Gauss’ curvature theory. Furthermore, known concepts of the theory of balanced reduction like the Hankel operator, Schmidt decomposition, etc., can be understood in a proper and general perspective.

Appendix

Proof of Proposition 2.2

In the differential geometric framework, duality of these spaces follows from the known contraction operation $i_{\xi } \alpha$ between vector fields and differential 1-forms. Since the inner products in Table 1 are positive definite, symmetric, bilinear forms on (the space of vector fields) $T_p\mathcal W$ (on the space of 1-forms $T_p ^*\mathcal W$) such products are admissible. The remaining details can be seen in [26] and in the more general proof of Proposition 5, pg. 55 assuming $\mathcal W$ an infinite-dimensional Sobolev space (without differential geometry) in [40]. From Definition 2.6, the duality product $\langle \xi , \alpha \rangle _{T\mathcal W \times T^{*}\mathcal W }$ is well defined since for a $\phi \in \mathcal W ^*$ the dual of $d\phi \in T^*\mathcal W$ is given by (the gradient) $\nabla \phi \in T\mathcal W$ satisfying $\phi (w)=\langle \nabla \phi , \dot{w} \rangle _{T\mathcal W }$, $\dot{w} \in T\mathcal W$. Coordinate invariance of $\langle \xi , \alpha \rangle _{T\mathcal W \times T^*\mathcal W }$ follows from known facts of differential geometry (see e.g. [6]).

Proof of Proposition 2.3

Since by construction the inner products satisfy Assumption 2.1 on dissipativity, semipositive definiteness, symmetry and boundedness are satisfied and such inner products are admissible. The remaining Hilbert manifold properties are inherited to this structure from $(\mathcal W , \langle \cdot , \cdot \rangle _{T\mathcal W })$ and $(\mathcal W ^*, \langle \cdot , \cdot \rangle _{T^*\mathcal W })$. Duality of $\alpha _{-}$ with $\xi ^{+}$ is identified by the abstract duality pairing $\langle \xi ^{+}, \alpha _{-} \rangle _{T\mathfrak B ^{-} \times T\mathfrak B ^{+}}$ for an isometry $\tilde{\Gamma }$ s.t. $\xi ^{+}=\tilde{\Gamma }^* \alpha _{-}$, with $\xi ^{+}$ dual to $\alpha _{+}$, which according to Definition 2.6 must satisfy $\langle \alpha _{-}, \alpha _{-} \rangle _{T\mathfrak B ^-}=\langle \xi ^{+}, \alpha _{-} \rangle _{T\mathfrak B ^{-} \times T\mathfrak B ^{+}}=\langle \xi ^{+}, \tilde{\Gamma }^* \,\alpha _{-} \rangle _{T\mathfrak B ^+}$, which is equivalent to $\Vert \alpha _{-}\Vert _{T\mathfrak B ^-}=\Vert \xi ^{+}\Vert _{T\mathfrak B ^+}$.

Proof of Proposition 2.4

1) This result is a consequence of the Definition 2.6 for the duality pairing $\langle \xi ^{+}, \alpha _{-} \rangle _{T\mathfrak B ^{-} \times T\mathfrak B ^{+}}$. By Definition 2.6, duality of $\alpha _{-}$ with $\xi ^{+}$ requires the preservation of the relationship

$$\begin{aligned} \langle \alpha _{-}, \alpha _{-} \rangle _{T\mathfrak B ^-}\!=\!\langle \xi ^{+}, \alpha _{-} \rangle _{T\mathfrak B ^{-} \times T\mathfrak B ^{+}}\!=\!\langle \xi ^{+}, \tilde{\Gamma }^* \,\alpha _{-} \rangle _{T\mathfrak B ^+},\quad \forall \alpha _{-} \in \mathfrak B ^-,\, \xi ^{+} \in \mathfrak B ^+.\nonumber \\ \end{aligned}$$

(41)

for inner products defined as in Proposition 2.3, for an isometric isomorphism $\tilde{\Gamma }:\mathfrak B ^- \rightarrow \mathfrak B ^+$ defined by Eq. (3), which from Definition 2.5 is seen to satisfy commutativity in diagram (15) with the following structure at $T\mathfrak B$:

$$\begin{aligned} \left[\begin{array}{c} \xi _{u} ^{+}(w)\\ \\ \xi _{y}^{+} (w) \end{array} \right]_{T\mathfrak B ^{+}}= \left[\begin{array}{c c} 0&\Gamma ^{\dagger }(w) |_* \\ \Gamma (w)|_*&0 \end{array} \right] \left[\begin{array}{c} \xi _{u}^{-} (w)\\ \xi _{y}^{-} (w) \end{array} \right], \end{aligned}$$

(42)

where the map $\Gamma : \mathcal U \rightarrow \mathcal Y$ has an adjoint map $\Gamma ^{\dagger }: \mathcal Y ^* \rightarrow \mathcal U ^*$.

2) Define by $\tilde{\Gamma }^\dagger :\mathfrak B ^+ \rightarrow \mathfrak B ^-$ an associated operator to the behavioral operator in Definition 2.3 with the following structure at $T\mathfrak B$:

$$\begin{aligned} \left[\begin{array}{l} \alpha _{-} ^{\hat{u}}(\hat{w})\\ \\ \alpha _{-} ^{\hat{y}} (\hat{w}) \end{array} \right]_{T\mathfrak B ^{-}}= \left[\begin{array}{ll} 0&\Gamma ^{\dagger }(\hat{w}) |^* \\ \Gamma (\hat{w})|^*&0 \end{array} \right] \left[\begin{array}{l} \alpha _{+} ^{\hat{u}} (\hat{w})\\ \alpha _{+} ^{\hat{y}} (\hat{w}) \end{array} \right], \end{aligned}$$

(43)

such operator is such that $\langle \xi , \Gamma _*\zeta \rangle _{T\mathfrak B ^-}=\langle \Gamma _*\xi , \zeta \rangle _{T\mathfrak B ^+}$, $\xi ,\zeta \in T\mathfrak B ^+$ satisfying Definition 2.12 and therefore is selfadjoint. Since $\tilde{\Gamma }$ is isometric, it satisfies $\langle \xi ^{-},\zeta ^{-} \rangle _{T\mathfrak B ^-}=\langle \tilde{\Gamma }_* \xi ^{-}, \tilde{\Gamma }_* \zeta ^{-} \rangle _{T\mathfrak B ^+}\equiv \langle \xi ^{+}, \zeta ^{+} \rangle _{T\mathfrak B ^+}$, $\xi ^{-},\zeta ^{-} \in T\mathfrak B ^-$, $\xi ^{+},\zeta ^{+} \in T\mathfrak B ^+$. Let the map $\tilde{\Gamma }^\dagger :\mathfrak B ^{+} \rightarrow \mathfrak B ^{-}$ satisfy $\langle \xi ^{+},\zeta ^{+} \rangle _{T\mathfrak B ^+}\mathop {=}\limits ^\mathrm{def}\langle \tilde{\Gamma }^\dagger _* \xi ^{+}, \tilde{\Gamma }^\dagger _* \zeta ^{+} \rangle _{T\mathfrak B ^-}$ hence isometric, with $\xi ^{-} \mathop {=}\limits ^\mathrm{def}\tilde{\Gamma }^\dagger _* \xi ^{+}$. Then Definition 2.11 is verified since $\langle \tilde{\Gamma }^\dagger _* \xi ^{+}, \zeta ^{-} \rangle _{T\mathfrak B ^-}=\langle \xi ^{+}, \tilde{\Gamma }_* \zeta ^{-} \rangle _{T\mathfrak B ^+}$.

3) Since we may write $\langle \xi ^{-}, \zeta ^{-} \rangle _{T\mathfrak B ^-}=\langle (\tilde{\Gamma }^\dagger \circ \tilde{\Gamma })_* \xi ^{-}, \zeta ^{-} \rangle _{T\mathfrak B ^-}=\langle \xi ^{-}, (\tilde{\Gamma }^\dagger \circ \tilde{\Gamma })_* \zeta ^{-} \rangle _{T\mathfrak B ^-}$ and $\langle \xi ^{+}, \zeta ^{+} \rangle _{T\mathfrak B ^+}=\langle \xi ^{+}, (\tilde{\Gamma }\circ \tilde{\Gamma }^\dagger )_* \zeta ^{+} \rangle _{T\mathfrak B ^+}=\langle (\tilde{\Gamma }\circ \tilde{\Gamma }^\dagger )_* \xi ^{+}, \zeta ^{+} \rangle _{T\mathfrak B ^+}$, the operators $\tilde{\Gamma }^\dagger \circ \tilde{\Gamma }(\cdot )$ and $\tilde{\Gamma }\circ \tilde{\Gamma }^\dagger (\cdot )$ are self-adjoint on $\mathfrak B ^-$ and $\mathfrak B ^+$, respectively, satisfying Definition 2.12.

4) Direct comparison of operators $\tilde{\Gamma }$ and $\tilde{\Gamma }^\dagger$ in Eqs. (3) and (16) shows that both are equivalent and their tangent maps Eqs. (42) and (43) satisfy Definition 2.12.

5) Denote by $\mathcal W |_\mathcal{U },\mathcal W |_\mathcal{Y }$ the restriction of a manifold $\mathcal W \supset \mathcal U \oplus \mathcal Y$ by their disjoint subsets and consider the restricted maps $\Gamma : \mathfrak B ^-|_{u} \rightarrow \mathfrak B ^+|_{y}$, $\Gamma ^\dagger : \mathfrak B ^-|_{y} \rightarrow \mathfrak B ^+|_{u}$. Since we may write $\langle \xi ^{-}, \zeta ^{-} \rangle _{T\mathfrak B ^-|_{u}}=\langle (\Gamma ^\dagger \circ \Gamma )_* \xi ^{-}, \zeta ^{-} \rangle _{T\mathfrak B ^-|_{u}}=\langle \xi ^{-}, ( \Gamma ^\dagger \circ \Gamma )_* \zeta ^{-} \rangle _{T\mathfrak B ^-|_{u}}$ and $\langle \xi ^{+}, \zeta ^{+} \rangle _{T\mathfrak B ^+|_{y}}=\langle \xi ^{+}, (\Gamma \circ \Gamma ^\dagger )_* \zeta ^{+} \rangle _{T\mathfrak B ^+|_{y}}=\langle (\Gamma \circ \Gamma ^\dagger )_* \xi ^{+}, \zeta ^{+} \rangle _{T\mathfrak B ^+|_{y}}$, the operators $\Gamma ^\dagger \circ \Gamma (\cdot )$ and $\Gamma \circ \Gamma ^\dagger (\cdot )$ are self-adjoint on $\mathfrak B ^-|_{u}$ and $\mathfrak B ^+|_{y}$, respectively, satisfying Definition 2.12. $\square$

Proof of Proposition 2.6

1) Consider the map $\Upsilon ^t: \mathbb R ^1 \times \mathcal W \rightarrow \mathcal W$ for the first eigenproblem $\Upsilon (\varrho (t))=\lambda \varrho (t)$ and some eigenvalue $\lambda$ along the path $\varrho (t) \in \mathcal W$. Such path is the integral trajectory of a vector field satisfying $\frac{d\varrho (t)}{dt}= \xi (\varrho (t))$. On the other side, time derivation at both sides of the equation defining the eigenvalue problem yields $\frac{\partial \Upsilon }{\partial x}\, \frac{d\varrho (t)}{dt} =\lambda \frac{d\varrho (t)}{dt}$ which is precisely $\Upsilon _*|_x\,\xi =\lambda \xi$ (a linear eigenvalue problem) and then the problem is well defined. The inverse implication is straightforward. Since a generator $\xi$ of $\Upsilon ^t$ acts on differentiable functions as $\xi f=\frac{\mathrm{d}}{\mathrm{d}t} f(\Upsilon ^t(x))|_{t=0}$ a similar reasoning follows for the proof of Proposition 2.6(2). $\square$

Proof of Proposition 2.7

1) Evident from the differentiable structure of $\tilde{\Gamma }$.

2) Express the map $\tilde{\Gamma }_*$ as a matrix with components $G_i ^j=[\partial _{w^j} \tilde{\Gamma }_{i}]$ and rewrite the eigenproblem (18) for each entry as $\hat{w}_i ^{j}(0)+ \int _{0} ^{\infty } \sum _{j} G_i ^j \beta _{j} ^i(w^{-}(t)) \,\mathrm{d}\mu (\tau )$ which defines a system of $1$st-kind Fredholm equations (for generic definitions see [30]).

3) Using Proposition 2.6(1) the eigenvalue problem in terms of $T\mathfrak B$ with tangent map $\Gamma _*:T\mathfrak B \rightarrow T\mathfrak B$ yields the eigenproblem presented above with tangent eigenvectorfield $\xi _{i} \in T\mathfrak B$.

Proof of Theorem 3.1

1) Since by Assumption 2.3, $\mathfrak g _\mathfrak{B } ^{-}(w^{-},0) \mathop {=}\limits ^\mathrm{def}S_r ^*(\hat{w}^0,r_r)=\langle \alpha _{-} , \alpha _{-} \rangle _{T\mathfrak B ^-}$, $\mathfrak g _\mathfrak{B } ^{+}(w^{+},0) \mathop {=}\limits ^\mathrm{def}S_a(w^0,r_a) =\langle \xi ^{+}, \xi ^{+} \rangle _{T\mathfrak B ^+ }$ which by the isometric isomorphism $\xi ^{+}=\tilde{\Gamma }^* \alpha _{-}$ transforms the future metrics in terms of the past metrics by $\langle \xi ^{+}, \xi ^{+} \rangle _{T\mathfrak B ^+ }=\langle \tilde{\Gamma }^* \alpha _{-} , \tilde{\Gamma }^* \alpha _{-} \rangle _{T\mathfrak B ^+ }=\langle \tilde{\Gamma }_* ^\dagger \tilde{\Gamma }^*\alpha _{-} , \alpha _{-} \rangle _{T\mathfrak B ^-}$, and therefore $A_\eta ^\mathfrak B =\tilde{\Gamma }_* ^\dagger \tilde{\Gamma }^*$.

2) Since the quotient (23) is such that $K(\varepsilon \alpha _{-} )=K(\alpha _{-} )$ for any scalar $0 \ne \varepsilon \in \mathbb R ^1$ and $\alpha _{-} \in T_p\mathfrak B ^{-}$, this implies that any tangent 1-form candidate solution to the eigenvalue problem satisfies $\langle \alpha _{-} , \alpha _{-} \rangle _{T\mathfrak B ^{-}}=1$. Therefore, denote by $\beta _{-}$ the elements of a sphere $\mathcal S \subset T\mathfrak B ^{-}$ defined by $\mathcal S =\{\beta _{-} \in T\mathfrak B ^{-} | \langle \beta _{-}, \beta _{-} \rangle _{T\mathfrak B ^{-}}=1\}$. By the Stone-Weierstrass Theorem (e.g. [40]) any continuous function supported on $\mathcal S$ (being a compact subset) attains in there its maximum and minimum. Express the numerator of (23) by $\phi :\mathcal S \rightarrow \mathbb R ^1$, i.e., $\phi (\beta _{-})\mathop {=}\limits ^\mathrm{def}II_\mathfrak{B }(\beta _{-}, \beta _{-})=\langle A_\eta ^\mathfrak B \beta _{-}, \beta _{-} \rangle _{T\mathfrak B ^{-}}$. A point $\beta _{-} ^0 \in T\mathfrak B ^{-}$ is a critical point in $\mathcal S$ if for any curve $\varrho :[-1,1] \rightarrow \mathcal S$, $\varrho (0)=\beta _{-} ^0$, satisfies $d \phi (\varrho )/d\tau |_{\tau =0}=0$. Performing this operation yields,

$$\begin{aligned} \left. \frac{\mathrm{d}}{\mathrm{d}\tau } \langle A_\eta ^\mathfrak B \varrho (\tau ), \varrho (\tau ) \rangle _{T\mathfrak B ^{-}} \right|_{\tau =0}&= \langle A_\eta ^\mathfrak B \dot{\varrho }(0), \varrho (0) \rangle _{T\mathfrak B ^{-}}+\langle A_\eta ^\mathfrak B \varrho (0), \dot{\varrho }(0) \rangle _{T\mathfrak B ^{-}} \\&=\langle A_\eta ^\mathfrak B \dot{\varrho }(0), \beta _{-} ^0 \rangle _{T\mathfrak B ^{-}}+\langle A_\eta ^\mathfrak B \beta _{-} ^0, \dot{\varrho }(0) \rangle _{T\mathfrak B ^{-}}\\&= 2 \langle A_\eta ^\mathfrak B \beta _{-} ^0, \dot{\varrho }(0) \rangle _{T\mathfrak B ^{-}} \end{aligned}$$

being the last equality due to selfadjointness of $A_\eta ^\mathfrak B$. This result implies that $\beta _{-} ^0$ is a critical point of $\phi (\beta _{-})$ on $\mathcal S$ if $A_\eta ^\mathfrak B \beta _{-} ^0$ is orthogonal to any tangent vector field $\dot{\varrho }(0) \in T\mathcal S$. Due to the geometry of the sphere $\mathcal S$, any tangent vector field on $T\mathcal S$ is orthogonal to a (possibly scaled) normal vector field on $\mathcal S$. Thus if $\beta _{-} ^0$ is an eigenform of $A_\eta ^\mathfrak B$ there is a $\kappa$ scaling $\beta _{-} ^0$ normal in $\mathcal S$, implying $\beta _{-}$ is a critical point of $\phi$, concluding the proof.

3) By induction. Let $\mathrm{dim} \mathcal W =\omega$ for $\omega =1$ the result is trivial. Assume it is true for $\omega =k$. Let $\omega =k+1$; from Theorem 3.1(2), there exists at least a unitary 1-form $\beta _{-} ^1 \in T_p \mathfrak B ^{-}$ of $A_\eta ^\mathfrak B$. By Remark 2.5, define locally an orthogonal complementary codistribution by $W_1 ^*=\{\alpha _{-} \,|\, \langle \beta _{-} ^1, \alpha _{-} \rangle _{T\mathfrak B ^{-}}=0,\, \alpha _{-} \in T_p\mathfrak B ^{-}\}$ then since $\langle A_\eta ^\mathfrak B \alpha _{-},\beta _{-} ^1 \rangle _{T\mathfrak B ^{-}}=\langle \alpha _{-}, A_\eta ^\mathfrak B \beta _{-} ^1 \rangle _{T\mathfrak B ^{-}}=\langle \alpha _{-},\lambda _1 \beta _{-} ^1 \rangle _{T\mathfrak B ^{-}}=\lambda _1 \langle \alpha _{-},\beta _{-} ^1 \rangle _{T\mathfrak B ^{-}}=0$, locally $W_1 ^*$ is an $A_\eta ^\mathfrak B$-invariant eigencodistribution, i.e., $A_\eta ^\mathfrak B W_1 ^* \subseteq W_1 ^*$. It is easy to see that the restriction of $A_\eta ^\mathfrak B$ to $W_1 ^*$, $A_\eta ^\mathfrak B |_{W_1 ^*}$ is also self-adjoint. Since $\mathrm{dim} W_1 ^*= \mathrm{dim} \mathcal W -1=k$ there are $k$-basis 1-forms $\{ \beta _{-} ^2, \ldots , \beta _{-} ^{\omega } \}$ for $W_1 ^*$ that are eigenforms of $A_\eta ^\mathfrak B |_{W_1 ^*}$. Though, each eigenform of $A_\eta ^\mathfrak B |_{W_1 ^*}$ must be an eigenform of $A_\eta ^\mathfrak B$. Conclude that the eigenforms of $A_\eta ^\mathfrak B$, $\{\beta _{-} ^1, \beta _{-} ^2, \ldots , \beta _{-} ^{\omega } \} \in T_p\mathfrak B ^{-}$ must be an orthonormal basis for $T_p\mathfrak B ^{-}$ at each $p \in \mathfrak B$ and thereby $T_p\mathfrak B ^{-}$ can be partitioned locally in eigencodistributions as $W_1 ^* \oplus \ldots \oplus W_\omega ^*$.

Proof of Proposition 3.1

Each component of $\mathbb G$ can be calculated from $g_{ij}=\langle \beta _i,\beta _j\rangle _{T\mathfrak B ^-}$. Since $\langle \beta _j,\eta \rangle _{T\mathfrak B ^{-}}=0$ for $\beta _j \in T\mathfrak B ^-$, $\eta \in (T\mathfrak B ^-)^\perp$, then after derivation $\partial _{\beta _i}\langle \beta _j,\eta \rangle _{T\mathfrak B ^-}=\langle \partial _{\beta _i}\beta _j,\eta \rangle _{T\mathfrak B ^-}+\langle \beta _j,\partial _{\beta _i}\eta \rangle _{T\mathfrak B ^-}=0$ yields $\langle \partial _{\beta _i} \beta _j,\eta \rangle _{T\mathfrak B ^-}=\langle \beta _j,-\partial _{\beta _i} \eta \rangle _{T\mathfrak B ^-}=\langle \beta _j,-\bar{\nabla }_\eta \beta _i \rangle _{T\mathfrak B ^-}$. This implies that $II_\mathfrak{B }(\beta _i,\beta _j):=\langle A_\eta ^\mathfrak B \beta _i, \beta _j \rangle _{T\mathfrak B ^-}=\langle \partial _{\beta _i} \beta _j,\eta \rangle _{T\mathfrak B ^-}$. In consequence the components of $\mathbb Q$ can be calculated from $q_{ij}=\langle \partial _{\beta _i} \beta _j,\eta \rangle _{T\mathfrak B ^-}$. Now since

$$\begin{aligned} q_{ij}\!=\!\langle A_\eta ^\mathfrak B \beta _i,\beta _j \rangle _{T \mathfrak B ^-}=\left< \sum _{k=1} ^{\omega } a^{ik}\beta _k , \beta _j \right>_{T\mathfrak B ^-}\!=\! \sum _{k=1} ^{\omega } a^{ik} \langle \beta _k, \beta _j \rangle _{T\mathfrak B ^-}= \sum _{k=1} ^{\omega } a^{ik} g_{kj}\nonumber \\ \end{aligned}$$

(44)

conclude that $\mathbb Q =A_\eta ^\mathfrak B \mathbb G$. Since $\mathbb G$ is a positive definite matrix then it is invertible and multiplication by its inverse yields the desired result. By linear algebra arguments $K(\beta )=\det \mathbb Q /\det \mathbb G$, is another expression for Gauss (total) curvature.

Proof of Proposition 3.2

Any vector field $\xi ^{+} \in T_w \mathfrak B ^{+}$ (1-form $\alpha _{-} \in T_w \mathfrak B ^{-}$) can be expressed in local coordinates by a combination of their (co-) frame elements $\xi ^{+}=\sum _{i=1} ^{\omega } a_i \zeta _i ^{+}$ ( resp. $\alpha _{-}=\sum _{i=1} ^{\omega } b_i \beta _{-} ^{i}$) for scalar functions $a_i, b_i \in C^{\infty }(\mathcal W )$, see e.g. Chapter 8 in [25]. In particular, let $\xi ^{+} \in T_w \mathfrak B ^{+}$ have an orthogonal projection $\xi _{\text{ op}} ^{+} \in \mathfrak B _{\text{ red}} ^{+} \subseteq \mathfrak B ^{+}$ on a submanifold satisfying $T\mathfrak B _{\text{ red}} ^{+} \mathop {=}\limits ^\mathrm{def}\mathrm{span} \{ \zeta _1 ^{+}, \zeta _2 ^{+}, \ldots \zeta _{\varpi } ^{+} \}$, $\varpi \le \omega$. Then their difference $\xi ^{+}- \xi _0 ^{+}$ must be orthogonal to each element of the frame (by duality, see Remark 2.6), i.e., $\langle \xi ^{+}- \xi _{\text{ op}} ^{+}, \zeta _i ^{+} \rangle _{T\mathfrak B ^{+}}=0$, $i=1, \ldots ,\varpi$ where orthogonality can be defined in terms of the duality pairing $\langle \xi ^{+}, \alpha _{-} \rangle _{T\mathfrak B ^{-} \times T\mathfrak B ^{+}}$, see Remark 2.5 and Definition 2.8. Express $\xi _{\text{ op}} ^{+}$ by the $\varpi$-elements of the frame. The $a_i$ satisfying $\langle \xi ^{+}- \sum _{i=1} ^{\varpi } a_i \zeta _i ^{+}, \zeta _i ^{+} \rangle _{T\mathfrak B ^{+}}=0$ is equal to the projection of each vector field on the elements of the frame, i.e., $a_i=\langle \xi ^{+},\zeta _i ^{+} \rangle _{T\mathfrak B ^{+}}$. An equivalent reasoning for $\alpha _{-} \in T_w \mathfrak B ^{-}$, results in $b_i=\langle \alpha _{-},\beta _{-} ^{i} \rangle _{T\mathfrak B ^{-}}$, concluding the proof.

Proof of Theorem 3.2

Under the orthonormal frame $\{ \beta _{-} ^i\}$ of Proposition 3.2 any $\alpha _{-} \in T_w \mathfrak B ^{-}$, $\Vert \alpha _{-} \Vert =1$, can be expressed as $\alpha _{-}=\sum _{i=1} ^{\omega } \langle \alpha _{-},\beta _{-} ^{i} \rangle _{T\mathfrak B ^{-}} \beta _{-} ^i$. Since $\Vert \alpha _{-} \Vert =1$ then $K(\alpha _{-})= \langle A_\eta ^\mathfrak B (\alpha _{-}),\alpha _{-} \rangle _{T\mathfrak B ^{-}}$ can be expressed as

$$\begin{aligned} K(\alpha _{-})&= \left< A_\eta ^\mathfrak B \sum _{i=1} ^{\omega } \langle \alpha _{-},\beta _{-} ^{i} \rangle _{T\mathfrak B ^{-}} \beta _{-} ^i,\sum _{i=1} ^{\omega } \langle \alpha _{-},\beta _{-} ^{i} \rangle _{T\mathfrak B ^{-}} \beta _{-} ^i \right>_{T\mathfrak B ^{-}}\\&= \left< \sum _{i=1} ^{\omega } \kappa _i(w) \langle \alpha _{-},\beta _{-} ^{i} \rangle _{T\mathfrak B ^{-}} \beta _{-} ^i,\sum _{i=1} ^{\omega } \langle \alpha _{-},\beta _{-} ^{i} \rangle _{T\mathfrak B ^{-}} \beta _{-} ^i \right>_{T\mathfrak B ^{-}} \end{aligned}$$

notice that $\left< \beta _{-} ^{i}, \beta _{-} ^{i}\right>_{T\mathfrak B ^{-}}=1$ (by orthogonality) and we may write

$$\begin{aligned} K(\alpha _{-})&= \sum _{i=1} ^{\omega } \kappa _i(w) \langle \alpha _{-},\beta _{-} ^{i} \rangle _{T\mathfrak B ^{-}} ^2 \int _{-\infty } ^{0} \beta _{j} ^i \beta _{j} ^i \;\mathrm{d}\mu (t)= \sum _{i=1} ^{\omega } \kappa _i(w) \langle \alpha _{-},\beta _{-} ^{i} \rangle _{T\mathfrak B ^{-}} ^2 \end{aligned}$$

and since the angle between $\alpha _{-}$ and $\beta _{-} ^{i}$ is given by $\theta _i=\mathrm{angcos}(\langle \alpha _{-},\beta _{-} ^{i} \rangle _{T\mathfrak B ^{-}})$, yields (29). $\square$

Proof of Proposition 3.3

1) From Eq. (30) $w_{-} ^i=\frac{1}{\sigma _i}\Gamma \circ \hat{w}_i(\tau )$ substitution in (31) yields $\Gamma ^\dagger \circ \Gamma \circ w_i ^{-}=\sigma _i^2 w_i ^{-}$ with $\lambda _i =\sigma _i ^2$ we obtain Eq. (32). Substitution of Eqs. (31) in (30) and the same steps verifies Eq. (33).

2) At the tangent space, Eqs. (32) and (33) are written as

$$\begin{aligned} (\tilde{\Gamma }^\dagger \circ \tilde{\Gamma })|^* \alpha _{-} ^{i}-\lambda _i \alpha _{-} ^{i}&= 0, \quad i=1, \ldots , \omega , \end{aligned}$$

(45)

$$\begin{aligned} (\tilde{\Gamma }\circ \tilde{\Gamma }^\dagger )|_* \xi _{i} ^{+}-\lambda _i \xi _{i} ^{+}&= 0, \quad i=1, \ldots , \omega . \end{aligned}$$

(46)

which (by commutativity of composition under the differential operation) is precisely the eigenvalue problem for the selfadjoint Shape operator in Theorem 3.1(1). By Theorem 3.1(2)–(3) the elements of the coframe $\mathrm{span} \{ \beta _{-} ^{1}, \ldots ,\beta _{-} ^{\omega }\}=T_w \mathfrak B ^{-}$ are solution to the eigenproblem (45). Notice that since $\zeta _{i} ^{+}=\tilde{\Gamma }^* \beta _{-} ^{i}$ the elements of the frame $\mathrm{span} \{ \zeta _1 ^{+}, \ldots ,\zeta _{\omega } ^{+} \}=T_w \mathfrak B ^{+}$ are solution of the eigenproblem (46).

3) From Proposition 3.2 the orthogonal projection of $\xi ^{+} \in T_w \mathfrak B ^{+}$ is given by Eq. (27). Since $\xi ^{+}=\tilde{\Gamma }^* \alpha _{-}$ is on $T_w \mathfrak B ^{+}$, we obtain $\tilde{\Gamma }^* \alpha _{-}=\sum _{i=1} ^{\omega } \langle \tilde{\Gamma }^* \alpha _{-},\zeta _i ^{+} \rangle _{T\mathfrak B ^{+}} \zeta _i ^{+}$ where after dualization from Remark 2.6, the inner product can be written as $\langle \tilde{\Gamma }^* \alpha _{-},\tilde{\Gamma }^* \beta _{-} ^i \rangle _{T\mathfrak B ^{-} \times T\mathfrak B ^{+}}=\langle \tilde{\Gamma }_* ^\dagger \tilde{\Gamma }^* \alpha _{-}, \beta _{-} ^i \rangle _{T\mathfrak B ^{-}}$. Since by Theorem 3.1(3), $A_{\eta } ^\mathfrak{B } \alpha _{-}=\tilde{\Gamma }_* ^\dagger \tilde{\Gamma }^* \alpha _{-}$, we may write $\tilde{\Gamma }^* \alpha _{-}=\sum _{i=1} ^{\omega } \langle \lambda _i \alpha _{-}, \beta _{-} ^{i} \rangle _{T\mathfrak B ^{-}} \zeta _i ^{+}=\sum _{i=1} ^{\omega } \lambda _i \langle \alpha _{-}, \beta _{-} ^{i} \rangle _{T\mathfrak B ^{-}} \zeta _i ^{+}$, i.e., Eq. (34). Departing from the orthogonal projection of $\alpha _{-}=\tilde{\Gamma }_* ^\dagger \xi ^{+}$ from Eq. (28) $\tilde{\Gamma }_* ^\dagger \xi ^{+}=\sum _{i=1} ^{\omega } \langle \Gamma _* ^\dagger \xi ^{+},\beta _{-} ^{i} \rangle _{T\mathfrak B ^{-}} \beta _{-} ^{i}$, $\alpha _{-} \in T_w \mathfrak B ^{-}$. Since $\langle \Gamma _* ^\dagger \xi ^{+},\beta _{-} ^{i} \rangle _{T\mathfrak B ^{-}}=\langle \xi ^{+}, \Gamma ^* \Gamma _* ^\dagger \zeta _{i} ^{+} \rangle _{T\mathfrak B ^{+}}$ implying $\langle \xi ^{+}, \lambda _i \zeta _{i} ^{+} \rangle _{T\mathfrak B ^{+}}=\lambda _i \langle \xi ^{+}, \zeta _{i} ^{+} \rangle _{T\mathfrak B ^{+}}$ and Eq. (35) is obtained. $\square$

Proof of Proposition 3.5

1) By condition (39) $S_r ^* (\hat{w}^0,r_r)$ and $S_a (w^0,r_a)$ are orthogonally separated on their respective spaces. Hence, associated with each integral invariant function $S_r ^i$ is an (exact) differential 1-form $\zeta _i ^{+}$ defining a distribution $\Delta _i$ with annihilator $\Delta _i ^{\perp }\mathop {=}\limits ^\mathrm{def}\{\alpha _{-} \in T_p \mathfrak B ^{-} \,|\, \langle \alpha _{-}, \zeta _i ^{+} \rangle _{T\mathfrak B ^{-} \times T\mathfrak B ^{+}}=0,\, \zeta _i ^{+} \in \Delta _i \}$ (dual to the distribution obtained in Proposition 3.1(3)). From Definition 3.5, the set of annihilators $\{ \Delta _1 ^{\perp } ,\ldots ,\Delta _\omega ^{\perp } \}$ is an orthogonal web. Similarly, associated with each integral invariant function $S_a ^{i}$ is an exact differential 1-form $\beta _{-} ^i$ and the family of annihilators $\{ \Omega _1 ^{\perp }, \ldots , \Omega _\omega ^{\perp } \}$, defined by $\Omega _i ^{\perp }\mathop {=}\limits ^\mathrm{def}\{\xi ^{+} \in T_p \mathfrak B ^{+} \,|\, \langle \beta _{-} ^i, \xi ^{+} \rangle _{T\mathfrak B ^{-} \times T\mathfrak B ^{+}}=0,\, \beta _{-} ^i \in \Omega _i \}$ defining the orthogonal web for $S_a (w^0,r_a)$.

2) Since by virtue of Theorem 3.1 each element in the set of orthogonal differential 1-forms $\{\beta _{-} ^{i}\} \in T^*\mathcal W$ has an associated integral functional $\hat{S}_i \in \mathcal W$, from the proof of Theorem 3.1(2), each principal direction $\beta _{-} ^{i}$ is normal to an orthogonal hypersurface $\hat{\mathcal{N }}_i$ defined by a (locally) integrable distribution (i.e. its annihilator) whose integral invariant function is precisely $S_i \in \mathcal W$. By duality, the set of tangent vector fields $\{\zeta _{i} ^{+}\} \in T\mathcal W$ is normal to an orthogonal hypersurface $\mathcal N _i$ defined by a (locally) integrable codistribution (i.e. its annihilator) whose integral invariant function is precisely $S_i \in \mathcal W$. $\square$

Proof of Lemma 3.1

Given a completely separated function $S(w)$ on $\mathcal W$, a Riemannian metric for $\mathcal W$ can be defined by $ds^2=\sum _{i=1} ^{n} S_i(w) \, ds_i ^2,$ and $\{(\mathcal W _i, S_i)\}$ defines a partition of unity. For a principal frame-balanced realization $S_r ^*(\hat{w}^0,r)$ and $S_a(w^0,r)$ can be expressed as in Eq. (36) and metrics defined by $ds_p^2=\sum _{i=1} ^{\omega } S^r _i(w) \, ds_i ^2$ and $ds_f ^2=\sum _{i=1} ^{\omega } S^a _i(w) \, ds_i ^2$, with partitions of unity $\{(\mathcal W _i, S^r _i)\}$ and $\{(\mathcal W _i, S^a _i)\}$ are well-defined Riemannian metrics since each $S_i ds_i ^2$ makes sense on all $\mathcal W$ and $S_i=0$ outside each $\mathcal W _i$.

Proof of Proposition 4.1

Steps 1) and 2) are slight modifications of the corresponding arguments presented in Part II of this paper (Propositions 3.1, 3.2 and 3.5 for internal signals) and therefore their proofs are omitted. For step 3), we depart from an extended group $\{ \Phi ^t(w)\}_{t}$, regular by assuming Eq. (38) satisfied. We are looking for a bijection $\mathcal R (w) \in C^\infty (\mathcal W )$ defining a regular equivalence relation compatible with the group, i.e., $w^1\mathcal R w^2, \Phi \in \{ \Phi ^t(w)\}_{t}$, $\Rightarrow$ $\Phi (w^1) \mathcal R \Phi (w^2)$, where $\mathcal R _i(w) \in C^\infty (\mathcal W ), \; i=1 \ldots \omega$ are components of a vector function $\mathcal R (w): \mathbb R ^\omega \rightarrow \mathbb R ^{\omega -r},\; w \in \mathcal W$ s.t. $\det [\partial \mathcal R (w)/\partial w] \ne 0, \forall w \in \mathcal W$. Since the orbit of $\{ \Phi ^t(w)\}_{t \ge 0}$ is given by the set $\mathcal O \mathop {=}\limits ^\mathrm{def}\{ w \in \mathcal O | \Phi ^t(w) \in \mathcal O , \}$ the natural equivalence relation for all trajectories on the orbit $\mathcal O$ is precisely given by the natural projection $\pi : \mathcal W \rightarrow \mathcal W /\mathcal R$ since if two points $w^1, w^2 \in \mathcal W$ lie on the same orbit of $\Phi \in \{ \Phi ^t(w)\}_{t}$ such that $\Phi \cdot w \in \mathcal W$, the induced equivalence relation among them is given by $\pi (w^1)=\pi (w^2)$ s.t. $\pi (\Phi \cdot w)=\pi (w)$. In such regular equivalence relation, every equivalence class in the quotient set $\mathcal W /\Phi$ is an $r$-dimensional submanifold $\mathcal R _i(w)=k_i ,\; i=1, \ldots , r$, for constants $k_i \in \mathbb R ^1$, [6, 24] (see also [34]). It is a bijection since if $w \in \mathcal W /\Phi$, then its corresponding orbit in $\mathcal W$ is given by $\pi ^{-1}\{w\}$. Finally, since by assumption $\{ \Phi ^t(w)\}_{t}$ acts regularly on $\mathcal W$ and $\pi$ is open in $\mathcal W /\Phi$, then $\mathcal W /\Phi$ is endowed with the structure of a smooth manifold [6, 24].

4) Associated with each separated component function $S_r ^i (\hat{w})$ and $S_a ^i (w)$ is a subgroup $\Phi ^t _i (w)=-\nabla _w S_a ^i (w)$ whose orbit is defined on their corresponding orthogonal web as in Definition 3.5. This is natural since each generating function is an integral invariant of the groups it generates. There is thus a partition of the space s.t.

$$\begin{aligned} \begin{array}{c c l} \mathcal W _a&\mathop {=}\limits ^\mathrm{def}&\mathcal W _1\oplus \mathcal W _2 \oplus \cdots \oplus \mathcal W _r, \\ \mathcal W _b&\mathop {=}\limits ^\mathrm{def}&\mathcal W _{r+1}\oplus \mathcal W _{r+2} \oplus \cdots \oplus \mathcal W _{\omega }, \end{array} \;\; r \le \omega , \end{aligned}$$

(47)

Since associated with each subgroup there is a submanifold $\mathcal W _i \subset \mathcal W$ and a particular principal curvature $\kappa _i$, a local partial order for the submanifolds $\mathcal W _i$ can be provided by inequality relations for the principal curvatures (singular values): $\kappa _1 \ge \kappa _2 \ge \cdots \ge \kappa _\omega$.

5) Consider the submanifold partition in (47). By orthogonality a quotient manifold $\mathcal W /\mathcal W _b$ can be defined always and the reduced subsystem can be obtained with the use of the natural projection $\pi _b: \mathcal W \rightarrow \mathcal W /\mathcal W _b$, see [24]. In view of Proposition 3.2, the generator vector field $\xi _{\text{ op}} ^{+}$ restricted to the quotient space $\mathcal W /\mathcal W _b$ defines uniquely the reduced-order (quotient) system.

Proof of Proposition 4.2

Consider the set of past trajectories $\hat{w} \in \mathfrak B _{\text{ red}}^{-} \subset \mathfrak B ^{-}$ adapted to the behavior $\mathfrak B ^{-}$. From Eq. (28) any such adapted past trajectory has an adapted tangent vector field written as $\alpha _{-}=\sum _{i=1} ^{r} b_i \beta _{-} ^{i}$, $b_i \in C^{\infty }(\mathcal W )$, $\alpha _{-} \in \mathfrak B ^{-}$. Consider set of input vector fields given by $\bar{\alpha }_{-}=\alpha _{-}+b_{r+1} \beta _{-} ^{r+1}$ for $\Vert \tilde{\Gamma }-\tilde{\Gamma }_{\text{ red}} \Vert _\mathfrak{B }$. Since by construction $0 \ne b_{r+1}\in C^{\infty }(\mathcal W )$ can be selected such that $\bar{\alpha }_{-} \in \mathrm{Ker}\, \tilde{\Gamma }_{\text{ red}}$, $\Vert (\tilde{\Gamma }-\tilde{\Gamma }_{\text{ red}})\circ \hat{w}_{-} \Vert _\mathfrak{B ^{+}} ^2=\Vert \tilde{\Gamma }\circ \hat{w}_{-} \Vert _\mathfrak{B ^{+}} ^2$ using Eq. (34) and Euler’s formula (29) the squared norm becomes $\langle \tilde{\Gamma }^* \bar{\alpha }_{-},\tilde{\Gamma }^* \bar{\alpha }_{-} \rangle _{T\mathfrak B ^{+}}= \sum _{i=1} ^{r+1} \kappa _i(w) \langle \alpha _{-},\beta _{-} ^{i} \rangle _{T\mathfrak B ^{-}} ^2= \sum _{i=1} ^{r+1} \kappa _i(w) \langle b_i\beta _{-} ^{i},\beta _{-} ^{i} \rangle _{T\mathfrak B ^{-}} ^2$ which by orthonormality of $\beta _{-} ^{i}$ yields $\sum _{i=1} ^{r+1} \kappa _i(w) b_i ^2 \ge \kappa _{r+1}(w) \sum _{i=1} ^{r+1} b_i ^2=\kappa _{r+1}(w) \Vert w\Vert _\mathfrak{B ^{-}} ^2$, concluding the proof.