Seiberg–Witten invariant

Template:No footnotes The notion of orbit of a control system used in mathematical control theory is a particular case of the notion of orbit in group theory.

Definition

Let $\dot{q}=f(q,u)$ be a ${\mathcal{𝒞}}^{\mathrm{\infty }}$ control system, where $q$ belongs to a finite-dimensional manifold $M$ and $u$ belongs to a control set $U$. Consider the family $\mathcal{ℱ}=\{f(\cdot ,u)\mid u\in U\}$ and assume that every vector field in $\mathcal{ℱ}$ is complete. For every $f\in \mathcal{ℱ}$ and every real $t$, denote by $e^{tf}$ the flow of $f$ at time $t$.

The orbit of the control system $\dot{q}=f(q,u)$ through a point $q_0\in M$ is the subset ${\mathcal{𝒪}}_{q_0}$ of $M$ defined by

${\mathcal{𝒪}}_{q_0}=\{e^{t_k{f}_k}\circ e^{t_{k-1}{f}_{k-1}}\circ \cdots \circ e^{t_1{f}_1}(q_0)\mid k\in \mathbb{\mathbb{N}},t_1,\dots ,t_k\in \mathbb{ℝ},f_1,\dots ,f_k\in \mathcal{ℱ}\}.$

Remarks

The difference between orbits and attainable sets is that, whereas for attainable sets only forward-in-time motions are allowed, both forward and backward motions are permitted for orbits. In particular, if the family $\mathcal{ℱ}$ is symmetric (i.e., $f\in \mathcal{ℱ}$ if and only if $-f\in \mathcal{ℱ}$), then orbits and attainable sets coincide.

The hypothesis that every vector field of $\mathcal{ℱ}$ is complete simplifies the notations but can be dropped. In this case one has to replace flows of vector fields by local versions of them.

Orbit theorem (Nagano-Sussmann)

Each orbit ${\mathcal{𝒪}}_{q_0}$ is an immersed submanifold of $M$.

The tangent space to the orbit ${\mathcal{𝒪}}_{q_0}$ at a point $q$ is the linear subspace of $T_{q}M$ spanned by the vectors $P_{*}f(q)$ where $P_{*}f$ denotes the pushforward of $f$ by $P$, $f$ belongs to $\mathcal{ℱ}$ and $P$ is a diffeomorphism of $M$ of the form $e^{t_k{f}_k}\circ \cdots \circ e^{t_1{f}_1}$ with $k\in \mathbb{\mathbb{N}},t_1,\dots ,t_k\in \mathbb{ℝ}$ and $f_1,\dots ,f_k\in \mathcal{ℱ}$.

If all the vector fields of the family $\mathcal{ℱ}$ are analytic, then $T_q{\mathcal{𝒪}}_{q_0}={\mathrm{L}\mathrm{i}\mathrm{e}}_q\mathcal{ℱ}$ where ${\mathrm{L}\mathrm{i}\mathrm{e}}_q\mathcal{ℱ}$ is the evaluation at $q$ of the Lie algebra generated by $\mathcal{ℱ}$ with respect to the Lie bracket of vector fields. Otherwise, the inclusion ${\mathrm{L}\mathrm{i}\mathrm{e}}_q\mathcal{ℱ}\subset T_q{\mathcal{𝒪}}_{q_0}$ holds true.

Corollary (Rashevsky-Chow theorem)

If ${\mathrm{L}\mathrm{i}\mathrm{e}}_q\mathcal{ℱ}=T_{q}M$ for every $q\in M$ and if $M$ is connected, then each orbit is equal to the whole manifold $M$.

References

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