Characterization of a topological obstruction to reach control by continuous state feedback

Abstract

This paper studies a topological obstruction to solving the reach control problem (RCP) by continuous state feedback. Given a simplex and given an affine control system defined on the simplex, the RCP is to find a state feedback to drive closed-loop trajectories initiated in the simplex through an exit facet, without first exiting through other facets. We distill the problem as one of continuously extending a function that maps into a sphere from the boundary of a simplex to its interior. As such, we employ techniques from the extension problem of algebraic topology. Unlike previous work on the same problem, in this paper we remove unnecessary restrictions on the dimension of the simplex, the number of inputs of the system, and the particular geometry of the subset of the state space where the obstruction arises. Thus, the results of this paper represent the culmination of our efforts to characterize the topological obstruction. The conditions obtained in the paper are easily checkable and fully characterize the obstruction.

Appendix

Lemma 11

([18]) Suppose (A3) holds. Let $y\in {{\mathbb {S}}}^{m-1}$. Suppose $y\in {{\mathcal {C}}}_j$. The stereographic projection of ${{\mathcal {C}}}_j\backslash \{-y\}$ centered at $-y$ equals:

• (i) a closed half-space in ${{\mathbb {R}}}^{m-1}$ if $h_j\cdot y=0$,

• (ii) an $m-1$-dimensional closed ball in ${{\mathbb {R}}}^{m-1}$ if $h_j\cdot y<0$.

Lemma 12

Suppose (A3) and (A4) hold. ${{\mathcal {H}}}$ is locally contractible.

Proof

Since local contractibility is a local property, we need only study a neighborhood of any point in ${{\mathcal {H}}}$. To that end, let $x \in {{\mathcal {H}}}$ and suppose without loss of generality $x \in {{\mathcal {H}}}_1 \cap \cdots \cap {{\mathcal {H}}}_r$, and $x \notin {{\mathcal {H}}}_{r+1},\ldots ,{{\mathcal {H}}}_{\kappa +1}$. Since all ${{\mathcal {H}}}_j$’s are closed in ${{\mathbb {S}}}^{m-1}$ there is a neighborhood ${{\mathcal {W}}}$ of x such that ${{\mathcal {W}}}\cap {{\mathcal {H}}}= {{\mathcal {W}}}\cap \bigcup _{j=1}^r {{\mathcal {H}}}_j$. Now consider $-x$. It is certainly outside some neighborhood of x. We will shrink ${{\mathcal {W}}}$ so that $-x \not \in {{\mathcal {W}}}$. We will prove that ${{\mathcal {T}}}= \left( \bigcup _{j=1}^r {{\mathcal {H}}}_j\right) \backslash \{-x\}$ is locally contractible, from which it follows ${{\mathcal {W}}}\cap {{\mathcal {H}}}$ is locally contractible.

We use a stereographic projection centered at $-x$ of ${{\mathbb {S}}}^{m-1}\backslash \{-x\}$ into ${{\mathbb {R}}}^{m-1}$. By Lemma 11, this projection homeomorphically maps ${{\mathcal {C}}}_i \backslash \{-x\}$ to either a closed half-space in ${{\mathbb {R}}}^{m-1}$ (if $h_i \cdot x = 0$), or to a closed ball in ${{\mathbb {R}}}^{m-1}$, if $h_i\cdot x<0$. Since ${{\mathcal {H}}}_j = {{\mathcal {C}}}(x)$ for $x \in int ({{\mathcal {F}}}^{{\mathcal {O}}}_j)$, $j \in I_{{{\mathcal {O}}}_{{\mathcal {S}}}}$, each ${{\mathcal {H}}}_j \backslash \{-x\}$ is the intersection of sets ${{\mathcal {C}}}_i \backslash \{-x\}$, so ${{\mathcal {T}}}$ is the union of intersections of sets ${{\mathcal {C}}}_i\backslash \{-x\}$. By Lemma 11, each ${{\mathcal {C}}}_i \backslash \{-x\}$ is mapped by the same homeomorphism into a convex set: either a half-space or a closed ball. Thus, each ${{\mathcal {H}}}_j \backslash \{-x\}$ is mapped into a convex set. Finally, ${{\mathcal {T}}}$ is homeomorphically deformed into a finite union of convex sets. By Lemma 1, it is locally contractible. $\square$

Lemma 13

Suppose (A3) and (A4) hold. Also suppose ${{\mathcal {H}}}\ne {{\mathbb {S}}}^{m-1}$ and $\bigcap _{{{\mathcal {H}}}_j \ne {{\mathbb {S}}}^{m-1}} {{\mathcal {H}}}_j \ne \emptyset$. Then, ${{\mathcal {H}}}$ is contractible.

Proof

Let ${{\mathcal {Y}}}:= \cap _{{{\mathcal {H}}}_j \ne {{\mathbb {S}}}^{m-1}} {{\mathcal {H}}}_j$. Since each ${{\mathcal {H}}}_j$ is itself an intersection of ${{\mathcal {C}}}_j$’s, ${{\mathcal {Y}}}$ satisfies Lemma 5. Let $I' \subset I$ be the index set of ${{\mathcal {C}}}_j$’s whose intersection forms ${{\mathcal {Y}}}$. By Lemma 5 there exists $x \in {{\mathcal {Y}}}\subseteq {{\mathcal {H}}}$ such that $h_k \cdot x < 0$ for all $k \in I'$. Since $h_j \cdot (-x) > 0$ for all $k \in I'$, we know $-x \notin {{\mathcal {H}}}_j$ for any ${{\mathcal {H}}}_j \subset {{\mathcal {H}}}$. Thus, $-x \notin {{\mathcal {H}}}$. Consider geodesics on ${{\mathbb {S}}}^{m-1}$ coming out of x. Because the antipodal point $-x$ is not in ${{\mathcal {H}}}$, there exists a unique geodesic $f_{x'}$ between x and any point $x' \in {{\mathcal {H}}}_j$ for any ${{\mathcal {H}}}_j \subset {{\mathcal {H}}}$. Since each ${{\mathcal {H}}}_j$ is Robinson convex (see [9, 28]), the entire path of geodesic $f_{x'}$ lies inside some ${{\mathcal {H}}}_j \subseteq {{\mathcal {H}}}$, as both x and $x'$ are in ${{\mathcal {H}}}_j$. Thus, ${{\mathcal {H}}}$ is a star-shaped set with respect to geodesics on a sphere. By a repetition of the standard proof for star-shaped sets in Euclidean spaces, ${{\mathcal {H}}}$ is contractible. $\square$