Some Remarks on Controllability of the Liouville Equation

Let $\rho_{0}$ and $\rho_{1}$ be two probability densities on ${\mathbb{R}}^{n}$, which we will assume to be everywhere positive to keep things simple. Then a theorem of Brenier [Vil03, Thm. 2.12] guarantees the existence of a strictly convex function $h\mathrel{\mathop{\mathchar 58\relax}}{\mathbb{R}}^{n}\to{\mathbb{R}}$, such that its gradient $\psi=\nabla h$ pushes $\rho_{0}$ forward to $\rho_{1}$. Moreover, this mapping is optimal in the sense that

where the minimum is over all $\psi\mathrel{\mathop{\mathchar 58\relax}}{\mathbb{R}}^{n}\to{\mathbb{R}}^{n}$ that push $\rho_{0}$ forward to $\rho_{1}$. There is also a complementary dynamic formulation due to Benamou and Brenier [Vil03, Sec. 8.1]; in control-theoretic language, it guarantees the existence of a smooth feedback control law $u\mathrel{\mathop{\mathchar 58\relax}}[0,1]\times{\mathbb{R}}^{n}\to{\mathbb{R}}^{n}$, such that the trajectory of the Liouville equation

joins $\rho(0,\cdot)=\rho_{0}(\cdot)$ to $\rho(1,\cdot)=\rho_{1}(\cdot)$, and the performance index

is minimized. On the other hand, the corresponding controlled system $\dot{x}=u$ allows for maximum “control authority,” in the sense that all directions of motion are available to the controller at each time $t\in[0,1]$. The more difficult case of a general controlled system $\dot{x}=f(x,u)$ corresponds to the so-called nonholonomic setting [AL09, KL09], where the key question is how to relate the question of controllability (or obstructions to controllability) in the space of densities to structural properties of the family of vector fields $\{f(\cdot,u)\}_{u\in U}$ indexed by the elements of the control set $U$.

The case of a linear system

with $n$-dimensional state $x$ and $m$-dimensional input $u$ has been considered in [Bro07] under the assumption that the system is controllable, i.e., the columns of $B,AB,\dots,A^{n-1}B$ span ${\mathbb{R}}^{n}$. Let an orientation-preserving diffeomorphism $\psi\mathrel{\mathop{\mathchar 58\relax}}{\mathbb{R}}^{n}\to{\mathbb{R}}^{n}$ be given. By controllability, we can steer (2) from any fixed initial state $x(0)=x_{0}$ to $x(T)=\psi(x_{0})$ using the open-loop control

where

is the controllability Gramian of (2), which is positive definite since the system is controllable. The main idea in [Bro07] is to impose appropriate regularity conditions on $e^{-AT}\psi$ to ensure that the map

is injective for all $0<t<T$. If this is the case, then the closed-loop (feedback) control

can be used to carry out the transfer from $x(0)=x_{0}$ to $x(T)=\psi(x_{0})$ for every initial condition $x_{0}\in{\mathbb{R}}^{n}$.

In [Bro07], Brockett stated that one sufficient condition to ensure the above result is for the map $e^{-AT}\psi-{\rm id}$ to be a contraction (i.e., Lipschitz-continuous with constant strictly smaller than one). However, as pointed out recently by Abdelgalil and Georgiou [AG24], this is not enough. Indeed, the crux of the argument in [Bro07] is that the above contraction condition implies that the map

is invertible for every $t$. This, in turn, relies on the claim that, since $W(0,T)-W(0,t)$ is positive definite for every $0<t<T$ by controllability, it follows that $\|W(0,t)W(0,T)^{-1}\|\leq 1$. However, the latter claim is not valid: Since the matrix $W(0,t)W(0,T)^{-1}$ need not be symmetric, we can only guarantee that its spectral radius is bounded above by one. Following the line of argument in [AG24], we can easily show that Brockett’s argument goes through if we modify his contraction condition by replacing $\psi$ with

From this, it follows readily that the mapping

is invertible since $\|W(0,T)^{-1/2}W(0,t)W(0,T)^{-1/2}\|\leq 1$ (multiply the matrix inequality $W(0,t)\leq W(0,T)$ on the left and on the right by $W(0,T)^{-1/2}$). The desired conclusion then follows from the fact that

The above assumption on $\psi$ is fairly restrictive. For an arbitrary $\psi$, one could first represent it as a composition $\psi_{k}\circ\dots\circ\psi_{1}$ such that each $\psi_{i}$ satisfies (modified) Brockett’s condition and then apply the above construction for each $i$. However, the number $k$ will, in general, be very large, resulting in controls of very high complexity (we will come back to the issue of complexity later in the broader context of nonlinear systems). It was later shown by Hindawi et al. [HPR11] and Chen et al. [CGP17] that controllability is sufficient for any $\psi$ that can be written as the gradient of a convex function after a certain change of coordinates. These results can also be phrased in the setting of optimal transportation, as the resulting constructions are optimal for quadratic costs of the form $L(x,u)=x^{\mathsf{T}}Qx+u^{\mathsf{T}}Ru$ with symmetric positive-semidefinite $Q$ and symmetric positive-definite $R$ [HPR11].

Here, we will give a result of the same flavor that preserves the spirit of Brockett’s proof. The key concept we will need is that of a monotone mapping [RW98]: A mapping $\psi\mathrel{\mathop{\mathchar 58\relax}}{\mathbb{R}}^{n}\to{\mathbb{R}}^{n}$ is monotone if

Note that this definition makes no assumptions on differentiability of $\psi$ (in fact, it can be extended to set-valued mappings); however, when $\psi$ is differentiable, monotonicity is equivalent to the symmetric part of the Jacobian $D\psi$ being everywhere positive-semidefinite. In particular, any $\psi$ given by the gradient of a convex function is monotone.

To connect Theorem 1 to the problem of controlling a given initial density $\rho_{0}$ to a given final density $\rho_{1}$ using (2), let $\hat{\rho}_{0}$ be the pushforward of $\rho_{0}$ by the invertible linear transformation $x\mapsto W(0,T)^{1/2}x$ and, similarly, let $\hat{\rho}_{1}$ be the pushforward of $\rho_{1}$ by the invertible linear transformation $x\mapsto W(0,T)^{-1/2}e^{-AT}x$. By Brenier’s theorem, there is a monotone map $\hat{\psi}$ such that $\hat{\rho}_{1}=\hat{\psi}_{*}\hat{\rho}_{0}$. Then the map $\psi$ related to $\hat{\psi}$ via (5) evidently pushes $\rho_{0}$ forward to $\rho_{1}$, and in that case Theorem 1 tells us how to construct the desired feedback control $u(t,x)$. In particular, the Benamou–Brenier dynamic formulation of optimal transportation is a special case corresponding to $A=0$ and $B=I$ with $m=n$, cf. [HPR11, CGP17].

Let us now consider the case of a nonlinear system

whose state space $M$ is a smooth (say, $C^{\infty}$) closed finite-dimensional manifold. A smooth diffeomorphism $\psi\mathrel{\mathop{\mathchar 58\relax}}M\to M$ is given, and the problem is to determine whether there exists a control law that can steer every initial state $x(0)=x_{0}\in M$ to $x(T)=\psi(x_{0})$, for a fixed finite $T>0$. By analogy with the linear case, we hope to capitalize on some form of controllability of (7). Here, however, apart from the usual complications arising in the context of nonlinear controllability [HK77], a major difficulty is the lack of explicit expressions for control laws that transfer a given initial state to a given final state. Nevertheless, we can still aim for a structural result of some form.

One such result was obtained by Agrachev and Caponigro [AC09]. We will discuss it shortly in full generality, but for now we mention its corollary for driftless control-affine systems of the form

where $f_{1},\dots,f_{m}$ are smooth vector fields on $M$ and $u_{1},\dots,u_{m}$ are real-valued control inputs. Suppose that $\{f_{1},\dots,f_{m}\}$ is a bracket-generating family, i.e., for each $x\in M$ the set $\{g(x)\mathrel{\mathop{\mathchar 58\relax}}g\in{\rm Lie}(f_{1},\dots,f_{m})\}$, where ${\rm Lie}(f_{1},\dots,f_{m})$ is the Lie algebra generated by $f_{1},\dots,f_{m}$, coincides with the tangent space $T_{x}M$ to $M$ at $x$. Then, for any diffeomorphism $\psi\mathrel{\mathop{\mathchar 58\relax}}M\to M$ isotopic to the identity there exists a time-dependent feedback control law $u(t,x)=(u_{1}(t,x),\dots,u_{m}(t,x))$ that transfers $x(0)=x_{0}$ to $x(1)=\psi(x_{0})$ for every $x_{0}\in M$. (Two diffeomorphisms $\psi,\psi^{\prime}\mathrel{\mathop{\mathchar 58\relax}}M\to M$ are isotopic if there exists a smooth map $H\mathrel{\mathop{\mathchar 58\relax}}[0,1]\times M\to M$, such that $H(t,\cdot)\mathrel{\mathop{\mathchar 58\relax}}M\to M$ for each $t\in[0,1]$ is a diffeomorphism, $H(0,\cdot)=\psi(\cdot)$, and $H(1,\cdot)=\psi^{\prime}(\cdot)$ [Ban97]. We will denote by ${\rm Diff}_{0}(M)$ the family of all diffeomorphisms on $M$ that are isotopic to the identity.)

The general setting considered in [AC09] is as follows: Let a family $\mathcal{F}$ of smooth vector fields on $M$ be given. The control group of $\mathcal{F}$, defined by

is a subgroup of the group ${\rm Diff}(M)$ of smooth diffeomorphisms of $M$. Suppose that ${\mathbb{G}}(\mathcal{F})$ acts transitively on $M$, i.e., for any $x,y\in M$ there exists some $\psi\in{\mathbb{G}}(\mathcal{F})$ such that $y=\psi(x)$. Then we have the following [AC09]:

Note that the vector fields in (9) are rescaled by smooth real-valued functions on $M$, which explains the origin of time-varying feedback controls $u_{i}(t,x)$ in the above discussion of control-affine systems. It is also evident that these controls will be piecewise constant in $t$.

With this in mind, let us consider the system (8), where $\mathcal{F}=\{f_{1},\dots,f_{m}\}$ is bracket-generating. Let $\psi\in{\rm Diff}_{0}(M)$ be given. We will apply Theorem 2 to $\mathcal{F}$—because $\mathcal{F}$ is bracket-generating, ${\mathbb{G}}(\mathcal{F})$ acts transitively on $M$. Following [Cap11], we first establish a fragmentation property of $\psi$ relative to $\mathcal{O}$, i.e., show that there exist some $\psi_{1},\dots,\psi_{N}\in\mathcal{O}$ such that $\psi=\psi_{N}\circ\dots\circ\psi_{1}$. Since $\psi\in{\rm Diff}_{0}(M)$, there exists a smooth path $[0,1]\ni t\mapsto\varphi_{t}\in{\rm Diff}_{0}(M)$ such that $\varphi_{1}=\psi$ and $\varphi_{0}={\rm id}$. For every $N\in{\mathbb{N}}$, the maps

belong to ${\rm Diff}_{0}(M)$, and $\psi=\psi_{N}\circ\psi_{N-1}\circ\dots\circ\psi_{1}$. We can ensure that each $\psi_{i}\in\mathcal{O}$ by choosing $N$ large enough. Applying Theorem 2 to each $\psi_{i}$, we conclude that there exist $K=kN$ smooth functions $a_{1},\dots,a_{K}\in C^{\infty}(M)$, such that $\psi$ can be represented as

for some choice of indices $i_{1},\dots,i_{K}\in\{1,\dots,m\}$, cf. [Cap11, Prop. 4]. From this representation, it is straightforward to derive $m$ feedback controls $u_{i}\mathrel{\mathop{\mathchar 58\relax}}[0,1]\times M\to{\mathbb{R}}$ that are piecewise constant in $t$, with the number of pieces (switchings) equal to $K$. Thus, the integer $k$ in Theorem 2 is a lower bound on the number of switchings, which in turn is a natural measure of implementation complexity of a control law. Since the result of [AC09] has been used as a black-box device in subsequent works [Cap11, EZOP23], it is of interest to provide some quantitative estimates of $k$.

To that end, we first make some assumptions on $M$ and $\mathcal{F}$. We take $M$ to be a smooth compact manifold of dimension $n$ isometrically embedded in ${\mathbb{R}}^{d}$ for some $d>n$, and we will equip $M$ with the ambient metric $d(x,y)=|x-y|$, where $|\cdot|$ is the Euclidean norm on ${\mathbb{R}}^{d}$. We further assume that $M$ has positive reach $\tau>0$, where the reach of a set $A\subset{\mathbb{R}}^{d}$ is defined as the largest value of $\tau$, such that any point at a distance $0<r<\tau$ from $A$ has a unique nearest point in $A$ [Fed59]. For example, the unit sphere $S^{d-1}$ has reach $1$. We will assume that the vector fields $f\in\mathcal{F}$ are uniformly bounded in the $C^{1}(M)$ seminorm, in the following sense [AS04, Sec. 2.2]. For $r=0,1,\dots$, the $C^{r}(M)$ seminorm of a function $a\in C^{\infty}(M)$ is defined by

where $D_{1},\dots,D_{d}$ are the orthogonal projections on $M$ of the standard basis vector fields $\frac{\partial}{\partial x_{1}},\dots,\frac{\partial}{\partial x_{d}}$ on ${\mathbb{R}}^{d}$. The $C^{r}(M)$ seminorm of a vector field $f$ on $M$ is defined as

where $fa\in C^{\infty}(M)$ is the Lie derivative of $a$ along $f$. In terms of these definitions, we will assume that

We then have the following: