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A probabilistic argument for the controllability of conservative systems

We consider controllability for divergence-free systems that have a conserved quantity and satisfy a H\"ormander condition. It is shown that such systems are controllable, provided that the conserved quantity is a proper function. The proof of the result combines analytic tools with probabilistic arguments. While this statement is well-known in geometric control theory, the probabilistic proof given in this note seems to be new. We show that controllability follows from H\"ormander's condition, together with the a priori knowledge of an invariant measure with full topological support for a diffusion that `implements' the control system. Examples are given that illustrate the relevance of the assumptions required for the result to hold. Applications of the result to ergodicity questions for systems arising from non-equilibrium statistical mechanics and to the controllability of Galerkin approximations to the Euler equations are also given.


Introduction
In this note, we are interested in the controllability of systems of the forṁ (1.1) where f : R N → R N is a smooth vector field and the control u is only allowed to take values in a given linear subspace E of R N . Our aim is to study the possibility, given a starting point z 0 ∈ R N and a final point z 1 ∈ R N , of finding a control u taking values in the (possibly small) space E that allows to 'stir' the solution of (1.1) from z 0 into a small neighbourhood of z 1 . It is a well-known fact, see for example [Lob74] or [AS04,Chapter 3], that if this control system satisfies a Hörmander condition and f is Poisson stable, then (1.1) is approximately controllable in the sense that for any two points z 0 and z 1 in R N there exists a time T and a control u ∈ C ∞ ([0, T ], E) such that the solution z(T ) of (1.1) at time T is equal to z 1 . In this note, we show that this fact can be shown as a consequence of the fact that (1.1) is equivalent to a control system such that the corresponding stochastic differential equation has the strong Feller property and possesses an invariant measure with full topological support, see Proposition 3.5. In Section 4.5, we show how such an argument can be reversed in order to obtain the uniqueness of the invariant measure for an SDE where no invariant measure is known a priori, provided that it is equivalent in the sense of control systems to an SDE for which the invariant measure is known. This situation arises in non-equilibrium statistical mechanics, where a non-equilibrium system can be compared to the corresponding equilibrium system.
In the case when f is a polynomial vector field of odd degree, it is known (see for example [JK85] and references therein) that Hörmander's condition is both necessary and sufficient for the system (1.1) to be controllable. This is not the case in general. In Section 4.2, we provide an example of a non-controllable system which satisfies all of the conditions of the main theorem of the present article, except for the growth condition of the conserved quantity H. This illustrates the fact that the growth condition on H encodes global geometric information on f (Poincaré recurrence) that complements the local geometric information given by the Hörmander condition and is essential to the result. Applications to the controllability of the Euler equations and to an ergodicity problem coming from non-equilibrium statistical mechanics will be given in Sections 4.4 and 4.5.

Setting
We consider the controllability of systems that have a smooth conserved quantity called H, i.e. we assume throughout this paper that We assume that the state space R N splits in a natural way as E ⊕ E ⊥ and we denote its elements by (x, y). We would like to stress the fact that even though our prime interest is Hamiltonian systems, the splitting under consideration is not necessarily the standard splitting in position and momentum variables. Typically, x would consist only of part of the momentum variables and y would consist of all the other variables of the system. In particular, we allow dim E = dim E ⊥ , which is actually the most interesting case in our situation. Our first assumption is that the flow generated by f preserves the Lebesgue measure and that it has a conserved quantity H:

Assumption 1
The vector field f is divergence-free and there exists a smooth function H: R N → R such that (2.1) holds.
Our second assumption ensures that H grows at infinity:

Assumption 2
The level sets {z | H(z) ≤ K} are compact for every K > 0.
Recall that, given a vector field f on R N , we can identify it with the corresponding differential operator i f i (x)∂ x i . With this identification, the Lie bracket between two vector fields is simply the vector field corresponding to the commutator of the two differential operators. The Lie algebra generated by a family of smooth vector fields is the smallest subspace of the space of all vector fields that is closed under the Lie bracket operation.
We also introduce an extended phase space R N +1 , which includes time as an additional dimension, and extend f to a vector fieldf on R N +1 by settingf (x, t) = (f (x), 1). Our last assumption then essentially says that the differential operator ∂ t + L * (with L * defined as in (3.3) below) is hypoelliptic, see [Hör85].
Assumption 3 Given a basis {e 1 , . . . , e n } of E, the Lie algebra generated by {f , e i , . . . , e n } spans R N +1 at every point.
Note that the statement of Assumption 3 is actually independent of the particular choice of the basis of E. With these notations, the main result of this article is: Theorem 2.1 Under assumptions 1-3, for every initial condition z 0 ∈ R N and every terminal condition z 1 ∈ R N there exists a time T and a control u ∈ C ∞ ([0, T ], E) such that the solution z(T ) of (1.1) at time T is equal to z 1 .

Proof of the main result
The main idea in the proof of Theorem 2.1 is to consider the following Itô stochastic differential equation on R n : where g: R + → R is a function to be determined later. Here, w is an n-dimensional standard Wiener process. We will assume from now on that the function g is smooth, increasing, and that g(0) = 0. We will also assume that g grows sufficiently fast so that exp(−g • H) is integrable and we denote by Z the value of the integral. This can always be done thanks to Assumption 2. The following result is elementary: Lemma 3.1 Under the above assumptions, there exists a choice of function g such that (3.1) has a unique global strong solution for all times and such that µ H (dx, dy) = Z −1 exp(−(g • H)(x, y)) dx dy is an invariant probability measure for (3.1).
Proof. The existence of a unique local strong solution is a standard result for SDEs with smooth coefficients [Øks03]. To show that this solution can be continued for all times, we show that there exists a suitable choice of g such that H, evaluated at the solution to (3.1), grows at most exponentially fast. Using for any function h on R N the shortcut h s = h(ξ(s), η(s)), Itô's formula yields Since H is smooth and proper, we can now chose for g an increasing function that tends to +∞ sufficiently fast so that e −2g(H(x,y)) (∆ x H)(x, y) ≤ C + H(x, y) for some constant C and for every (x, y) ∈ R N . Using Gronwall's inequality, this ensures that for every t ≥ 0.
That µ H is an invariant probability measure for (3.1) follows immediately from the fact that where L * is the adjoint of the generator of the semigroup generated by (3.1), See e.g. [Has80,Øks03,RY99] for more details.
Note furthermore that the hypoellipticity assumption 3 implies that the transition probabilities corresponding to the solutions of (3.1) have a density with respect to Lebesgue measure that is smooth in all of its arguments (including time). This is an immediate consequence of the fact that the transition probabilities are a solution (in the sense of distributions) to the equation and that ∂ t + L * is hypoelliptic by Hörmander's theorem (see [Hör85]).
The result stated in Theorem 2.1 is now an almost immediate conclusion of the following two facts: 2. The measure µ H is the only invariant probability measure for (3.1) and is therefore ergodic. While the first claim is obvious, the second claim requires some more explanation. Surprisingly, it will turn out to be an almost immediate consequence of the first claim, once we realise that the hypoellipticity assumption 3 implies the following.
Lemma 3.2 Fix a time t > 0 and denote by P t (z, · ) the transition probabilities corresponding to (3.1). Then, they are continuous in the total variation topology. In particular, for every z ∈ R N , there exists δ z > 0 such that for every z ′ such that |z ′ − z| < δ z . Here, · TV denotes the total variation distance between probability measures (normalised in such a way that the distance between two mutually singular probability measures is 2).
Proof. This is an immediate consequence of the smoothness of the transition probabilities.
Recall that the topological support supp µ of a probability measure µ is the smallest closed set of full µ-measure. Equivalently, it is characterised as the set of points z such that every neighbourhood of z has positive µ-measure. As an immediate consequence of Lemma 3.2 and the fact that distinct ergodic invariant measures are mutually singular we have Since the S i 's are disjoint closed sets satisfying Corollary 3.4, the only way in which they can cover R N is by having only one ergodic invariant measure with support R N . We have thus shown that µ H is the only invariant probability measure for (3.1) and as a consequence is ergodic.
Let us now turn to the Consider now the solution of (3.1) with initial condition (ξ 0 , η 0 ) = z ∈ R N and define the stopping time T z = inf{t > 0 | (ξ(t), η(t)) ∈ A}. It follows immediately from Birkhoff's ergodic theorem and the fact that µ H (A) > 0 that the set satisfies µ H (B 0 ) = 1, so that B 0 is dense in R N . Furthermore, a consequence of Lemma 3.2 is that if B(z, δ z ) ∩ B 0 = φ then P(T z < ∞) ≥ 1 2 . Combining these two statements shows that for every z ∈ R N there exists a time t ≥ 0 such that P t (z, A) > 0. The support theorem [SV72], combined with the fact that the control problem associated to (3.1) is equivalent to (1.1) (since ∇ x H takes values in E) allows us to conclude that B = R N .
Retracing the argument of the proof, one sees that we have actually proven the following weaker fact:

Proposition 3.5 Consider a control system of the forṁ
where z ∈ R n , u is a smooth control taking values in R d , and A: R n → R n×d . If the corresponding stochastic differential equation

Examples and applications
In this section, we present a number of examples which explore to which extent the conditions formulated in this paper are necessary to the result. We also present a few applications in which our result may prove to be useful.

Dropping the hypoellipticity assumption
It is clear from [JK85] that Assumption 3 is crucial for any result of the type of Theorem 2.1, as can be seen from the following very easy example. Consider the Hamiltonian and define f as the corresponding Hamiltonian vector field. If we define E to be the linear subspace of R 4 corresponding to the variable p 1 , it is clear that all of our assumptions are satisfied, except for Assumption 3. However, q 2 2 + p 2 2 is an integral of motion of this system that cannot be perturbed by acting on the variable p 1 , so that the conclusion of Theorem 2.1 does not hold.
One can see immediately that, whatever the values of u and x are, one hasẏ > 0 for y ∈ (1, 2) andẏ < 0 for y ∈ (−2, −1). This shows that the conclusions of Theorem 2.1 cannot hold in this situation. However, the system (4.1) is hypoelliptic since the commutator between ∂ x and −x∂ x + g(y + g(x))∂ y − yϕ(y)∂ y is given by This vector field always has a non-zero component in the y-direction because of the assumption that g ′ is strictly positive. Furthermore, the orbits of (4.1) in the absence of control stay bounded for all times and the corresponding diffusion process has global strong solutions and a smooth invariant probability measure. (The above arguments actually show that it has at least two distinct ergodic smooth invariant probability measures.) The missing point in this argument of course is the fact that we have no explicit expression for any of these invariant measures and therefore no a priori knowledge about their support.
One may argue on the other hand that it is possible to find a Hamiltonian function H(x, y) such that ∂ x H(x, y) = g(y +g(x))−yϕ(y) and that the control systeṁ is equivalent (from the point of view taken in this paper) to (4.1), so that Assumption 1 and Assumption 3 are satisfied. This example shows that the condition that the level sets of H are compact is essential for Theorem 2.1 to hold and not just a technical condition that ensures that exp(−g • H) can be made integrable.

Controllability in finite time
Let us show by a simple example that, unlike in the polynomial case studied in [JK85], it is not reasonable in general, under the conditions of Section 2, to expect the existence of a time T independent of z and A such that (1.1) can be driven from z to A in time T . Consider the function H(x, y) = 1 + x 2 + y 2 and the control systemẋ It is a straightforward exercise to check that all the conditions from the previous section are satisfied, so that Theorem 2.1 applies. It is equally straightforward to check that |∂ x H(x, y)| ≤ 1 for every value of x and y. Therefore, whatever control u is used to stir (4.2) from z = (x, y) into A, it will always require a time T larger than inf (x ′ ,y ′ )∈A |y − y ′ |.

The Euler equations
The three-dimensional Euler equations on the torus are given bẏ where x ∈ T 3 . (Note that the algebraic condition on the divergence of u determines p in a unique way.) If we assume that u 0 (x) dx = 0 and expand this equation in Fourier modes, we obtaiṅ subject to the algebraic conditions k · u k = 0 and u −k =ū k . Here, the index k takes values in Z 3 \ {0} (which reflects the fact that u(x, t) dx = 0 for all times) and u k ∈ R 3 . Furthermore, the dot · denotes the usual scalar product in R 3 . Since in this note we are only interested in finite-dimensional systems, we fix a (large) value N ⋆ and impose u k = 0 for every k such that one of its components is larger than N ⋆ in absolute value.
It is a straightforward exercise to check that the total energy |u k | 2 is a conserved quantity for this system and that the right-hand side of (4.3) is divergencefree. This allows to recover immediately a weak form of the controllability results obtained in [Rom04, Section 6] and [AS05]. The same argument allows to show that the finite-dimensional Galerkin approximations for the two-dimensional Euler equations are controllable under the conditions presented in [HM04].

Non-equilibrium statistical mechanics
The articles [EPR99a, EPR99b, EH00] considered a mechanical system coupled to heat baths at different temperatures T i . This situation can be modelled by the following system of SDEs: The interpretation of this equation is that a Hamiltonian system with N degrees of freedom described by H S (p, q) is coupled to M heat baths with internal states r i that are maintained at temperatures T i . The functions F i (p, q) and the constants γ i and λ i describe coupling between the Hamiltonian system and the ith heat bath, as well as the relaxation times of the heat baths. The control problem corresponding to this system is given bẏ which is of the form (1.1) with conserved quantity (Note that one could actually add any function of r to H and it would still be a conserved quantity.) Provided that the level sets of H S are compact, it is easy to check that all of the assumptions of Section 2 are satisfied, except for Assumption 3 which has to be checked on a case by case basis. It can for example be checked for the chain of anharmonic oscillators studied in the abovementioned works, provided that the nearest-neighbour coupling potential does not contain any infinitely degenerate point.
Note that if all the T i appearing in (4.4) are equal, then this equation is similar to (3.1) and one can check that e −H(p,q,r)/T dp dq dr is its invariant probability measure. However, the mere existence of an invariant probability measure for (4.4) with arbitrary temperatures is an open problem in general. (In the case of a chain of oscillators it was shown in [EPR99a, EH00, RBT02] to exist, provided that the nearest-neighbour interaction dominates the on-site potential at high energies.) The controllability result shown in this article, combined with the support theorem [SV72] and the smoothness of transition probabilities for (4.4) immediately implies the following: Proof. The conditions of Theorem 2.1 are satisfied by assumption (note that one can always add to H a function of r that grows sufficiently fast at infinity, thus ensuring that Assumption 2 holds). This immediately implies that every invariant probability measure for (4.4) has the whole phase space as its support. The fact that every invariant measure has a smooth density allows to conclude by the same argument used to show that the measure µ H is the only invariant measure for (3.1).