---
title: Lagrangian Controllability
url: https://www.emergentmind.com/topics/lagrangian-controllability
type: topic
---

# Lagrangian Controllability

Lagrangian controllability denotes a family of non-equivalent control-theoretic notions in which the word *Lagrangian* refers either to a running cost embedded into the controllability criterion, or to the Lagrangian description of motion through particle trajectories and flow maps, or to the geometric mechanics of controlled Lagrangian systems. In one line of work, the nonnegative running cost \(l\) is built directly into a Hamiltonian decrease condition that yields both asymptotic controllability and an upper bound on the value function [1210.4281]. In another, Lagrangian controllability means steering material particles, flow maps, or material sets rather than only Eulerian fields, as in Euler, Navier–Stokes, Stokes, KdV, and compressible Navier–Stokes settings [1403.5369], [1602.03041], [1602.03045], [1611.02899], [2407.19210]. A further group of works uses Lagrangian reduction, active constraints, or Euler–Lagrange reformulations to analyze controlled mechanical systems [1702.04824], [2108.06644], [2307.13402].

## 1. Semantic range and common structure

Across these usages, the controlled object is not merely a terminal state in a finite-dimensional phase space. It may instead be a cost-augmented state, a material surface transported by a flow, a set of free coordinates driven by active constraints, or an extremal trajectory generated by an Euler–Lagrange equation. The recurring feature is that controllability is formulated through trajectories, geometric transport, or variational structure.

In the optimal-control sense developed for asymptotic target approach, the Lagrangian \(l\) appears in the minimized Hamiltonian
\[
H(x,p_0,p)=\inf_{a\in A}\{p_0\,l(x,a)+\langle p,f(x,a)\rangle\},
\]
so the same inequality controls both the decrease of a Lyapunov-like function and the accumulation of running cost [1210.4281]. In fluid mechanics, the term usually refers to control of the flow map
\[
\partial_t \phi_t(x)=u(t,\phi_t(x)),
\]
or of the image of a material set under that flow, rather than only to the terminal velocity field [1403.5369], [1602.03041]. In controlled mechanical systems, the expression often indicates that the dynamics, constraints, or reduction procedure are written in Lagrangian form, even when the main theorem concerns tracking, stabilization, or optimality rather than classical reachability [1702.04824], [2307.13402].

A plausible implication is that “Lagrangian controllability” is best treated as a thematic label rather than a single formal definition. Its precise meaning is set by the controlled quantity: running cost, particle transport, geometric shape motion, or variational extremal.

## 2. Running-cost-based asymptotic controllability

A particularly explicit theory of Lagrangian controllability is developed for control systems
\[
\dot z(t)=f(z(t),\alpha(t)),\qquad z(0)=x,
\]
with target a closed set \(\mathbf C\subset\mathbb R^n\), admissibility defined by asymptotic approach
\[
\lim_{t\to T_z^-}\mathbf d(z(t),\mathbf C)=0,
\]
and cost
\[
\mathcal J_{z,\alpha}=\int_0^{T_z} l(z(t),\alpha(t))\,dt,
\qquad l:\mathbf C^c\times A\to [0,+\infty),
\]
with no assumptions on the zero level set of \(l\) [1210.4281]. The objective is not finite-time hitting of the target at all costs, but asymptotic steering with minimal accumulated nonnegative Lagrangian.

The central device is the **Minimum Restraint Function** \(U\), a continuous, locally semiconcave, positive definite, proper function on \(\overline{\mathbf C^c}\) such that for some \(\bar p_0\ge 0\),
\[
H(x,\bar p_0,D^*U(x))<0
\qquad \forall x\in \mathbf C^c,
\]
where \(D^*U(x)\) is the set of limiting gradients and
\[
H(x,p_0,p)=\inf_{a\in A}\{p_0\,l(x,a)+\langle p,f(x,a)\rangle\}.
\]
If \(U\) were \(C^1\), this becomes
\[
\inf_{a\in A}\{\bar p_0\,l(x,a)+\langle \nabla U(x),f(x,a)\rangle\}<0.
\]
Because \(l\ge 0\), this is stronger than the usual Control Lyapunov Function decrease condition. The paper emphasizes that the Lagrangian is not merely part of the objective; it enters the controllability inequality itself [1210.4281].

The main theorem states that existence of a Minimum Restraint Function implies global asymptotic controllability to \(\mathbf C\), and if \(\bar p_0>0\), then the value function satisfies
\[
\mathcal V(x)\le \frac{U(x)}{\bar p_0}
\qquad \forall x\in \mathbf C^c.
\]
A sharper estimate shows that strict negativity of the Hamiltonian can be strengthened locally to
\[
H(x,\bar p_0,D^*U(x))\le -m(U(x)),
\]
with \(m\) continuous, strictly increasing, and positive on \(]0,+\infty[\), yielding both \(U(z(t))\to 0\) and the cost estimate. In the smooth heuristic formulation, if \(\dot z_0=l(z,\alpha)\) is the hidden cost state, then
\[
\frac{d}{dt}\big(U(z(t))+\bar p_0 z_0(t)\big)<0
\]
simultaneously drives \(z(t)\) toward the target and bounds \(\int l\) [1210.4281].

This formulation is notable because it does **not** assume \(l\ge \eta>0\), and therefore cannot in general be reduced to a minimum-time problem by time reparametrization. The paper explicitly treats degenerate nonnegative Lagrangians for which \(l\) may vanish away from the target. In that sense, Lagrangian controllability means the existence of a nonsmooth Lyapunov-type certificate that controls both asymptotic approach and accumulated running cost.

## 3. Particle transport, flow maps, and material-set control

In fluid mechanics and related PDEs, Lagrangian controllability usually means controlling trajectories of particles or the flow map generated by a velocity field. For the 3D Navier–Stokes system on the torus, a finite-dimensional control force can simultaneously approximately control the Eulerian velocity field and the induced Lagrangian flow of diffeomorphisms [1403.5369]. The exact formulation uses the flow map
\[
\partial_t \phi_t(x)=u(t,\phi_t(x)),\qquad \phi_0(x)=x,
\]
and the main controllability statement couples a target final velocity \(u_1\) with a target \(\zeta\in \mathrm{SDiff}(\mathbb T^3)\). The result is approximate, and the mechanism combines finite-dimensional low-mode forcing with a continuity estimate of the flow map in a relaxation norm [1403.5369].

For incompressible Euler flows, a constructive harmonic/potential-flow approach yields Lagrangian approximate controllability of material curves and surfaces [1602.03041]. The key interface operator is
\[
\Lambda_\gamma v=\nabla\Psi_v\cdot n_{12}\big|_\gamma,
\]
where \(\Psi_v\) solves a Neumann problem with boundary control supported on \(\Gamma\subset\partial\Omega\). The dense range property
\[
\overline{R(\Lambda_\gamma)}=H^{-1/2}_m(\gamma)
\]
shows that any admissible normal velocity on an interior interface can be approximated by a harmonic flow generated from the controlled boundary. In two dimensions this is made more explicit through Runge approximation of holomorphic functions by rational functions [1602.03041].

At low Reynolds number, the same Lagrangian viewpoint is developed for the stationary Stokes equation with boundary control on an open part \(\Sigma\subset\partial\Omega\) [1602.03045]. The main theorem gives approximate Lagrangian controllability of a smooth Jordan curve in \(2\)D or a Jordan surface in \(3\)D. The flow \(\phi^u\) associated with a time-dependent quasi-static Stokes velocity transports \(\gamma_0\) approximately to \(\gamma_1\), while remaining inside \(\Omega\). The proof relies on a weak Runge–Walsh-type approximation theorem for Stokes fields and, in a second variant, on a density theorem for traces on the moving interface [1602.03045].

For a 1D viscous compressible barotropic flow on \([0,\pi]\), local exact Lagrangian controllability is formulated directly in terms of the flow map
\[
\Phi[f](0,x)=x,\qquad \partial_t\Phi[f](t,x)=u\bigl(t,\Phi[f](t,x)\bigr),
\]
and exact transport of an interval \(I=[\alpha_1,\alpha_2]\) onto \(J=[\beta_1,\beta_2]\) by a localized force \(f\) supported in \(\omega\subset(1,\pi)\) [2407.19210]. The theorem constructs a two-parameter control \(f=\varepsilon_1 f_1+\varepsilon_2 f_2\) from adjoint solutions and proves local surjectivity of the endpoint map
\[
\Theta(\varepsilon_1,\varepsilon_2)=\bigl(\Phi[\varepsilon_1f_1+\varepsilon_2f_2](T,\alpha_1),\Phi[\varepsilon_1f_1+\varepsilon_2f_2](T,\alpha_2)\bigr).
\]
This is exact Lagrangian transport of selected material markers, not exact controllability of the full final Eulerian state [2407.19210].

A more recent no-slip setting concerns 3D incompressible Navier–Stokes equations in perforated domains with many small obstacles [2509.14913]. There the controllability statement is approximate in measure:
\[
\mathcal L\big(P_1\setminus \Phi^\varepsilon(T_c,P_0^\varepsilon)\big)\le C\eta,
\]
with remote forcing supported away from the initial and target material patches. Depending on the hole geometry, homogenization yields an effective Euler or Darcy dynamics, and approximate Lagrangian controllability follows in explicit asymptotic regimes [2509.14913]. This paper identifies a mechanism by which no-slip microgeometry can coexist with positive Lagrangian controllability.

## 4. One-dimensional PDEs and Lagrangian-coordinate hyperbolic systems

A different one-dimensional use arises for the Korteweg–de Vries equation on \([0,L]\), where Lagrangian controllability refers to the flow generated by the scalar field itself [1611.02899]. With boundary controls
\[
y(0,t)=u(t),\qquad y(L,t)=v(t),\qquad y_x(L,t)=w(t),
\]
the extended field \(\hat y\) defines a flow
\[
\frac{\partial \Phi}{\partial t}(x,t)=\hat y(\Phi(x,t),t),\qquad \Phi(x,0)=x.
\]
The system is called **small-time Lagrangian controllable** if controls can be chosen so that
\[
\Phi(x,T)\ge L,\qquad \forall x\in[0,L].
\]
Using \(N\)-soliton solutions and local exact Eulerian controllability in \(H^2\), the paper proves that for every \(L>0\) and \(T>0\), one can drive the system from rest to rest while ejecting all particles initially in \([0,L]\) to the right of the domain by time \(T\) [1611.02899]. Here Lagrangian controllability means material transport, not state-to-state controllability of the field \(y\).

For the non-isentropic 1D Euler equations in **Lagrangian coordinates**, the issue is different [1304.4090]. The state is
\[
u=(\tau,v,P),
\]
with \(\tau=1/\rho\) the specific volume, and the system has characteristic speeds
\[
\lambda_1=-\sqrt{\frac{\gamma P}{\tau}},\qquad \lambda_2=0,\qquad \lambda_3=\sqrt{\frac{\gamma P}{\tau}}.
\]
The paper proves exact boundary controllability toward constant states in the class of weak entropy \(BV\) solutions, provided
\[
1<\gamma<\frac53
\]
and the target constant satisfies the entropy compatibility condition
\[
S(\bar u_1)>S(\bar u_0).
\]
The proof uses front tracking, two strong shocks, cancellation and correction waves, and Glimm-type functionals. The zero second characteristic speed is the central obstacle, since \(2\)-contacts do not propagate toward the boundary. This is resolved indirectly through nonlinear interactions with strong \(1\)- and \(3\)-shocks [1304.4090].

These two one-dimensional examples show that the adjective *Lagrangian* may indicate either particle transport by a scalar PDE flow, as for KdV, or the use of Lagrangian coordinates in a hyperbolic conservation law, as for compressible Euler.

## 5. Mechanical systems, active constraints, and adjacent uses

In mechanical control, one usage concerns systems actuated by **active constraints** rather than external forces [1702.04824]. If some coordinates \(u\) are directly prescribed and the remaining free coordinates are \(q\), the reduced equations take the form
\[
\binom{\dot q}{\dot p}
=
\binom{Ap}{-\frac12 p^\top Bp}
+
\binom{K}{-p^\top C}\dot u
+
\dot u^\top \binom{0}{D}\dot u.
\]
Because the right-hand side depends both linearly and quadratically on \(\dot u\), the system is not control-affine. The paper introduces the differential inclusion
\[
\dot q \in K(q,u)\dot u + \Gamma(q,u),
\]
with
\[
\Gamma(q,u)=\operatorname{co}\{A(q,u)(w^\top D(q,u)w):w\in\mathbb R^m\},
\]
and proves that every trajectory of the inclusion can be uniformly approximated by a trajectory of the original Lagrangian system on a sufficiently large time interval, starting at rest; under a stronger hypothesis, the same terminal point can be reached exactly [1702.04824]. Here controllability is mediated through an averaged first-order inclusion derived from the Lagrangian structure.

For underactuated aerial manipulators, Lagrangian reduction has been used to obtain the lowest dimensional equations of motion, with symmetry-breaking potential energy terms resolved using advected parameters so that full \(SE(3)\) reduction is achieved at the cost of additional advection equations [2108.06644]. The reduced equations highlight the shifting center of gravity due to manipulation and are brought into control-affine form. Using Sussmann’s sufficient condition, the system is shown to be small-time locally controllable near equilibrium, requiring Lie bracket motions up to degree three [2108.06644]. In this setting, *Lagrangian* refers to the reduction procedure rather than to a special notion of reachable set.

Related works lie near, but not within, classical controllability. For fully actuated lossless Lagrangian systems, observer-less output-feedback global tracking has been established using only position measurements and approximate differentiation, yielding uniform global asymptotic stability of the closed-loop tracking error rather than a reachability theorem [1307.4659]. For fully actuated Lagrangian systems in obstacle-scattered environments, a hybrid CLF-CBF-QP framework guarantees safety and asymptotic stabilization through a sequence of reach-avoid stages, but the paper explicitly does not study classical controllability in the Kalman or Lie-algebraic sense [2009.02148]. These works are best viewed as neighboring notions of global trackability, constrained maneuverability, or safe reachability for Lagrangian systems.

## 6. Variational, optimization, and numerical reformulations

Several papers connect controllability questions to Lagrangian or Euler–Lagrange formalisms without redefining controllability itself. For control-affine second-order mechanical systems with quadratic running cost, a new Lagrangian approach rewrites the optimal control problem on an enlarged configuration space so that the necessary conditions become Euler–Lagrange equations equivalent to Pontryagin’s maximum principle [2307.13402]. The new formulation is regular and enables symplectic discretisation via variational integrators in a straightforward way [2307.13402]. This is a structural reformulation of optimal control, not a controllability theorem.

A related preliminary line for LTI systems treats the system as a module over \(\mathbb R[d/dt]\), with controllability characterized by freeness of the system module [2410.07040]. Once controllability holds, any basis of the free module gives a flat output, and a quadratic Lagrangian in those coordinates leads to a linear Euler–Lagrange equation for open-loop trajectory generation, with assignable endpoints and horizon selection [2410.07040]. The paper explicitly states that it does not introduce a standard notion called “Lagrangian controllability”; rather, it uses controllability to make a Lagrangian trajectory-planning method effective [2410.07040].

At the numerical-analysis level, null controllability of linear parabolic equations and systems has been reformulated by classical Lagrangian and Augmented Lagrangian techniques applied to weighted constrained extremal problems with Carleman-inspired weights that blow up as \(t\to T^{-}\) [2411.14031]. The augmented Lagrangian
\[
\mathcal L_{R,K}(y,v,q)=\mathcal L_R(y,v,q)+\frac K2\|Mv+\bar y-y\|_{L^2(Q)}^2
\]
leads to Uzawa-type iterations and exact null controllability of numerical approximations in two- and three-dimensional experiments [2411.14031]. Here the word *Lagrangian* refers to the optimization formalism used to compute controls.

Finally, in discrete-time dynamic optimization, controllability and observability of the linearized dynamics have been shown to imply uniform regularity properties of the optimization Lagrangian and hence exponential decay of sensitivity [2101.06350]. Uniform controllability implies uniform LICQ of the dynamic constraints, while observability of \((A_i,Q_i)\), where \(Q_i\) is a block of the Lagrangian Hessian, implies positivity of the reduced Lagrangian Hessian. The resulting estimate
\[
\| w^\dag_i(d)-w^\dag_i(d')\|
\le
\sum_{j=-1}^{N}\Upsilon \rho^{|i-j|}\|d_j-d'_j\|
\]
shows how controllability of linearized dynamics can be translated into decay properties of the Lagrangian KKT system [2101.06350].

Taken together, these reformulations suggest a broad but technically precise landscape. In some works, Lagrangian controllability is literally controllability mediated by a running Lagrangian; in others it is control of Lagrangian particle motion; in yet others it names the Lagrangian structure used to derive, reduce, or compute controlled trajectories. The unifying theme is that control is organized around trajectories, material transport, or variational geometry rather than around a purely Eulerian or terminal-state viewpoint.

Source: https://www.emergentmind.com/topics/lagrangian-controllability