Lagrangian Controllability

• Lagrangian controllability is defined through trajectories that integrate running costs, particle transport, and variational structures to influence control outcomes.
• It employs a running-cost framework where a minimum restraint function ensures asymptotic steering and provides bounds on the value function.
• The approach extends to fluid dynamics and mechanical systems by controlling flow maps and leveraging Lagrangian reduction for optimized, trajectory-based control.

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 (Motta et al., 2012). 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 (Nersesyan, 2014, Horsin et al., 2016, Glass et al., 2016, Gagnon, 2016, Koike et al., 2024). A further group of works uses Lagrangian reduction, active constraints, or Euler–Lagrange reformulations to analyze controlled mechanical systems (Bressan et al., 2017, Wei et al., 2021, Leyendecker et al., 2023).

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 (Motta et al., 2012). 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 (Nersesyan, 2014, Horsin et al., 2016). 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 (Bressan et al., 2017, Leyendecker et al., 2023).

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 $l$ (Motta et al., 2012). 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 $l$0 such that for some $l$1,

$l$2

where $l$3 is the set of limiting gradients and

$l$4

If $l$5 were $l$6, this becomes

$l$7

Because $l$8, 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 (Motta et al., 2012).

The main theorem states that existence of a Minimum Restraint Function implies global asymptotic controllability to $l$9, and if $$H(x,p_0,p)=\inf_{a\in A}\{p_0\,l(x,a)+\langle p,f(x,a)\rangle\},$$0, then the value function satisfies

$$H(x,p_0,p)=\inf_{a\in A}\{p_0\,l(x,a)+\langle p,f(x,a)\rangle\},$$1

A sharper estimate shows that strict negativity of the Hamiltonian can be strengthened locally to

$$H(x,p_0,p)=\inf_{a\in A}\{p_0\,l(x,a)+\langle p,f(x,a)\rangle\},$$2

with $$H(x,p_0,p)=\inf_{a\in A}\{p_0\,l(x,a)+\langle p,f(x,a)\rangle\},$$3 continuous, strictly increasing, and positive on $$H(x,p_0,p)=\inf_{a\in A}\{p_0\,l(x,a)+\langle p,f(x,a)\rangle\},$$4, yielding both $$H(x,p_0,p)=\inf_{a\in A}\{p_0\,l(x,a)+\langle p,f(x,a)\rangle\},$$5 and the cost estimate. In the smooth heuristic formulation, if $$H(x,p_0,p)=\inf_{a\in A}\{p_0\,l(x,a)+\langle p,f(x,a)\rangle\},$$6 is the hidden cost state, then

$$H(x,p_0,p)=\inf_{a\in A}\{p_0\,l(x,a)+\langle p,f(x,a)\rangle\},$$7

simultaneously drives $$H(x,p_0,p)=\inf_{a\in A}\{p_0\,l(x,a)+\langle p,f(x,a)\rangle\},$$8 toward the target and bounds $$H(x,p_0,p)=\inf_{a\in A}\{p_0\,l(x,a)+\langle p,f(x,a)\rangle\},$$9 (Motta et al., 2012).

This formulation is notable because it does not assume $\partial_t \phi_t(x)=u(t,\phi_t(x)),$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 $\partial_t \phi_t(x)=u(t,\phi_t(x)),$1 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 (Nersesyan, 2014). The exact formulation uses the flow map

$\partial_t \phi_t(x)=u(t,\phi_t(x)),$2

and the main controllability statement couples a target final velocity $\partial_t \phi_t(x)=u(t,\phi_t(x)),$3 with a target $\partial_t \phi_t(x)=u(t,\phi_t(x)),$4. 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 (Nersesyan, 2014).

For incompressible Euler flows, a constructive harmonic/potential-flow approach yields Lagrangian approximate controllability of material curves and surfaces (Horsin et al., 2016). The key interface operator is

$\partial_t \phi_t(x)=u(t,\phi_t(x)),$5

where $\partial_t \phi_t(x)=u(t,\phi_t(x)),$6 solves a Neumann problem with boundary control supported on $\partial_t \phi_t(x)=u(t,\phi_t(x)),$7. The dense range property

$\partial_t \phi_t(x)=u(t,\phi_t(x)),$8

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 (Horsin et al., 2016).

At low Reynolds number, the same Lagrangian viewpoint is developed for the stationary Stokes equation with boundary control on an open part $\partial_t \phi_t(x)=u(t,\phi_t(x)),$9 (Glass et al., 2016). The main theorem gives approximate Lagrangian controllability of a smooth Jordan curve in $\dot z(t)=f(z(t),\alpha(t)),\qquad z(0)=x,$0D or a Jordan surface in $\dot z(t)=f(z(t),\alpha(t)),\qquad z(0)=x,$1D. The flow $\dot z(t)=f(z(t),\alpha(t)),\qquad z(0)=x,$2 associated with a time-dependent quasi-static Stokes velocity transports $\dot z(t)=f(z(t),\alpha(t)),\qquad z(0)=x,$3 approximately to $\dot z(t)=f(z(t),\alpha(t)),\qquad z(0)=x,$4, while remaining inside $\dot z(t)=f(z(t),\alpha(t)),\qquad z(0)=x,$5. 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 (Glass et al., 2016).

For a 1D viscous compressible barotropic flow on $\dot z(t)=f(z(t),\alpha(t)),\qquad z(0)=x,$6, local exact Lagrangian controllability is formulated directly in terms of the flow map

$\dot z(t)=f(z(t),\alpha(t)),\qquad z(0)=x,$7

and exact transport of an interval $\dot z(t)=f(z(t),\alpha(t)),\qquad z(0)=x,$8 onto $\dot z(t)=f(z(t),\alpha(t)),\qquad z(0)=x,$9 by a localized force $\mathbf C\subset\mathbb R^n$0 supported in $\mathbf C\subset\mathbb R^n$1 (Koike et al., 2024). The theorem constructs a two-parameter control $\mathbf C\subset\mathbb R^n$2 from adjoint solutions and proves local surjectivity of the endpoint map

$\mathbf C\subset\mathbb R^n$3

This is exact Lagrangian transport of selected material markers, not exact controllability of the full final Eulerian state (Koike et al., 2024).

A more recent no-slip setting concerns 3D incompressible Navier–Stokes equations in perforated domains with many small obstacles (Higaki et al., 18 Sep 2025). There the controllability statement is approximate in measure: $\mathbf C\subset\mathbb R^n$4 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 (Higaki et al., 18 Sep 2025). 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 $\mathbf C\subset\mathbb R^n$5, where Lagrangian controllability refers to the flow generated by the scalar field itself (Gagnon, 2016). With boundary controls

$\mathbf C\subset\mathbb R^n$6

the extended field $\mathbf C\subset\mathbb R^n$7 defines a flow

$\mathbf C\subset\mathbb R^n$8

The system is called small-time Lagrangian controllable if controls can be chosen so that

$\mathbf C\subset\mathbb R^n$9

Using $\lim_{t\to T_z^-}\mathbf d(z(t),\mathbf C)=0,$0-soliton solutions and local exact Eulerian controllability in $\lim_{t\to T_z^-}\mathbf d(z(t),\mathbf C)=0,$1, the paper proves that for every $\lim_{t\to T_z^-}\mathbf d(z(t),\mathbf C)=0,$2 and $\lim_{t\to T_z^-}\mathbf d(z(t),\mathbf C)=0,$3, one can drive the system from rest to rest while ejecting all particles initially in $\lim_{t\to T_z^-}\mathbf d(z(t),\mathbf C)=0,$4 to the right of the domain by time $\lim_{t\to T_z^-}\mathbf d(z(t),\mathbf C)=0,$5 (Gagnon, 2016). Here Lagrangian controllability means material transport, not state-to-state controllability of the field $\lim_{t\to T_z^-}\mathbf d(z(t),\mathbf C)=0,$6.

For the non-isentropic 1D Euler equations in Lagrangian coordinates, the issue is different (Glass, 2013). The state is

$\lim_{t\to T_z^-}\mathbf d(z(t),\mathbf C)=0,$7

with $\lim_{t\to T_z^-}\mathbf d(z(t),\mathbf C)=0,$8 the specific volume, and the system has characteristic speeds

$\lim_{t\to T_z^-}\mathbf d(z(t),\mathbf C)=0,$9

The paper proves exact boundary controllability toward constant states in the class of weak entropy $$\mathcal J_{z,\alpha}=\int_0^{T_z} l(z(t),\alpha(t))\,dt, \qquad l:\mathbf C^c\times A\to [0,+\infty),$$0 solutions, provided

$$\mathcal J_{z,\alpha}=\int_0^{T_z} l(z(t),\alpha(t))\,dt, \qquad l:\mathbf C^c\times A\to [0,+\infty),$$1

and the target constant satisfies the entropy compatibility condition

$$\mathcal J_{z,\alpha}=\int_0^{T_z} l(z(t),\alpha(t))\,dt, \qquad l:\mathbf C^c\times A\to [0,+\infty),$$2

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 $$\mathcal J_{z,\alpha}=\int_0^{T_z} l(z(t),\alpha(t))\,dt, \qquad l:\mathbf C^c\times A\to [0,+\infty),$$3-contacts do not propagate toward the boundary. This is resolved indirectly through nonlinear interactions with strong $$\mathcal J_{z,\alpha}=\int_0^{T_z} l(z(t),\alpha(t))\,dt, \qquad l:\mathbf C^c\times A\to [0,+\infty),$$4- and $$\mathcal J_{z,\alpha}=\int_0^{T_z} l(z(t),\alpha(t))\,dt, \qquad l:\mathbf C^c\times A\to [0,+\infty),$$5-shocks (Glass, 2013).

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 (Bressan et al., 2017). If some coordinates $$\mathcal J_{z,\alpha}=\int_0^{T_z} l(z(t),\alpha(t))\,dt, \qquad l:\mathbf C^c\times A\to [0,+\infty),$$6 are directly prescribed and the remaining free coordinates are $$\mathcal J_{z,\alpha}=\int_0^{T_z} l(z(t),\alpha(t))\,dt, \qquad l:\mathbf C^c\times A\to [0,+\infty),$$7, the reduced equations take the form

$$\mathcal J_{z,\alpha}=\int_0^{T_z} l(z(t),\alpha(t))\,dt, \qquad l:\mathbf C^c\times A\to [0,+\infty),$$8

Because the right-hand side depends both linearly and quadratically on $$\mathcal J_{z,\alpha}=\int_0^{T_z} l(z(t),\alpha(t))\,dt, \qquad l:\mathbf C^c\times A\to [0,+\infty),$$9, the system is not control-affine. The paper introduces the differential inclusion

$l$0

with

$l$1

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 (Bressan et al., 2017). 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 $l$2 reduction is achieved at the cost of additional advection equations (Wei et al., 2021). 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 (Wei et al., 2021). 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 (Loria, 2013). 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 (Barbosa et al., 2020). 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 (Leyendecker et al., 2023). The new formulation is regular and enables symplectic discretisation via variational integrators in a straightforward way (Leyendecker et al., 2023). 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 $l$3, with controllability characterized by freeness of the system module (Join et al., 2024). 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 (Join et al., 2024). 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 (Join et al., 2024).

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 $l$4 (Fernandez-Cara et al., 2024). The augmented Lagrangian

$l$5

leads to Uzawa-type iterations and exact null controllability of numerical approximations in two- and three-dimensional experiments (Fernandez-Cara et al., 2024). 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 (Shin et al., 2021). Uniform controllability implies uniform LICQ of the dynamic constraints, while observability of $l$6, where $l$7 is a block of the Lagrangian Hessian, implies positivity of the reduced Lagrangian Hessian. The resulting estimate

$l$8

shows how controllability of linearized dynamics can be translated into decay properties of the Lagrangian KKT system (Shin et al., 2021).

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.