Luenberger Program and Kinetic Observers

• The Luenberger Program is a collection of control-theoretic and data assimilation methods that extend the classical observer to nonlinear, infinite-dimensional, and measure-theoretic settings.
• It employs kinetic lifting, convex analysis, and duality-based techniques to transform complex nonlinear PDE observer problems into linear assimilation frameworks.
• The framework demonstrates robust convergence under complete, partial, and noisy observations, with applications in meteorology, oceanography, and engineering control systems.

The Luenberger Program encompasses a collection of control-theoretic and data assimilation methodologies that generalize the original Luenberger observer—from finite-dimensional linear systems to nonlinear, infinite-dimensional, kinetic, and measure-theoretic contexts, with extensions to constrained optimization, parameter estimation, and quantum systems. Central to the program is the exploitation of observer (or filter) structures and duality-based approaches for robust estimation, optimization, and control, often leveraging convex analysis, fixed-point iteration, or variational principles. The Luenberger observer concept underpins these developments as a unifying strategy for stabilizing error dynamics, incorporating partial observations, and synthesizing computationally efficient assimilation or control schemes.

1. Kinetic Luenberger Observers for Hyperbolic Conservation Laws

A paradigmatic extension of the Luenberger Program to nonlinear PDEs is the kinetic Luenberger observer for hyperbolic conservation laws (Boulanger et al., 2013). Here, a (possibly nonlinear) conservation law is 'lifted' into kinetic form by assigning to each macroscopic variable $u(t,x)$ a kinetic density $f(t,x,\xi)$, with $\xi$ representing a free velocity parameter. The underlying PDE,

$$\partial_t f(t,x,\xi) + a(\xi) \cdot \nabla_x f(t,x,\xi) = \lambda (M(t,x,\xi) - f(t,x,\xi)),$$

resembles a BGK (Bhatnagar-Gross-Krook) type equation. The target density $M(t,x,\xi)$ is reconstructed from observations, often chosen as a Gibbs equilibrium (for hydrodynamic systems, $M$ encodes the macroscopic fields such as water height and velocity).

This framework yields a 'linear' kinetic equation for $f$, sidestepping the difficulties associated with direct observer design in the original nonlinear conservation law. The observer's macroscopic estimate is recovered via

$u(t,x) = \int_\mathbb{R} f(t,x,\xi)\,d\xi.$

2. Convergence Theory Under Complete, Partial, and Noisy Observations

For complete observations and smooth solutions, the kinetic Luenberger observer admits explicit representation formulas (using, e.g., variation of constants), leading to exponential convergence:

$$\|f(t) - M(t)\| \le \|f(0) - M(0)\| e^{-\lambda t}.$$

By integrating in $\xi$, one deduces exponential convergence of the macroscopic estimate to the true state in $L^1$.

In the nonsmooth (shocked or discontinuous) case, convergence rates are governed by additional defect/collision measures; the error bound becomes

$$\|f(T) - M(T)\|_{V_1} \le e^{-\lambda T}\|f_0 - f_0^\lambda\|_{V_1} + \omega(T,1/\lambda),$$

with the modulus $\omega$ and modified initial condition $f_0^\lambda$ both controlled as $\lambda \to \infty$.

In scenarios with partial or time-sampled observations, a mollified nudging term is used:

$$\partial_t f + a(\xi) \cdot \nabla_x f + \lambda \sum_k \varphi_\sigma(t-t_k) f(t_k,x,\xi) = \lambda \sum_k \varphi_\sigma(t-t_k) M(t_k,x,\xi),$$

where $\varphi_\sigma$ is a smoothing kernel centered at observation times $t_k$. Convergence persists as the mollification $\sigma \to 0$ and sampling becomes dense. For noisy data, analytical results establish an optimal nudging parameter $\lambda_\mathrm{opt}$, balancing synchronization speed against error amplification due to noise, with explicit estimates in fractional Sobolev spaces.

3. Numerical Implementation: Burgers' Equation and Shallow Water (Saint-Venant) System

Numerical experiments validate the kinetic observer's efficacy on both scalar and system-level conservation laws. For Burgers' equation, an upwind finite volume scheme ensures stability under a $\lambda$-dependent CFL condition:

$$\partial_t f + \xi \partial_x f = \lambda (\chi(\xi, u) - f), \qquad u(t,x) = \int_\mathbb{R} f(t,x,\xi)\,d\xi,$$

where $\chi$ is a characteristic function derived from the kinetic representation.

In the presence of noisy data, the error (measured in appropriate norms) exhibits a non-monotonic dependence on $\lambda$, with an optimal value observed numerically, in agreement with theoretical predictions.

For the Saint-Venant system, the observer reconstructs the velocity when only water height $H$ is observed, by constructing an auxiliary target density $\tilde M$ using the observer-estimated velocity,

$$\tilde M(t,x,\xi) = \frac{H(t,x)}{c} \chi\left(\frac{\xi - \hat{u}(t,x)}{c}\right), \quad c = \sqrt{\frac{g H(t,x)}{2}},$$

and deploying

$$\partial_t f + \xi \cdot \nabla_x f - g \nabla_x z_b \cdot \nabla_\xi f = \lambda(\tilde M - f).$$

Simulations confirm robust synchronization, even under incomplete or noisy observations.

4. Methodological Advantages and Trade-Offs

This kinetic observer framework introduces key structural and computational benefits:

• Reduction of nonlinear observer synthesis to a linear kinetic assimilation problem, simplifying both analysis and numerical implementation;
• Direct transfer of energy estimates and convergence results via explicit solution representation;
• Decoupling of the observer algorithm from the nonlinear flux structure, as macroscopic fields are recovered purely by averaging over kinetic states;
• Flexibility to handle partial and noisy data by modifying the nudging term or incorporating mollifiers, yielding provable convergence in weak Sobolev norms.

A critical design parameter is the nudging/relaxation coefficient $\lambda$: higher values accelerate synchronization but can amplify measurement noise, especially for noisy or non-smooth data. Analyses provide guidance for choosing $\lambda$ optimally.

5. Summary of Principal Formulas

Context	Key Formula
Kinetic observer equation	$$\partial_t f + a(\xi)\cdot\nabla_x f = \lambda(M - f)$$
Macroscopic recovery	$u(t,x) = \int_\mathbb{R} f(t,x,\xi)\,d\xi$
Gibbs equilibrium (Saint-Venant)	$M = [H/c]\ \chi((\xi-u)/c),\ c = \sqrt{gH/2}$
Partial/time-sampled observation	as above, with mollifier $\varphi_\sigma$ in nudging terms
Smooth-case error estimate	$\|f(t) - M(t)\| \le \exp(-\lambda t) \|f_0 - M_0\|$
Noisy-case error bound	$$\|\rho_\psi\|_{\dot{H}^{\gamma/2}}^2 \le K_1 \|E\|^{(2-\gamma)/(1+\alpha)} \|Q\|^{(2\alpha+\gamma)/(1+\alpha)}$$

6. Implications and Applications

The kinetic Luenberger observer provides a practical, theoretically rigorous path to robust data assimilation for nonlinear conservation law systems where traditional Luenberger observers fail (due to lack of dissipation or strong nonlinearity). Applications include:

• Data assimilation in meteorology and oceanography (e.g., shallow water or atmospheric flows);
• Synchronization and control of hyperbolic PDEs in engineering systems;
• Filtering and estimation for models characterized by shocks and discontinuities, where smoothing via kinetic lifting yields tractable convergence properties.

The approach naturally accommodates both complete and partial observations, and is amenable to online and off-line implementation styles. The explicit treatment of noise-induced trade-offs in the nudging parameter allows tailoring observer aggressiveness to measurement quality.

7. Theoretical and Computational Outlook

This kinetic observer strategy establishes a blueprint for extending Luenberger-type programs to a range of nonlinear, non-dissipative, or high-dimensional systems, moving beyond linear or finite-dimensional dynamics. It demonstrates how reformulating a nonlinear observer synthesis problem at a higher 'lifted' level can dramatically expand the tractability of convergence analysis and algorithm design. Moreover, it provides a framework for rigorous, scalable, and stable data assimilation tools in the context of hyperbolic conservation laws and related nonlinear evolutions, with concrete numerical validation to support theoretical results (Boulanger et al., 2013).