Hydrodynamics can determine the optimal route for microswimmer navigation

The study addresses how self-propelled microswimmers should steer to minimize travel time to a target in environments where hydrodynamic interactions with walls and obstacles are significant. Unlike macroscopic navigation problems (e.g., airplanes in wind fields) traditionally handled by optimal control theory, microswimmer navigation occurs at low Reynolds number with overdamped dynamics, thermal fluctuations, and long-ranged hydrodynamic interactions. The central question is whether and how hydrodynamics modifies the optimal route compared to the shortest path, and how fluctuations impact optimal strategies. Understanding this is relevant for biological processes (e.g., foraging, immune cell search) and for applications such as targeted delivery by artificial micro/nanorobots in complex vascular or obstacle-laden environments.

Classical optimal control problems for macroagents are well-established. In active matter, recent works explored optimal navigation and search strategies for dry active particles and in external flows, often using reinforcement learning to learn steering policies and smart gravitaxis, and analytical treatments for optimal navigation of active particles. Deep RL has been applied to microswimmer navigation in mazes and obstacle arrays with varying environmental knowledge. Complementary research optimized swimming strokes and body deformations with optimal control theory for deformable swimmers. However, the role of hydrodynamic interactions with boundaries (nonconservative, long-ranged) and their interplay with fluctuations in shaping optimal routes for microswimmers remained largely unexplored, motivating the present study.

The authors formulate a minimal optimal control problem: a microswimmer can freely steer its orientation but not its intrinsic speed and must reach a prescribed target. Dynamics are overdamped. Starting with planar motion (xz-plane), they derive deterministic kinematics including far-field hydrodynamic interactions with boundaries and later add translational noise. Two complementary solution frameworks are used. 1) Lagrangian (calculus of variations) approach: Eliminate the steering angle from the equations of motion to express the travel time T as a line integral T = ∫ L(x, z(x), z'(x)) dx over a path z(x), then solve the Euler–Lagrange equation with boundary conditions via shooting methods to obtain the time-optimal trajectory. Swimmer models: - Source dipole near a planar wall: In nondimensional units, ẋ = (1 − 4 z^{-3}) cos ψ, ż = (1 − s z^{-3}) sin ψ with s = sgn(σ). Eliminating ψ yields a Lagrangian L_SD and an Euler–Lagrange ODE solved numerically; trajectories depend on the sign of σ but not its magnitude after nondimensionalization. - Force dipole (pushers/pullers) near a planar wall: ẋ = cos ψ + (3 s sin 2ψ)/(8 z^2), ż = sin ψ + (3 s [1 − 3 cos 2ψ])/(16 z^2). After algebra, they obtain a Lagrangian L_FD that is invariant under sgn(a), implying identical optimal paths for pushers and pullers; the boundary value problem is solved by shooting. - Force quadrupole near a planar wall: Analogous elimination of ψ leads to L_FQ (involving higher-order polynomials) and numerical solution of the Euler–Lagrange equation. - Spherical obstacles and obstacle fields: Hydrodynamic interactions with a rigid sphere are computed using the Stokes Green’s function; the swimmer velocity is decomposed into orientation and wall-induced components with coefficients P(h; R, σ) and Q(h; R, σ) depending on the gap h and sphere radius R. For two widely separated spheres, a superposition approximation sums contributions of each obstacle. The resulting Lagrangian L_SP(x, z, z′) is evaluated and fitted numerically, and the Euler–Lagrange problem is solved (e.g., Matlab ode45). 2) Optimal control (Pontryagin’s maximum principle): For time-dependent swimmers (time-varying propulsion speed and/or source dipole strength), the authors construct the Hamiltonian H(r, p, γ, t) = F · p, minimize over the control (steering angle) γ, and integrate the Hamiltonian state–costate equations as a boundary value problem with end-time transversality condition H(T) = 1. Time is rescaled so the end-time is unity; T is treated as an additional state (ȧ = 0) for shooting. Noise and strategies: To study fluctuations, translational Brownian noise with coefficient D (dimensionless D*) is added (3D simulations). Two strategies are compared: (i) Straight steering directly toward the target at each instant; (ii) Optimal steering: repeatedly re-compute and follow the instantaneous optimal-noise-free direction from the current position. Statistics are obtained from ensembles (e.g., 5000 trajectories), with the target considered reached upon entering a small spherical neighborhood.

• Shortest path is not fastest: In the presence of walls/obstacles, hydrodynamic interactions make the time-optimal trajectories deviate markedly from the geometric shortest path, even without external force or flow fields. Swimmers take systematic excursions (detours) that minimize time. - Dependence on swimmer type and wall distance: Source dipole swimmers with σ > 0 (s = +1) slow down near walls and optimally bend away from boundaries; those with σ < 0 (s = −1) speed up near walls and optimally approach them. For flat and spherical boundaries, this produces qualitatively different detours that can be faster than the straight line despite being longer. - Pusher–puller identity in optimal paths: For force dipole swimmers (pushers vs pullers), the Lagrangian for travel time is invariant under sign reversal of the dipole strength. Consequently, pushers and pullers have identical optimal trajectory shapes, although their optimal steering protocols ψ(t) differ. - Quadrupolar swimmers: The optimal paths bend toward or away from the wall depending on the sign of the quadrupole coefficient; parabolic-like excursions are observed. - Obstacles and landscapes: Near spherical obstacles and arrays, hydrodynamics qualitatively alters fastest routes compared to dry active particles. For σ > 0 swimmers, detours that maintain a favorable distance from obstacles are faster; for σ < 0, hugging obstacles is advantageous. - Fluctuations favor optimal detours: Across a broad noise range (dimensionless D* from 0 to ≈0.15), the re-planned optimal strategy consistently outperforms the straight strategy. The relative advantage increases with noise. Mechanisms: For s = −1, optimal steering guides swimmers into faster near-wall regions that counteract noise; straight steering can drift into slow regions. For s = +1, optimal paths keep swimmers away from slow near-wall regions, reducing trapping by noise. Improvements can amount to up to about a factor of two faster than heading straight. - Time-dependent swimmers: With oscillatory propulsion and/or flow-field strength (e.g., g1(t), g2(t) out of phase), optimal trajectories exhibit step–plateau features in wall distance, adjusting distance during slow phases to optimize subsequent faster phases. Travel time shows nonmonotonic dependence on driving frequency and amplitude, with extrema when the target is reached before a full driving cycle; the global minimum occurs at ω* = π in an example. - Parameter relevance: Estimates for typical microorganisms (e.g., E. coli with measured force dipoles) indicate that hydrodynamic effects become important at distances of order 1–2 body lengths, implying that detours should be significant in realistic settings. Typical noise levels (D* ~ 10⁻²–10⁻¹) fall within the regime where optimal strategies substantially reduce travel time.

The findings demonstrate that hydrodynamic interactions fundamentally change the optimal navigation problem for microswimmers compared to dry active particles or macroscopic vehicles. Nonconservative, wall-mediated flows alter the swimmer’s effective speed as a function of wall distance, making time-optimality require deliberate detours to modulate this speed field. The analysis shows when swimmers should avoid or exploit proximity to boundaries based on their flow singularity type. Under fluctuations, continually re-optimizing the direction based on the current state protects swimmers from noise-induced trapping in slow regions and enhances overall performance. The results imply that microorganisms could gain survival advantages by actively regulating their distance to boundaries (e.g., surfaces or interfaces) while foraging or navigating, and they provide reference solutions to benchmark machine-learning approaches and guide the programming of synthetic microswimmer controllers in complex environments.

Hydrodynamic interactions with remote boundaries can determine the fastest routes for microswimmers, making the shortest path suboptimal. Optimal trajectories involve strategic detours whose shapes depend on swimmer type (source/force/quadrupole) and boundary geometry; notably, pushers and pullers share identical optimal paths. In fluctuating environments, re-planned optimal steering significantly outperforms straight steering, with benefits increasing with noise strength. Time-dependent propulsion further enriches optimal strategies, producing nontrivial path structures and frequency-dependent travel times. These insights supply a theoretical foundation and benchmarks for optimal microswimmer navigation in complex settings, inform biological hypotheses about active regulation of wall distance, and can aid the development and validation of machine-learning-based navigation and control schemes. Future work could solve the full Hamilton–Jacobi–Bellman equation with noise, extend to more complex, nonlocally known environments, include more detailed swimmer–boundary hydrodynamics, and validate predictions experimentally.

• Far-field hydrodynamic models are used for swimmer–boundary interactions; near-field effects and lubrication forces are not explicitly resolved. - For multiple obstacles, a superposition approximation is employed, valid for widely separated obstacles. - Many analyses begin with 2D planar motion; full 3D steering complexities are addressed mainly in stochastic simulations. - Noise modeling primarily includes translational Brownian noise with space-independent D; rotational diffusion and control delays are discussed qualitatively but not modeled in detail. - Time-dependent case studies consider specific functional forms (e.g., sinusoidal g1, g2), which illustrate but do not exhaust general possibilities. - Numerical solutions rely on shooting methods and fitted Lagrangians for complex geometries; explicit closed forms are often intractable. - Experimental validation is not presented; parameter relevance is argued from estimates.