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Time-optimal problem in the space of probabilities measures

This paper investigates the time-optimal control problem for a continuity equation in the space of probability measures by deriving its dynamic programming principle, proving that the Kruzhkov transform of the value function is the unique discontinuous viscosity solution to the associated Hamilton-Jacobi equation, and establishing the $\Gamma$ -convergence of the value function under perturbations.

Original authors: Yurii Averboukh, Ekaterina Kolpakova

Published 2026-03-03

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Yurii Averboukh, Ekaterina Kolpakova

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are the captain of a massive fleet of thousands of tiny, identical boats floating on a vast, foggy ocean. You can't steer each boat individually; instead, you have a "wind controller" that changes the wind direction for the entire fleet at once.

Your goal is to get the entire fleet to a specific destination island (let's call it "Target Island") as quickly as possible. The tricky part is that the boats influence each other: if the fleet bunches up, the water gets choppy, which changes how the boats move. This is a "nonlocal" system—every boat feels the presence of every other boat.

This paper is a mathematical guide on how to solve this "Time-Optimal" problem. Here is the breakdown of what the authors did, translated into everyday language:

1. The Problem: Steering a Cloud, Not a Single Boat

Usually, when we think of control problems, we imagine steering a single car or a single robot. But here, the "object" being controlled is a cloud of probability. Think of it like a swarm of bees or a crowd of people.

The Challenge: You need to figure out the best wind pattern (control) to push this whole cloud to the target island in the shortest time.
The Complication: The cloud moves according to a complex rule (the continuity equation) where the movement of the whole depends on the shape of the cloud itself.

2. The Strategy: The "Dynamic Programming" Map

The authors use a concept called Dynamic Programming. Imagine you are trying to find the fastest route home.

Instead of planning the whole trip from start to finish at once, you ask: "If I am at this specific spot right now, what is the best move I can make for the next 10 seconds to get home fastest?"
By answering this question for every possible spot the cloud could be in, you build a "map" (called the Value Function) that tells you the minimum time required to reach the target from anywhere.

3. The Secret Weapon: The "Kruzhkov Transform"

The "Value Function" (the map of time) is tricky because it can have sharp edges or sudden jumps (discontinuities). For example, if you are just outside the target island, it might take 1 second. If you are just inside, it takes 0 seconds. That jump makes standard math tools break.

To fix this, the authors use a mathematical magic trick called the Kruzhkov Transform.

The Analogy: Imagine you have a jagged, bumpy mountain range (the time map). It's hard to climb or analyze. The Kruzhkov transform is like taking a photo of that mountain and turning it into a smooth, rolling hill (using the formula $1 - e^{-time}$ ).
Why it helps: Once the map is smooth, they can use powerful tools from calculus (specifically, Viscosity Solutions) to analyze it, even if the original map was bumpy.

4. The Hamilton-Jacobi Equation: The "Rulebook"

The authors prove that this smoothed-out map follows a specific set of rules known as the Hamilton-Jacobi equation.

Think of this equation as the "Law of Physics" for your time map. It says: "The speed at which your time-to-target changes depends on how you steer the wind right now."
They prove that their map is the unique solution to this law. This means there is only one correct answer to the problem, and their method finds it.

5. Handling the "Rough Edges" (Viscosity Solutions)

In the real world, things aren't always perfectly smooth. Sometimes the best path involves hitting a wall or making a sudden turn.

The authors developed a special way to handle these "rough edges" using Viscosity Solutions.
The Analogy: Imagine trying to roll a ball down a hill that has a few potholes. A standard mathematician might say, "The ball can't roll here because the ground is broken." A "Viscosity" mathematician says, "Let's pretend the pothole is a tiny, smooth dip, see where the ball goes, and then take the limit as the dip gets smaller." This allows them to find the solution even when the terrain is messy.

6. Stability: What if the Wind Changes Slightly?

Finally, the authors asked: "What if our model of the wind isn't 100% perfect? What if there's a tiny error?"

They proved that if you make a small change to the wind rules (a "perturbation"), the resulting best time doesn't explode or go crazy. It stays close to the original answer.
The Analogy: If you tweak the wind controller slightly, the time it takes to reach the island changes only slightly. This proves the solution is robust and reliable for real-world applications.

Summary

In short, this paper takes a very complex problem—steering a massive, interacting crowd of particles to a target as fast as possible—and:

Proves an optimal strategy exists.
Creates a mathematical "map" (Value Function) for that strategy.
Smooths out the map's rough edges using a clever transformation.
Shows that this map follows a specific, unique set of rules (Hamilton-Jacobi).
Proves the solution is stable even if the rules change slightly.

It's like giving a captain a reliable, mathematically proven GPS for steering a chaotic swarm of boats through a storm to safety.

1. Problem Statement

The paper investigates a time-optimal control problem for a system governed by a nonlocal continuity equation in the space of probability measures with finite second moments ( $P_2(\mathbb{R}^d)$ ).

Dynamics: The system evolution is described by:
$\partial_t m(t) + \text{div}\left( \int_U f(x, m(t), u) \xi(du|t) \cdot m(t) \right) = 0$
where $m(t) \in P_2(\mathbb{R}^d)$ is the distribution of particles, $U$ is a compact metric space of controls, and $\xi$ represents a relaxed control (a measure on $[0, \infty) \times U$ ). The vector field $f$ depends on the state $x$ , the measure $m(t)$ (nonlocal interaction), and the control $u$ .
Objective: Given an initial distribution $\mu$ and a closed target set $M \subset P_2(\mathbb{R}^d)$ , the goal is to find a control $\xi$ that steers the system to $M$ in minimal time.
Value Function: The value function is defined as:
$\text{Val}(\mu) = \inf \{ \tau(m(\cdot; \mu, \xi)) : \xi \in \mathcal{U} \}$
where $\tau$ is the first hitting time of the set $M$ .

2. Methodology

The authors employ a combination of optimal control theory, nonsmooth analysis in Wasserstein spaces, and viscosity solution theory.

Relaxed Controls: The study utilizes relaxed controls (generalized controls) to ensure the existence of optimal solutions and the compactness of the control space.
Kruzhkov Transform: To handle the potentially infinite value of the time-optimal problem, the authors apply the Kruzhkov transform:
$\phi(\mu) = 1 - e^{-\text{Val}(\mu)}$
This maps the value function to the bounded interval $[0, 1]$ , transforming the problem into a Dirichlet problem for a Hamilton-Jacobi (HJ) equation.
Nonsmooth Analysis Framework:
- Sublinear Variations: A key methodological innovation is the restriction of admissible variations to Borel functions satisfying a sublinear growth condition ($SL(m)$), rather than the standard square-integrable ( $L^2$ ) condition used in many existing works (e.g., [5, 15]).
- Dual Space: Consequently, the sub- and superdifferentials are defined in the dual space $LD(m)$ (functions $s$ such that $\int \|s(x)\|(1+\|x\|)m(dx) < \infty$ ). This allows for elements that grow faster than $L^2$ functions, which is crucial for the specific structure of the time-optimal problem.
Viscosity Solutions: The authors define viscosity sub- and supersolutions for the HJ equation using these generalized sub/superdifferentials.

3. Key Contributions and Results

A. Existence and Properties of the Value Function

Existence of Optimal Control: The authors prove the existence of an optimal relaxed control whenever the value function is finite (Theorem 3.7).
Lower Semicontinuity: The value function is proven to be lower semicontinuous (Proposition 3.9).
Dynamic Programming Principle (DPP): A rigorous DPP is derived for the value function on the closure of the domain $G = P_2(\mathbb{R}^d) \setminus M$ (Theorem 3.10).

B. Hamilton-Jacobi Characterization

Dirichlet Problem: The transformed value function $\phi$ is shown to satisfy the following Dirichlet problem in the viscosity sense:
$H(m, \nabla \phi(m)) + 1 - \phi(m) = 0, \quad m \in G$
$\phi(m) = 0, \quad m \in \partial G$
where the Hamiltonian is $H(m, s) = \inf_{u \in U} \int \langle s(x), f(x, m, u) \rangle m(dx)$ .
Viscosity Solution: The paper proves that the Kruzhkov transform of the value function is a viscosity solution to this Dirichlet problem (Theorem 6.5).

C. Uniqueness and Comparison Principle

Comparison Principle: A major theoretical contribution is the proof of a comparison principle (Theorem 5.1) for the viscosity solutions of the Dirichlet problem. This relies on:
- The differentiability properties of the squared Wasserstein distance.
- A moment gauge function $V_g(m)$ (Lemma 5.4) constructed to regularize the difference between sub- and supersolutions. This function ensures compactness of sublevel sets and handles the unbounded nature of the domain.
Uniqueness: The comparison principle implies the uniqueness of the viscosity solution to the Dirichlet problem (Corollary 5.2). This is a significant result, as many previous works on mean-field control only established existence or lacked uniqueness for time-optimal problems.

D. Continuity and Stability

Continuity Condition: The authors establish a sufficient condition for the continuity of the value function: if the viscosity solution is continuous on the boundary $\partial G$ , it is continuous everywhere on $\text{cl } G$ (Theorem 7.1). This is linked to the "small-time local attainability" property.
$\Gamma$ -Convergence: The paper proves that the value functions of perturbed time-optimal problems (where the dynamics $f_n$ converge to $f$ ) $\Gamma$ -converge to the value function of the original problem (Theorem 7.4). This ensures the stability of the optimal control problem under perturbations of the dynamics.

4. Significance and Impact

Novel Definition of Viscosity Solutions: By moving beyond $L^2$ variations to sublinear growth variations, the authors provide a framework better suited for time-optimal problems where the value function may be discontinuous and the dynamics have specific growth properties. This definition is "stronger" in the sense that it admits a broader class of test functions for the sub/superdifferentials.
Uniqueness in Time-Optimal Control: While mean-field control literature often focuses on finite-horizon problems (Cauchy problems), this paper successfully addresses the Dirichlet problem (time-optimal) and establishes uniqueness of the solution, a result often missing in similar contexts.
Robustness: The $\Gamma$ -convergence result provides a theoretical foundation for the numerical approximation and stability analysis of such control systems, which is vital for applications in crowd dynamics, opinion formation, and swarm robotics.
Generalization: The results apply to general nonlocal continuity equations with external forces, extending previous work that often assumed specific structures (e.g., hyperplane targets or specific interaction kernels).

5. Example

The paper concludes with an explicit example in 1D where the target is the set of measures with zero mean. The authors derive the explicit value function and the corresponding viscosity solution, demonstrating that the value function is indeed lower semicontinuous but not necessarily continuous everywhere, validating the necessity of the viscosity solution framework.

In summary, this paper provides a rigorous mathematical foundation for time-optimal control in the space of probability measures, resolving issues of existence, uniqueness, and stability through a novel viscosity solution framework tailored to the specific growth conditions of the problem.