Rigorous long-time asymptotics for the optimal control of conditioned processes (the Achdou–Laurière–Lions ansatz)

Achdou, Laurière and Lions (J. Math. Pures Appl. 148 (2021) 308–341) studied the finite-horizon optimal control of a diffusion killed at the boundary of a bounded domain, with a cost built from expectations conditioned on survival and controls acting in feedback (Markovian) form. The optimality system couples a forward Fokker–Planck equation and a backward Hamilton–Jacobi–Bellman equation, both with Dirichlet boundary conditions and nonlocal terms normalized by the survival probability, which decays as time grows. Formal asymptotics and numerical experiments suggest that, as the horizon $T \to \infty$ and away from the initial and terminal layers, the normalized conditioned density, the rescaled value function, the survival decay rate, and the optimal feedback converge to a stationary system coupling a principal-eigenvalue problem for a nonsymmetric elliptic operator with a stationary Bellman equation. The authors state that proving this ansatz rigorously is an open problem, the main obstruction being that other asymptotic regimes, for example time-periodic ones, cannot currently be excluded; they also leave open the identification, in the limit of unbounded controls, of a measure $\gamma$ with the normalized conditional energy. The problem remains open.

Background

Let $\Omega \subset \mathbb{R}^d$ be a bounded domain with smooth boundary and let $\sigma > 0$. For a feedback (also called closed-loop or Markovian) control — a deterministic function $b(t,x)$ of time and state — consider the diffusion $dX_t = -b(t,X_t)\,dt + \sigma\,dW_t$ started with a smooth density $p_0$ supported in $\Omega$, and let $\tau^b$ be the first exit time from $\Omega$. Achdou, Laurière and Lions studied the finite-horizon control problem in which the cost is computed from expectations conditioned on survival,

$$J(b) = \int_0^T \mathbb{E}\big[L(X_t, b(t,X_t)) \,\big|\, \tau^b > t\big]\,dt + \mathbb{E}\big[g(X_T) \,\big|\, \tau^b > T\big],$$

in a generalized version where the running and terminal costs may also depend on the conditional law of the process through differentiable functionals, together with an optional term $-\varepsilon \ln \mathbb{P}(\tau^b > T)$ that penalizes the decay of the survival probability. The problem was proposed by P.-L. Lions in his November 2016 lectures at the Collège de France, as recounted in the papers cited below.

The killed process has a sub-probability density $p_b(t,x)$ (the law of $X_t$ restricted to the event of survival) which solves a Fokker–Planck equation — the forward equation for the law of a diffusion — with a Dirichlet boundary condition on $\partial\Omega$, i.e. the requirement that the solution vanish on the boundary, encoding the absorption of the process there; its mass $S(t) = \int_\Omega p_b(t,x)\,dx = \mathbb{P}(\tau^b > t)$ is the survival probability, which decays as $t$ grows. Rewriting the conditional cost with $p_b$ turns the problem into a deterministic control problem driven by this Fokker–Planck equation, in the spirit of mean field type control (the control of McKean–Vlasov dynamics through the PDE for their law) and structurally close to mean field game systems (forward–backward PDE systems arising for large symmetric populations of optimizing agents). The optimality conditions couple this forward equation with a backward Hamilton–Jacobi–Bellman equation (the dynamic-programming equation for the value function) for an adjoint function $u$, both with Dirichlet boundary conditions and with nonlocal terms normalized by $S(t)$; the paper proves existence of optimal controls and of weak solutions of this forward–backward system.

For an uncontrolled killed diffusion, the classical theory of quasistationary distributions — invariant laws for the dynamics conditioned on survival — states that the conditioned law converges to a quasistationary distribution given by the principal eigenfunction of the Dirichlet generator (the positive eigenfunction associated with the eigenvalue of maximal real part), while the survival probability decays at the rate given by that principal eigenvalue. Numerical experiments in the paper suggest that the optimally controlled system obeys the same picture: away from the initial and terminal time layers (a turnpike-type behavior, i.e. the phenomenon whereby finite-horizon optimal solutions spend most of the horizon near a stationary state), the solutions appear to converge, as the horizon $T \to \infty$, to a stationary regime in which the survival decay rate is an eigenvalue. Substituting this ansatz into the forward–backward system yields a stationary system coupling a principal-eigenvalue problem for a nonsymmetric elliptic operator (the stationary Fokker–Planck operator with the optimal drift) with a stationary Bellman equation. The paper proves that this stationary system has a solution, and confirms the ansatz numerically in one and two dimensions, but does not prove the convergence.

The difficulty is that the long-time behavior of coupled forward–backward control systems is not covered by existing theory: the exponential turnpike results known for mean field game and mean-field-type control systems rely on monotonicity, weak-monotonicity-plus-smallness, or linear-quadratic structure, on domains without a killing boundary, while related linear-quadratic mean-field systems are known to admit time-periodic long-time behavior in some regimes — precisely the alternative the authors could not exclude. The paper also studies, separately, the limit in which the uniform bound $M$ on the controls tends to infinity; the limit optimality system contains a positive Radon measure $\gamma$ in place of the effects of the bound, and the authors note that identifying $\gamma(t)$ with the normalized conditional energy $\int_\Omega p(t,x)\big(\tfrac{1}{2}|b(t,x)|^2 + f(x)\big)dx / \big(\int_\Omega p(t)\big)^2$ (for the quadratic cost $L(x,b) = \tfrac12 |b|^2 + f(x)$) seems possible by crossed-regularity arguments — estimates obtained by testing each equation against the other equation's solution, a technique from the theory of weak solutions of Fokker–Planck and Hamilton–Jacobi–Bellman equations developed by Porretta — but they do not carry this out.

Problem Statement

Setting (Achdou–Laurière–Lions, J. Math. Pures Appl. 148 (2021), eqs. (5)–(17)). $\Omega \subset \mathbb{R}^d$ is bounded with smooth boundary; $\sigma > 0$; $p_0$ is smooth, nonnegative, supported in $\Omega$; the Lagrangian $L(x,b)$ is continuous, strictly convex and continuously differentiable in $b$; the cost may include Fréchet-differentiable bounded-below functionals of the conditional law and the survival penalization $-\varepsilon \ln S(T)$ with $\varepsilon \ge 0$; the controls are feedback functions bounded in $L^\infty$. For each horizon $T$, let $\big(p_b^{(T)}, u^{(T)}, b_{opt}^{(T)}\big)$ solve the finite-horizon optimality system (11)–(12) of the paper: the Dirichlet Fokker–Planck equation for the sub-probability density $p_b^{(T)}$ coupled with the backward Dirichlet Hamilton–Jacobi–Bellman equation for $u^{(T)}$ with its nonlocal normalization terms involving the survival mass $\int_\Omega p_b^{(T)}(t)$.

Part A. Prove, or disprove, the long-time ansatz: that there exist $\eta \in (0, 1/2)$ and a solution $(\tilde p, \tilde u, \lambda, \tilde b)$ of the stationary system such that, as $T \to \infty$, for all $t$ with $t/T \in (\eta, 1-\eta)$,

$$\frac{p_b^{(T)}(t,x)}{\int_\Omega p_b^{(T)}(t,y)\,dy} \longrightarrow \tilde p(x), \qquad \Big(\int_\Omega p_b^{(T)}(t,y)\,dy\Big)\, u^{(T)}(t,x) \longrightarrow \tilde u(x),$$ $$\frac{1}{\int_\Omega p_b^{(T)}(t,y)\,dy}\int_\Omega \frac{\partial p_b^{(T)}}{\partial t}(t,y)\,dy = \frac{d}{dt} \ln \int_\Omega p_b^{(T)}(t,y)\,dy \longrightarrow -\lambda, \qquad b_{opt}^{(T)}(t,x) \longrightarrow \tilde b(x),$$

where the stationary system is

$$-\frac{\sigma^2}{2}\Delta \tilde p - \mathrm{div}\big(\tilde p \, H_\xi(\cdot, D\tilde u)\big) = \lambda \tilde p \ \ \text{in } \Omega, \qquad \tilde p = 0 \ \ \text{on } \partial\Omega, \qquad \tilde p \ge 0, \quad \int_\Omega \tilde p = 1,$$ $$-\frac{\sigma^2}{2}\Delta \tilde u + H(\cdot, D\tilde u) = \lambda \tilde u + F[\tilde p] + c_1 \ \ \text{in } \Omega, \qquad \tilde u = 0 \ \ \text{on } \partial\Omega, \qquad \int_\Omega \tilde u \, \tilde p \, dx = -\varepsilon,$$

with $\tilde b = H_\xi(\cdot, D\tilde u)$; here $H$ is the Hamiltonian (the Legendre-type transform of the Lagrangian appearing in the dynamic-programming equation), $H_\xi$ its derivative with respect to the momentum variable, $F$ is the gradient of the cost functional acting on the conditional law, and $c_1 = -\int_\Omega \tilde p \big(L(\cdot, H_\xi(\cdot, D\tilde u)) + F[\tilde p]\big)\,dx$, the nonlocal constants following eqs. (16)–(17) of the paper. The source states the ansatz formally, with bare limits for $t/T \in (\eta, 1-\eta)$, and does not fix the topology of the four limits: the four limits are understood in a topology that is part of the asserted result — for instance narrow convergence of the normalized densities as probability measures on $\bar{\Omega}$, weak, strong $L^q$, or locally uniform convergence of the rescaled value function and of the optimal feedback, and ordinary convergence of the scalar decay rate — stronger topologies giving stronger statements. Likewise, the source writes the ansatz for the associated optimal control and notes (Remark 2.3 of the paper) that, due to the lack of convexity of $J$, uniqueness of the minimizer is not known; as in the source, the ansatz is therefore understood without a fixed selection rule among possible optimizers. The argument must in particular settle whether alternative asymptotic regimes of the optimally controlled system — for example asymptotically time-periodic behavior, or non-convergent families of optimizers — can occur, this being precisely the obstruction the source identifies. A result covering only special cases (special dimension or geometry, restricted cost structures, restricted control classes) is a partial answer unless the generality of the stated setting is retained.

Part B. In the limit $M \to \infty$ of the uniform control bound, under the standing hypotheses of Section 5 of the paper — $f$ smooth and nonnegative on $\bar{\Omega}$, $p_0 \ln p_0 \in L^1(\Omega)$, and $\varepsilon > 0$, the latter ensuring a lower bound for the survival mass $\int_\Omega p_b(t)$ uniform with respect to $M$ — the limit optimality system of Theorem 5.2 contains a positive Radon measure $\gamma$ on $[0,T]$. Prove, or disprove, the identification

$$\gamma(t) = \frac{\int_\Omega p(t,x)\big(\tfrac{1}{2}|b(t,x)|^2 + f(x)\big)\,dx}{\big(\int_\Omega p(t,x)\,dx\big)^2}$$

for the quadratic cost $L(x,b) = \tfrac12 |b|^2 + f(x)$, i.e. that $\gamma$ is the conditional energy normalized by the survival probability.

Relationship between the parts. Part A is the paper's central open problem; Part B is a related but technically distinct limit-justification question ($M \to \infty$ rather than $T \to \infty$) raised in the same work.

References

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