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Article Dans Une Revue ESAIM: Proceedings and Surveys Année : 2023

Mortensen Observer for a class of variational inequalities - Lost equivalence with stochastic filtering approaches

Résumé

We address the problem of deterministic sequential estimation for a nonsmooth dynamics governed by a variational inequality. An example of such dynamics is the Skorokhod problem with a reflective boundary condition. For smooth dynamics, Mortensen introduced in 1968 a nonlinear estimator based on likelihood maximisation. Then, starting with Hijab in 1980, several authors established a connection between Mortensen's approach and the vanishing noise limit of the robust form of the so-called Zakai equation. In this paper, we investigate to what extent these methods can be developed for dynamics governed by a variational inequality. On the one hand, we address this problem by relaxing the inequality constraint by penalization: this yields an approximate Mortensen estimator relying on an approximating smooth dynamics. We verify that the equivalence between the deterministic and stochastic approaches holds through a vanishing noise limit. On the other hand, inspired by the smooth dynamics approach, we study the vanishing viscosity limit of the Hamilton-Jacobi equation satisfied by the Hopf-Cole transform of the solution of the robust Zakai equation. In contrast to the case of smooth dynamics, the zero-noise limit of the robust form of the Zakai equation cannot be understood in our case from the Bellman equation on the value function arising in Mortensen's procedure. This unveils a violation of equivalence for dynamics governed by a variational inequality between the Mortensen approach and the low noise stochastic approach for nonsmooth dynamics.
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Dates et versions

hal-03659066 , version 1 (04-05-2022)
hal-03659066 , version 2 (22-02-2023)

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Louis-Pierre Chaintron, Álvaro Mateos González, Laurent Mertz, Philippe Moireau. Mortensen Observer for a class of variational inequalities - Lost equivalence with stochastic filtering approaches. ESAIM: Proceedings and Surveys, 2023, 73, pp.130-157. ⟨10.1051/proc/202373130⟩. ⟨hal-03659066v2⟩
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