Mortensen observer for a class of variational inequalities – lost equivalence with stochastic filtering approaches. (English. French summary) Zbl 1541.37105
Summary: 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, R. E. Mortensen [J. Optim. Theory Appl. 2, 386–394 (1968; Zbl 0177.36004)]
introduced in 1968 a nonlinear estimator based on likelihood maximisation. Then, starting with O. Hijab [in: Advances in filtering and optimal stochastic control, Proc. IFIP-WG 7/1 Work. Conf., Cocoyoc/Mex.Lect. Notes Contr. Inf. Sci. 42, 170–176 (1982; Zbl 0503.93057)] 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.
MSC:
| 37N40 | Dynamical systems in optimization and economics |
| 37N35 | Dynamical systems in control |
| 49J40 | Variational inequalities |
| 49J52 | Nonsmooth analysis |
| 93E10 | Estimation and detection in stochastic control theory |
| 93E03 | Stochastic systems in control theory (general) |
Keywords:
deterministic sequential estimation; nonsmooth dynamics governed; variational inequality; Zakai equation; Bellman equationReferences:
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