Abstract
We consider the existence and first order conditions of optimality for a stochastic optimal control problem inspired by the celebrated FitzHugh–Nagumo model, with nonlinear diffusion term, perturbed by a linear multiplicative Brownian-type noise. The main novelty of the present paper relies on the application of the rescaling method which allows us to reduce the original problem to a random optimal one.
Highlights
Consider the following problem⎧ ⎪⎨dv(t, ξ ) = Δγ (v(t, ξ )) − I ion(v(t, ξ )) − f (ξ )v(t, ξ ) + F(t, ξ ) dt + v(t, ξ )d W (t), ξ ∈ O⎪⎩vγ((0v,(ξt,)ξ=)) v0(ξ ) =0, on (0, T ) × ∂O (1)γ : R → R being a monotone, increasing continuous function, v = v(t, ξ ) represents the transmembrane electrical potential, O ⊂ Rd, d = 2, 3, is a bounded and open set with smooth boundary ∂O
We indicate with Δξ the Laplacian operator with respect to the spatial variable ξ, while ε and δ are positive constants representing phenomenological coefficients, f (ξ ) is a given external forcing term, while
The present paper addresses the problem of existence and uniqueness of a strong solution, in a sense to be better specified in a while, to Eq (1)
Summary
The present paper addresses the problem of existence and uniqueness of a strong solution, in a sense to be better specified in a while, to Eq (1) We will further consider the problem of existence of an optimal control for the nonlinear FHN equation. It must be stressed that the techniques used in the present work, presenting some similarity with [12,22], as for instance the usage of the Ekeland principle to treat the cubic nonlinearity typical of the FHN equation, are in general different. The nonlinear diffusion γ affects the main technique used in proving the existence of an optimal control since a suitable trasnformation to reduce the problem to a random equation must be applied.
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