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Application of nonlinear risk-sensitive optimal control equations to excitable noise system

In this work, the risk-sensitive optimal control equations for polynomial stochastic systems of third degree with exponential criterion to be minimized and parameter of diffusion into the state equations has been applied to the Fitz Hugh-Nagumo (Bon Hoeffervan der Pol) model. This model represents an excitable system with driven noise, which could be associated with diverse processes, from the kinetic one of chemical reactions and physics of the solid state up to biological. In some applications of Fitz Hugh-Nagumo model, the supervenience depends of maintaining certain noise amplitude. The goal of this work is to apply both methodologies of control, risk-sensitive and traditional control equations for polynomial stochastic systems of third degree. This system has the characteristic that is of third degree and needs to maintain certain level of noise, which make it very difficult to handle. Exponential-quadratic cost criterion J is evaluated in both methods, taking the value of J in final time in each method. Some results have been obtained, illustrating the performance of both methodologies, for some values of the parameters.