Control in Probability for SDE Models of Growth Population

Abstract

In this paper, we consider a (control) optimization problem, which involves a stochastic dynamic. The model proposes selecting the best control function that keeps bounded a stochastic process over an interval of time with a high probability level. Here, the stochastic process is governed by a stochastic differential equation affected by a stochastic process. This setting becomes a chance-constrained control optimization problem, where the constraint is given by the probability level of infinitely many random inequalities. Since such a model is challenging, we discretize the dynamic and restrict the space of control functions to piecewise mappings. On the one hand, it transforms the infinite-dimensional optimization problem into a finite-dimensional one. On the other hand, it allows us to provide the well-posedness of the problem and approximation. Finally, the results are illustrated with numerical results, where classical model for the growth of a population are considered.

Keywords: Chance constrained optimization; Control in Probability; Growth population models.

Fig. 1

Numerical approximation of the logistic growth model without control $u^N=(0,\dots ,0)$ (upper panel) and fully controlled $u^N=(1,\dots ,1)$ (lower panel). In dashed bold line we plot the empirical mean over $M=3\times {10}^4$ simulations. The carrying capacity is $k=100[kt]$ (maximal biomass supported by the environment), and the upper bound safety level or limit biomass used for defining the probability function $\phi$ is $\overline{b}=80[kt]$. The growth factor function and its corresponding regular cut-off function are $\sigma (u,z)={10}^{-10}+\varphi (0.5z-2u)$ and $\varphi (s)=(1+tanh(s))/2$, and the diffusion parameter on $\xi (t)$ is $\alpha =0.5$