TY - JOUR
T1 - Robust Statistic Estimation of Constrained Optimal Control Problems of Pollution Accumulation (Part I)
AU - Escobedo-Trujillo, Beatris Adriana
AU - López-Barrientos, José Daniel
AU - Higuera-Chan, Carmen Geraldi
AU - Alaffita-Hernández, Francisco Alejandro
N1 - Publisher Copyright:
© 2023 by the authors.
PY - 2023/2/1
Y1 - 2023/2/1
N2 - In this paper, we study a constrained optimal control on pollution accumulation where the dynamic system was governed by a diffusion process that depends on unknown parameters, which need to be estimated. As the true values are unknown, we intended to determine (adaptive) policies that maximize a discounted reward criterion with constraints, that is, we used Lagrange multipliers to find optimal (adaptive) policies for the unconstrained version of the optimal control problem. In the present context, the dynamic system evolves as a diffusion process, and the cost function is to be minimized by another function (typically a constant), which plays the role of a constraint in the control model. We offer solutions to this problem using standard dynamic programming tools under the constrained discounted payoff criterion on an infinite horizon and the so-called principle of estimation and control. We used maximum likelihood estimators by means of a minimum least square error approximation in a pollution accumulation model to illustrate our results. One of the advantages of our approach compared to others is the intuition behind it: find optimal policies for an estimated version of the problem and let this estimation tend toward the real version of the problem. However, most risk analysts will not be as used to our methods as they are to, for instance, the model predictive control, MATLAB’s robust control toolbox, or the polynomial chaos expansion method, which have been used in the literature to address similar issues.
AB - In this paper, we study a constrained optimal control on pollution accumulation where the dynamic system was governed by a diffusion process that depends on unknown parameters, which need to be estimated. As the true values are unknown, we intended to determine (adaptive) policies that maximize a discounted reward criterion with constraints, that is, we used Lagrange multipliers to find optimal (adaptive) policies for the unconstrained version of the optimal control problem. In the present context, the dynamic system evolves as a diffusion process, and the cost function is to be minimized by another function (typically a constant), which plays the role of a constraint in the control model. We offer solutions to this problem using standard dynamic programming tools under the constrained discounted payoff criterion on an infinite horizon and the so-called principle of estimation and control. We used maximum likelihood estimators by means of a minimum least square error approximation in a pollution accumulation model to illustrate our results. One of the advantages of our approach compared to others is the intuition behind it: find optimal policies for an estimated version of the problem and let this estimation tend toward the real version of the problem. However, most risk analysts will not be as used to our methods as they are to, for instance, the model predictive control, MATLAB’s robust control toolbox, or the polynomial chaos expansion method, which have been used in the literature to address similar issues.
KW - consistent estimators
KW - control with restrictions
KW - discounted cost
KW - least square errors
KW - maximum likelihood estimators
UR - http://www.scopus.com/inward/record.url?scp=85148946251&partnerID=8YFLogxK
U2 - 10.3390/math11040923
DO - 10.3390/math11040923
M3 - Artículo
AN - SCOPUS:85148946251
SN - 2227-7390
VL - 11
JO - Mathematics
JF - Mathematics
IS - 4
M1 - 923
ER -