Error analysis of Haar wavelet-based Galerkin numerical method with application to various nonlinear optimal control problems

ABSTRACT

First, this paper defines a general nonlinear optimal control problem with state/control constraints and its approximation problem as the Haar wavelet Galerkin optimal control problem (HWGOCP). Then, a Haar wavelet-based Galerkin numerical method has been developed, which converts it to a nonlinear optimization problem. We theoretically prove that a Haar wavelet feasible solution of HWGOCP will exist. We also show that the approximate solutions of HWGOCP are consistent and converge to the optimal solution of the problem. A variety of application problems have been considered, which include optimal control of tumour growth using Chemotherapy drugs, optimal control of infection via the SIS model using treatment, the Brachistochrone problem in mechanics, optimal control of mold using a fungicide, optimal control of pH value of a chemical reaction to determine the quality of a product, etc.

KEYWORDS:

• Haar wavelets
• numerical method
• optimal control problem
• bounded control
• state constraints

MATHEMATICS SUBJECT CLASSIFICATIONS 2010:

• 33F05
• 49J15
• 65T60

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The first author acknowledges the financial support (UGC-Ref. No. : 1309/(CSIR-UGC NET JUNE 2019)) by the University Grants Commission, New Delhi, India.