A Numerical Algorithm for Constrained Optimal Control Problems with Applications to Harvesting

Abstract

We are interested in optimal control problems with constraints involving the state and control variables for all values of the independent variable (time say) in some interval. These problems while substantially discussed in the literature from a theoretical viewpoint (see for example Cesari (1983)) and an algorithmic viewpoint (see Goh and Teo (1987), (1988), Miele (1975), Miele et al (1970), Miele et al (1986), Teo and Goh (1987), (1989), Wong et al (1986)) continue to be examined with the view to improving the computational algorithms in both efficiency and stability. We have chosen the control parameterization route (Goh and Teo (1988) and Teo and Goh (1989)) but we note that there is a family of gradient restoration algorithms due to Miele and his co-workers. A general purpose optimal control software package, MISER, (see Goh and Teo (1987)) has been developed based on the control parameterization technique and we use this development as the starting point to improve efficiency and stability. In the current version of MISER, a particular constraint transcription proposed by Teo and Goh (1987) is used to handle inequality continuous (in time) state constraints involving state and control variables. A disadvantage of this constraint transcription is that the usual constraint qualification is not satisfied as the gradient of the constraint is zero at all feasible solutions. Hence convergence of the numerical algorithm is not guaranteed and some oscillation can occur in computation close to the solution.