A symplectic pseudospectral method for constrained time-delayed optimal control problems and its application to biological control problems

Time-delays are often involved in modelling biological processes, which arise more difficulties for the control task. In fact, once the control goal is given, a constrained optimal control problem can be formulated to describe the biological control mission. However, it's almost impossible to obtain analytical solutions for such complicated problems, especially for those with high nonlinearity and complex constraints. In this paper, a symplectic pseudospectral method for solving nonlinear optimal control problems of constrained stated-delayed system is developed. The original nonlinear problem is transformed into a series of linear-quadratic optimal control problems by the successive convexification technique at the first implementation. By applying the local Legendre-Gauss-Lobatto pseudospectral method and the parametric variational principle, each of the transformed linear-quadratic problems is converted into a standard linear complementarity problem which can be solved easily by the Lemke's method. The cost functional, system dynamics and constraints can be generally nonlinear functions of state, control and time variables. and three kinds of inequality constraints, i.e. pure-state, pure-control and mixed state-control, can be treated under a uniform framework. The developed symplectic pseudospectral method is finally validated by an optimal harvesting problem and an optimal vaccination problem.