Abstract
This paper describes an optimal control for a class of nonlinear systems dxi(t)/dt=xi(t)Fi(xj(t))-bifi(t)u(t)xi(t), i, j=1, 2(i≠j) which represent some kinds of interactions, for example predator-prey, symbiotic or mutual, and competitive interactions.
We can classify optimal control laws into two cases by the system's global characteristics, i.e., the signs of dFi/dxj, dfi/dt, bi, etc. One case is that the optimal control is a Bang-Bang one, and the other case is that the optimal control is a singular one.
The optimal control is generally a Bang-Bang one. But when each element has the control input (bi>0, i=1, 2), and the system satisfies the following conditions, fi'=dfi/dt>0 (i=1, 2), fi'·fj'<0 (i, j=1, 2, i≠j), or fi'≤0 (i=1, 2) and some additional conditions, the optimal control may be the singular one. In that case, the performance index must be the function of input u(t) only. It is remarkable that this separation is independent on the values of the function fi(t), and dependent on only those of fi'(t) (in some cases, it is independent on even the values of fi'(t)).
The function fi(t) is not the only time variant coefficient with control input. The variation of the effect of control, which is caused by the change of the system structure brought about by the inputs, (especially it is important for the systems in nature), can be considered in this system by means of both the function fi(t) and the results required in this paper.
Finally the optimal control of the predatorprey Volterra model including a resistance to an insecticide is shown as an example.
