PALM: FISHP:RY REGULATION VIA CONTROL THEORY 



because the optimal values of the control variable 

 lie at its boundaries. Thus the control variable 

 switches between a lower value (usually zero) and 

 an upper value which might be difficult to specify. 



There are advantages as well as limitations with 

 the linear-quadratic approach as compared with 

 the bang-bang control approach. With the linear- 

 quadratic approach, quantities such as yield and 

 present value of profits are not directly maximized 

 to obtain the feedback control function, as is done 

 with the bang-bang approach. Rather, the 

 maximization is first done with static methods, 

 and then a feedback control function is construct- 

 ed to keep the system near the resulting 

 equilibrium condition. To do this, the system 

 equations are linearized about the equilibrium. If 

 disturbances carry the system far from 

 equilibrium, the linearization breaks down. 

 However, this is generally not a serious limitation, 

 since the feedback control function is designed to 

 counteract disturbances and to keep the system 

 near equilibrium. The method is not restricted to 

 equilibrium analysis, and frequently the two 

 approaches are combined by using bang-bang 

 control methods, instead of static methods, to 

 compute an optimal "open-loop" control function. 

 Linearization of the model around the resulting 

 trajectory enables the linear-quadratic method to 

 be used to synthesize a closed-loop (feedback) con- 

 trol function to keep the system on the optimal 

 trajectory (Ho and Bryson 1969). 



A significant advantage of the linear-quadratic 

 approach is that it allows the use of linear control 

 theory, whose techniques are more highly 

 developed and easier to apply than the nonlinear 

 techniques required for bang-bang control 

 analysis. Powerful methods of compensating for 

 incomplete information, uncertainties in 

 measurements, model parameters, and model 

 structure are available for the linear-quadratic 

 approach but are scarce for the bang-bang control 

 approach. Also, solutions to bang-bang control 

 problems are extremely difficult to obtain if the 

 model contains more than two variables. 



First a single-variable model is used to illustrate 

 the basic concepts. A model with three variables is 

 then analyzed to show how the techniques are 

 easily extended to the general multivariable case. 

 The details of the general method are in the Ap- 

 pendix. There it is also shown in more detail why 

 static optimization methods, such as linear 

 programming, and dynamic optimization 

 methods, such as optimal control theory, should not 



be treated as competing methods but rather should 

 be used together as part of the total approach to 

 the problem because they are mutually 

 complementary methods. This is mentioned 

 because there is a tendency among economics- 

 oriented analysts to use static methods, whereas 

 analysts with control- theory backgrounds tend 

 toward dynamic methods. 



It is assumed that the reader is familiar with the 

 fundamentals of differential equations and ma- 

 trix operations. A matrix will be denoted by 

 brackets [ ]; a matrix transpose by [ ]^; and a 

 column vector by a bar underneath, as x. 



SINGLE- VARIABLE MODEL 



The following model is the Schaefer or logistic 

 model: 



dN 



^=-aN-bN'~-qfN, (1) 



at 



where A^ is the biomass or number of catchable fish 

 in the fishery, t is time, q is the catchability 

 coefficient, and/is the fishing effort. The constant 

 a is the intrinsic rate of natural increase of the 

 population, and the constant h is related to the 

 carrying capacity of the environment c by the 

 relation: b — ale. The system's equilibrium (A' 

 /eq) is found by setting the derivative in Equation 

 (1) equal to zero: 



= oA^eq - ^^'-eq - ^/'eq^eq- 



The equilibrium yield Y ^^ is: 



>^eq = '//eq^'eq = (« "^A^eq ) ^^eq • 



To find the maximum equilibrium yield, we 

 differentiate Y ^ with respect to A'^^, and set this 

 result equal to zero. 



eq 



Solving this for the population size and fishing 

 effort corresponding to maximum equilibrium 

 yield, we obtain: 



N^ = a/26. 



.' eq 



{a - />A\,, ) A^,„ a 



«) ' eq 



q N 



eq 



2q. 



831 



