ic Optimal Control Problem. In order to set limits 

 on fishing effort which are functions of system 

 variables such as population density or mean or- 

 ganism weight, it is necessary to measure these 

 variables. Any measurement process is stochastic 

 or noisy, and it is necessary to compensate for this 

 in the design of a feedback regulation scheme. In 

 many engineering applications this has been suc- 

 cessfully accomplished by the use of the Kalman- 

 Bucy filter (Athans 1971). In addition, it may be 

 impossible even to measure some variables. This 

 problem of incomplete information has been 

 frequently solved by the use of the Observer 

 Theory (Kwakernaak and Sivan 1972). 



Also there will be uncertainties in the deter- 

 mination of the model constants. In fact the "con- 

 stants" may not be constants at all, Init merely the 

 representation of several effects lumped together. 

 Thus there is also error in the model structure, 

 since the model constants are actually variables 

 dependent upon a variety of effects. For the 

 Schaefer model these effects would be interspecies 

 interactions, age structure, availability and 

 vulnerability of the age groups, and physical en- 

 vironment influences on the biological processes. 

 Such difficulties are amenable to solution by add- 

 ing a "noise" term to the model equations and by 

 modifying the linear-quadratic techniques to ac- 

 commodate these stochastic effects (Athans 1971). 

 It should also be pointed out that compensation for 

 modeling errors is one of the purposes of feedback 

 control. 



The change in model parameters with time can 

 be compensated for by regularly recomputing the 

 feedback gains as more data becomes available. 

 Finally, while no pretense is made of being able to 

 predict exact time paths, the methods described in 

 this paper should prove useful in pro\iding 

 management guidelines. The effects of stochastic 

 pr(K-esses and uncertainties can be handled in a 

 manageable way by computer simulation, and 

 l)rediction of the future course of the managed 

 fishery, in an average sense, can be made with 

 appropriate error bands placed on the predictions. 



ACKNOWLEDGMENT 



I thank Saul Saila for bringing (loh's results lo 

 my attention, and the referees for their comments 

 on the bang-bang control approach and the effects 

 of uncertainties in model parameters. 



FISHERY BULLETIN: VOL. 73, NO. 4 



LITERATURE CITED 



.\THANS, M. 



1971. Thf rcile and u.se of the .stochastic linear-(|uadratic- 

 gaussian iirohlem in control sy.stem de.sign. Inst. Electr. 

 Electron. Eng., Trans. Autom. Control. AC-1(5:529-.5.51. 



Beverton. R. .]. H.. .AND S. J. Holt. 



1957. On the dynamics of e.xploited fish populations. Fish 

 Invest. Minist. Agric. Fish. Food (G. B.), Ser. II, 19, 533 p. 

 Carrothers, W. A. 



1941. The British Columbia fisheries. Univ. Toronto Press, 

 Toronto, 136 p. 

 Clark, C, G. Edwards, and M. Friedlander. 



1973. Beverton-Holt model of a commercial fishery: Optimal 

 dynamics. J. Fish. Res. Board Can. 30:1629-1640. 

 Clark, C.W. 



1973. The economics of overexploitation. Science (Wash., 

 D.C.) 181:630-634. 

 Fox, W. W., -Jr. 



1970. An e.xponential surplus-yield model for optimizing 

 exploited fish populations. Trans. Am. Fish. Soc. 99:80-88. 

 GoH.B.S. 



1969. Optima! control of the fish resource. Malayan Sci. 



5:65-70. 

 1973. Optimal control of renewable resources and popula- 

 tions. (Summary of paper presented at the 6th Hawaii 

 International Conference on System Sciences) 5 p. 

 Ho, Y. C, and a. E. Bryson. 



1969. Applied optimal control: Optimization, estimation and 

 control. Blaisdell, Waltham, Mass., 481 p. 

 Kwakernaak, H., and R. Sivan. 



1972. Linear optimal control systems. Wiley - Interscience, 

 N.Y.,.575p. 



Larkin, p. a. 



1963. Interspecific competition and exploitation. J. Fish. 

 Res. Board Can. 20:647-678. 



1966. Ex])loitation in a t\-pe of predator-prey rela- 

 tionship. .J. Fish. Res. Board Can. 23:349-3.56. 

 Palm, W.J. 



1975. An application of control theory to population 

 dynamics. In E. 0. Roxin. P. T. Liu, and R. L. Sternberg 

 (editors), Differential games and control theory, p. 59-70. 

 Marcel-Dekker, N.Y. 



PELLA, ,J. J., AND P. K. TOMLINSON. 



1969. A generalized stocl; production model. Inter-Am. 

 Trop. Tuna Comm,, Bull. 13:419-496. 

 Ricker, W. E. 



1958. Handbook of computations for biological statistics of 

 fi.sh populations. Fish. Res. Board Can., Bull. 119, 300 p. 

 Saila, S. B. 



In press. Some applications of ojitiinal control theory to 

 fisheries management. Trans. Am. Fish. Soc. 

 Schaefer, M. B. 



19.54. Some aspects of the dynamics of ))()pulations impor- 

 tant to the management of commercial marine 

 fisheries. Inter-Am. Trop. Tuna Comm., Bull. l:2.5-,56. 

 1968. Methods of estimating effects of fishing on fish 

 populations. Trans. Am. Fish. Soc. 97:231-241. 

 TiMiN, M. E., and B. 1). Collier. 



1971. A model incorporating energy utilization for the 

 dynamics of single species poinilations. Theor. Popul. 

 Biol. 2:237-251. 



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