A representation theorem for generators of BSDEs with finite or infinite time intervals and linear-growth generators

A structured population the individuals of which are divided into n age or typical groups x_1,…,x_n. is considered. We assume that at any time moment k, k = 0,1,2… the size of the population x(k) is determined by the normal autonomous system of difference equations x(k+1)=F(x(k)), where F(x)=col(f_1 (x),…,〖 f〗_n (x) ) are given vector functions with real non-negative components f_i (x), i=1,…n. We investigate the case when it is possible to influence the population size by means of harvesting. The model of the exploited population under discussion has the form x(k+1)=F((1-u(k) )x(k) ), where u(k)= (u_1 (k),…,u_n (k))∈〖[0; 1]〗^n is a control vector, which can be varied to achieve the best result of harvesting the resource. We assume that the cost of a conventional unit of each of n classes is constant and equals to C_i≥0, i=1,…,n. To determine the cost of the resource obtained as the result of harvesting, the discounted income function is introduced into consideration. It has the form H_α (u ̅,x(0))=∑_(j=0)^∞▒〖∑_(i=1)^n▒〖C_i x_i (j) u_i (j) e^(-αj) 〗,〗 where α>0 is the discount coefficient. The problem of constructing controls on finite and infinite time intervals at which the discounted income from the extraction of a renewable resource reaches the maximal value is solved. As a corollary, the results on the construction of the optimal harvesting mode for a homogeneous population are obtained (that is, for n = 1).

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Arrow-type sufficient conditions for optimality of age-structured control problems

Open Mathematics ◽

10.2478/s11533-013-0219-7 ◽

2013 ◽

Vol 11 (6) ◽

Author(s):

Vladimir Krastev

Keyword(s):

Boundary Conditions ◽

Sufficient Conditions ◽

Control Problems ◽

Infinite Time ◽

Time Intervals ◽

Sufficient Conditions For Optimality ◽

Age Structured

AbstractWe consider a class of age-structured control problems with nonlocal dynamics and boundary conditions. For these problems we suggest Arrow-type sufficient conditions for optimality of problems defined on finite as well as infinite time intervals. We examine some models as illustrations (optimal education and optimal offence control problems).

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Comment

Comparative Studies in Society and History ◽

10.1017/s001041750000548x ◽

1969 ◽

Vol 11 (4) ◽

pp. 426-430 ◽

Cited By ~ 2

Author(s):

Thomas S. Kuhn

Keyword(s):

Industrial Revolution ◽

Linear Growth ◽

Central Portion ◽

Productive Capacity ◽

Time Intervals ◽

Local Fluctuations ◽

Or Economics ◽

Life On Earth ◽

Do So

Possessing little knowledge or competence in either demography or economics, I am in no position to comment on the central portion of Professor Dovring's paper. Fortunately, I feel no call to do so, for I very much doubt that it can be faulted. No refinement of data or analysis is likely to set aside his basic conclusions. If the time intervals analyzed are made long enough to eliminate local fluctuations, accelerated rather than linear growth, whether of population or productive capacity, has characterized man's life on earth since at least the conquest of fire. Only historical myopia can account for the view that an increasing tempo of change dates only from the Industrial Revolution, that our current condition is, with respect to the existence of acceleration, essentially new.

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