TY - JOUR
T1 - On the control of certain interacting populations
AU - Albrecht, Felix
AU - Gatzke, Harry
AU - Haddad, Abraham
AU - Wax, Nelson
PY - 1976/3
Y1 - 1976/3
N2 - We study the time optimal control of the system x ̇1 = x1f{hook}1(x1, x2) + u1(t) g1(x1), x ̇2 = x2f{hook}2(x1, x2) + u2(t)g2(x2), where x1 is the size of the population of one species, x2 is the population size of the second species, f{hook}1 and f{hook}2 are the fractional growth rates of the respective species, g1 and g2 are nowhere vanishing functions of class C1(0, + ∞), and the control u(t) = (u1(t), u2(t)) takes on values in a closed rectangle. The functions f{hook}1 and f{hook}2 are chosen to represent prey-predator, competitive, and symbiotic interactions. We show, for the various interactions, that a time optimal control, if it exists, must be "bang-bang," and give sufficient conditions for the controllability, and for the existence, of time optimal controls of the above system.
AB - We study the time optimal control of the system x ̇1 = x1f{hook}1(x1, x2) + u1(t) g1(x1), x ̇2 = x2f{hook}2(x1, x2) + u2(t)g2(x2), where x1 is the size of the population of one species, x2 is the population size of the second species, f{hook}1 and f{hook}2 are the fractional growth rates of the respective species, g1 and g2 are nowhere vanishing functions of class C1(0, + ∞), and the control u(t) = (u1(t), u2(t)) takes on values in a closed rectangle. The functions f{hook}1 and f{hook}2 are chosen to represent prey-predator, competitive, and symbiotic interactions. We show, for the various interactions, that a time optimal control, if it exists, must be "bang-bang," and give sufficient conditions for the controllability, and for the existence, of time optimal controls of the above system.
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U2 - 10.1016/0022-247X(76)90094-9
DO - 10.1016/0022-247X(76)90094-9
M3 - Article
AN - SCOPUS:0016928470
SN - 0022-247X
VL - 53
SP - 578
EP - 603
JO - Journal of Mathematical Analysis and Applications
JF - Journal of Mathematical Analysis and Applications
IS - 3
ER -