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Optimal control of prey-predator systems with Lagrange type and Bolza type cost functionals. (English) Zbl 0541.49003

The form of the optimal control function U(t) included in a modification of a Volterra system: \[ \dot x_ 1\quad =\quad x_ 1(\lambda_ 1+u\mu_ 1x_ 2),\quad \dot x_ 2 = x_ 2(\lambda_ 2+u\mu_ 2x_ 1),\quad 0\leq u\leq 1 \]
\[ \lambda_ 1>0,\quad \lambda_ 2<0,\quad \mu_ 1<0,\quad \mu_ 2>0,\quad t\in [0,T] \] where u(t) represents the degree of mixing of the preys and predators, is investigated. The form of the cost functional is either of Lagrange type: \(J_ L[u]=- \int^{T}_{0}(x_ 1(\tau)+x_ 2(\tau))d\tau\) of of Bolza type: \[ J_ B[u]\quad =\quad -(x_ 1(T)+x_ 2(T))-\int^{T}_{0}(\rho_ 1x_ 1(\tau)+\rho_ 2x_ 2(\tau))d\tau,\quad \rho_ i>0. \] The form of optimal control in both cases depends on the sign of \(\mu_ 1+\mu_ 2\). It is explicitly obtained in all cases. The paper contains also an investigation of the optimal critical response and its perturbation.

MSC:

49J15 Existence theories for optimal control problems involving ordinary differential equations
92D25 Population dynamics (general)
49K15 Optimality conditions for problems involving ordinary differential equations
93C10 Nonlinear systems in control theory
93C15 Control/observation systems governed by ordinary differential equations
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