Since there exists an optimal control for maximizing the functional (10) subject to equations (6)-(9), we use Pontryagin's Maximum Principle to derive the necessary conditions for this optimal control. Pontryagin's Maximum Principle introduces adjoint functions that allow us to attach our state system (of differential equations), to our objective functional. After first showing existence of optimal controls, this principle can be used to obtain the differential equations for the adjoint variables, corresponding boundary conditions and the characterization of an optimal control u v * and u m * . This characterization gives a representation of an optimal control in terms of the state and adjoint functions. Also, this principle converts the problem of minimizing the objective functional subject to the state system into minimizing either the Lagrangian or the Hamiltonian with respect to the controls (bounded measurable functions) at each time t [40] . The Lagrangian is defined as where w 11 (t) ≥ 0, w 12 (t) ≥ 0 are penalty multipliers satisfying w 11 (t)(a 11u v (t)) + w 12 (t)(u v (t)b 11 ) at the optimal u v * , and w 21 (t) ≥ 0, w 22 (t) ≥ 0 are penalty multipliers satisfying w 21 (t)(a 22u m (t)) + w 22 (t)(u m (t)b 22 ) at the optimal u m * . Given optimal controls u v * and u m * , and solutions of the corresponding state system (6) with transversality conditions l i [t f ] = 0, for i = 1, 2, 3, 4. To determine the interior maximum of our Lagrangian, we take the partial derivatives of L with respect to u v and u m , respectively, and set it to zero. Thus, To determine an explicit expression for our controls u m * , u m * (without w 11 , w 12 , w 21 , w 22 ), a standard optimality technique is utilized. The following cases are considered to determine a specific characterization of the optimal control. 