end{aligned} Hence, (J_{varepsilon } u^{0}_{varepsilon },y^{0}_{varepsilon })<+infty) (which implies ((u^{0}_{varepsilon },y^{0}_{varepsilon })inXi_{Delta})) and ((u^{0}_{varepsilon },y^{0}_{varepsilon })) is an optimal pair to the corresponding optimization problem (4.1).
(4.10) It remains to show that ((u^{0}_{varepsilon },y^{0}_{varepsilon })inXi_{Delta}) and ((u^{0}_{varepsilon },y^{0}_{varepsilon })) is an optimal pair to the constrained minimization problem (4.1) for a given (varepsilon >0).
Let ({ u_{varepsilon }^{0},y_{varepsilon }^{0}) }_{varepsilon >0}subsetXi_{Delta}) be a sequence of optimal pairs to the corresponding fictitious problem (4.1).
Theorem 7.1 Let us suppose that f ∈ W − 1, q , y d ∈ W 0 1, p , and A ad ≠ ∅ are given with p ≥ 2. Let ( u 0, y 0 ) ∈ L ∞ × W 0 1, p be an optimal pair to problem (3.1 - 3.3 3.1 - 3.3
The above implies that if we let x ε = x ( ⋅ ; x 0 ε, u ε , then ( x ε , u ε is an optimal pair for the problem where the state equation is (2.1) and the cost functional is J ε ( x 0, u . Now, we derive the necessary conditions for ( x ε , u ε.
Constraints (1d -(1e) force variables z ij s to be one if in the optimal solution to the problem there exist a pair of adjacent vertices i j ∈ V having a different value at the s-th coordinate.
The bad news is that government regulations aren't the optimal solution to the problem.
Let u ∗ be the optimal control to the problem ( P ).
which is an optimal solution to the problem by definition.
This algorithm is the optimal solution to the problem.
