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RETARDED FUNCTIONAL DIFFERENTIAL EQUATIONS
STRONG LOCAL MINIMA
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adjoint equation attainability cone Banach space bounded linear operator chain rule chapter chemical reactor closed and convex codimension Colonius Condition 2.5 consider constraint qualification constraints of Problem contains a subspace continuously Frechet differentiable controlled Hopf bifurcation converges convex cone Corollary defined Ekeland's Variational Principle equivalent exists finite codimension forcing term function f functional differential equations Furthermore global maximum principle Hence holds Hopf bifurcation Hypothesis 1.5 implies Lagrange multipliers Lemma locally proper maximum principle minimum of Problem n-Test necessary optimality conditions neighborhood optimal control optimal periodic control optimal solution order conditions order necessary optimality ordinary differential equations periodic control problem periodic solution Pontryagin function Problem 1.1 Proof Proposition relaxed minimum Relaxed Problem 1.2 Remark result RnxRm satisfying the constraints second order necessary Section steady state problem strong local minimum subspace of finite system equation t,+r Theorem 2.1 theory twice continuously Frechet unique solution x-periodic