By J.L.Kelly.

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6. Jump behaviour with a pseudoconcave objective An optimal control problem whose objective is pseudoconcave, but not concave, may show jump behaviour in the control, not associated with a boundary of a feasible region. A simple example is proposed in Islam and Craven (2004); it is given here, with numerical results. Mathematics of Optimal Control 21 Consider the simple example: T MAX U (u(t))dt subject to: 0 x(0) = x0 , x(t) ˙ = γx(t) − u(t) (0 ≤ t ≤ T ), x(T ) = xT . ) may be unconstrained, or there may be a lower bound (∀t)u(t) ≥ ulb .

The vector adjoint equation is then: ˙ Λ(t) = fx (x(t), u(t), t) + Λ(t)mx (x(t), u(t), t), Λ(T ) = Φ (x(T )). , t; Λ(t)) → PMAX over [a(t), b(t)] at the optimal u(t). 34 Chapter 2 The maximum in Pontryagin’s principle becomes here a Pareto maximum. If the constraints on the controls are inactive, then optimizing the vector Hamiltonion implies (for two objectives) that: (∃τ ≥ 0, τ = 0)τ Hu (x(t), u(t), t; Λ(t)) = 0, at the optimal x(t), u(t), Λ(t). ) If multipliers τ and λ are known, then Λ(t) may be constructed from τ T Λ(t) = λ(t).

Is approximated by 0 λ(t)β(t)dt. ). The vector adjoint equation is then: ˙ Λ(t) = fx (x(t), u(t), t) + Λ(t)mx (x(t), u(t), t), Λ(T ) = Φ (x(T )). , t; Λ(t)) → PMAX over [a(t), b(t)] at the optimal u(t). 34 Chapter 2 The maximum in Pontryagin’s principle becomes here a Pareto maximum. If the constraints on the controls are inactive, then optimizing the vector Hamiltonion implies (for two objectives) that: (∃τ ≥ 0, τ = 0)τ Hu (x(t), u(t), t; Λ(t)) = 0, at the optimal x(t), u(t), Λ(t). ) If multipliers τ and λ are known, then Λ(t) may be constructed from τ T Λ(t) = λ(t).